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Chapter 3: Biochemical Technology and Biostatistics

In the early 1900s, the Carlsberg Laboratory in Copenhagen had a problem that was costing it money. Beer brewing depends on yeast and on enzymes, and batches that should have behaved identically were turning out differently. Something in the brewing liquid was changing from batch to batch, but nobody could say exactly what, because nobody had a convenient way to measure it. Chemists knew that acidity mattered. They simply had no agreed number for it. Concentrations of hydrogen ions in a fermenting liquid are extremely small, and writing them out as decimals with long strings of zeros made comparison awkward and error-prone. In 1909, the head of the laboratory’s chemistry department, Søren Sørensen, proposed a solution that was almost embarrassingly simple: instead of quoting the hydrogen ion concentration itself, quote its negative logarithm. A messy number such as 0.0000001 became a tidy 7. He called this new quantity pH. Along with the scale, Sørensen’s group also worked out mixtures that could hold acidity steady while an experiment ran — what we now call buffers. A question raised by a brewery had produced two of the most-used tools in every biology laboratory in the world. Neither of them is a chemical. Both are ways of controlling and describing a number.

Diagram

Fig. 3.1 – A split illustration: on the left, a fermentation vessel with a pH probe dipped in it and a reading on the display; on the right, the 0–14 pH scale as a coloured strip with acidic, neutral and alkaline zones labelled, and typical biological examples (gastric juice, saliva, blood, intestinal fluid) marked at their positions.

NoteCuriosity Corner

Q1. Why do scientists write acidity as pH instead of simply writing the hydrogen ion concentration? Q2. What exactly is inside a buffer that lets it “absorb” added acid without its pH shooting up or down? Q3. If a buffer is so good at holding pH steady, why does it eventually fail when you keep adding acid to it? Q4. A fermenter is a sealed steel tank. How do the microbes inside it get oxygen, nutrients and an even temperature all the way through? Q5. Ten students score 15, 18, 16, 14, 15, 15, 12, 17, 90 and 95. Their average is 30.7 — yet no student scored anywhere near 30.7. Is the average lying, and what should we use instead?

This chapter will answer each of these questions.

3.1 Fundamentals of Biochemical Technology

Long before anyone had heard of a microorganism, people were already using them. Curd was set, dough was leavened, toddy was fermented and vinegar was soured — all by microbial activity that nobody could see and nobody could explain. The results were reliable enough to be trusted, but the reasons behind them were unknown, so the process could never be improved in any deliberate way.

Biochemical technology is what happens when those reasons become known. Once we understand which organism is doing the work, which enzyme it uses, and what conditions that enzyme needs, we stop merely hoping for a good batch and start engineering one. Living cells and their cellular components are then used on purpose, on a large scale, to manufacture useful products at a lower cost.

NoteDefinition

Biochemical technology is the branch of biotechnology that applies the chemical and metabolic activities of living cells, or of components obtained from them, to manufacture useful products on a commercial scale under controlled conditions.

Notice the phrase “or of components obtained from them”. A whole yeast cell can be used to ferment sugar, but an isolated enzyme extracted from a cell can also be used on its own — as when the enzyme rennet, obtained from calf stomach or from microbes, is used to clot milk for cheese. Both count as biochemical technology.

Why the Environment Must Be Tightly Controlled

Every one of these processes runs on the metabolic activity of living cells. That single fact decides almost everything else about how a biochemical process is designed. A chemical reaction in a beaker will proceed as long as the reactants are present; a biochemical process will proceed only as long as the cells inside it are alive, healthy and dividing.

Cells therefore need a nutrient substrate — the raw material that supplies carbon, nitrogen, minerals and energy — supplied continuously and in the right proportions. They also need their optimum conditions of temperature and pH maintained throughout, not just at the start.

Among these variables, pH is the one that punishes carelessness fastest. Enzymes are proteins, and a protein’s shape depends on the charges carried by its side chains. Shift the pH and those charges shift with it, altering the shape of the active site. The enzyme slows down, the metabolic pathway it belongs to stalls, and product yield falls. Push the pH further and the protein denatures altogether. This is why every fermenter carries a pH probe, and why the very first laboratory skills an ISC student learns in Paper II are measuring pH with a pH meter or pH paper and preparing buffers of a chosen pH.

Q. If the cells themselves produce acids during growth, doesn’t the pH change on its own?

A. Yes — and that is exactly the danger. Fermenting cells release organic acids and carbon dioxide, so a medium that starts at the correct pH will drift downwards as growth proceeds. Left uncorrected, the culture poisons its own enzymes. The medium is therefore buffered, and in large vessels acid or alkali is also added automatically to hold the set point.

Purity and Economy of the Substrate

A second requirement is that the culture medium must be free from impurities. Unwanted chemicals can react with the substrate or inhibit enzymes, and unwanted organisms are worse still — a contaminating microbe competes for the same nutrients, grows faster than the production strain, and may release substances that spoil the product. This is why sterilisation of media and aseptic handling are treated as non-negotiable steps rather than as precautions.

Diagram

Fig. 3.2 – A central culture vessel with four labelled arrows feeding into it — nutrient substrate, optimum temperature, optimum pH, sterile (contamination-free) conditions — and one arrow leading out labelled “product”. Show a pH probe and a thermometer inserted into the vessel.

Finally, the process must make economic sense. A substrate that is cheap, easily available and, where possible, reusable keeps production costs down. Many industrial fermentations deliberately use agricultural or industrial by-products — molasses from sugar mills, corn steep liquor, whey from cheese making — as their carbon source. What was waste for one industry becomes raw material for another, which lowers cost and reduces the burden of disposal at the same time.

The overall shape of any biochemical process, from a laboratory flask to an industrial vessel, is summarised in FC 3.1.

Flowchart

FC 3.1 – General sequence of a biochemical process: select the organism or enzyme → prepare the nutrient substrate → sterilise the medium and vessel → adjust pH and temperature to the optimum → inoculate with the culture → maintain controlled conditions during growth → monitor pH, temperature and product formation → recover and purify the product → quality checks before release

Everything that follows in this chapter serves this diagram. Sections 3.2 to 3.5 deal with holding the chemical and physical conditions steady; Section 3.6 deals with the vessel in which the process is run; and Sections 3.7 to 3.11 deal with how the numbers coming out of such a process are collected, summarised and judged.

3.2 Acids, Bases and the Concept of pH

A living cell is a narrow-minded chemist. It will carry out its reactions beautifully within a small window of conditions and refuse to cooperate outside it. Of all the conditions that define that window — temperature, salt concentration, oxygen supply — the one that changes fastest and matters most is acidity. Before we can control acidity, we need a clear idea of what makes a solution acidic in the first place.

Acids and Bases as Proton Traders

The most useful way to think about acids and bases in biology is the Brønsted–Lowry view, which describes them by what they do with protons rather than by how they taste or feel. A hydrogen ion, \(\text{H}^+\), is simply a proton, because a hydrogen atom that has lost its only electron has nothing left but its nucleus.

NoteDefinition

An acid is a substance that donates a proton (\(\text{H}^+\)) to another substance during a chemical reaction.

NoteDefinition

A base is a substance that accepts a proton (\(\text{H}^+\)) from another substance during a chemical reaction.

When hydrochloric acid dissolves in water, it hands its proton over to a water molecule:

\[\text{HCl} + \text{H}_2\text{O} \longrightarrow \text{H}_3\text{O}^+ + \text{Cl}^-\]

Here HCl is the acid, because it donates the proton, and water is the base, because it accepts one. Acetic acid, \(\text{CH}_3\text{COOH}\), behaves the same way in solution and is therefore also an acid.

Water, however, does not always play the same role. When ammonia dissolves in it, the traffic runs the other way:

\[\text{NH}_3 + \text{H}_2\text{O} \longrightarrow \text{NH}_4^+ + \text{OH}^-\]

Ammonia accepts a proton and is the base; water donates one and is behaving as an acid. This flexibility is not a flaw in the definitions — it is the point. An acid and a base are defined by their behaviour in a particular reaction, not by a permanent label.

Conjugate Acid–Base Pairs

Look again at what is left behind after the proton has moved. When an acid donates its proton, the species that remains can, in principle, take that proton back — so it is itself a base. It is called the conjugate base of that acid. Likewise, when a base accepts a proton, the species formed can give it up again, and is the conjugate acid of that base.

Every acid–base reaction therefore involves two conjugate pairs. In the hydrochloric acid reaction above:

  1. HCl donates a proton and becomes \(\text{Cl}^-\); so HCl is the acid and \(\text{Cl}^-\) is its conjugate base.
  2. \(\text{H}_2\text{O}\) accepts a proton and becomes \(\text{H}_3\text{O}^+\); so \(\text{H}_2\text{O}\) is the base and \(\text{H}_3\text{O}^+\) is its conjugate acid.

The two conjugate pairs are written as \(\text{HCl}/\text{Cl}^-\) and \(\text{H}_2\text{O}/\text{H}_3\text{O}^+\). Notice that the members of a pair differ by exactly one proton — nothing else. Keep this idea in view, because a buffer, which is the practical tool this whole discussion is building towards, is nothing but a weak acid kept in the same solution as its own conjugate base.

Q. Why does the Brønsted–Lowry description suit biology better than the older idea that acids release \(\text{H}^+\) and bases release \(\text{OH}^-\)?

A. Because many biologically important bases never release a hydroxide ion at all. Ammonia is the classic case, and so are the nitrogen-containing side chains of amino acids. Defining a base as a proton acceptor covers all of them, and it also works in non-aqueous solvents, where there may be no water available to supply \(\text{OH}^-\).

The pH Scale

Water itself splits very slightly into ions, so even pure water contains a small concentration of hydrogen ions — about \(1 \times 10^{-7}\) mol per litre. Biological fluids are not far from this value. Writing such quantities out as decimals is clumsy and invites mistakes, and comparing \(0.00000001\) with \(0.0000001\) by eye is an unreliable business.

The solution is to quote the negative logarithm of the concentration instead. This quantity is called pH — the letters standing for power (or potential) of hydrogen.

NoteDefinition

pH is the negative logarithm to base 10 of the hydrogen ion concentration of a solution: \(\text{pH} = -\log_{10}[\text{H}^+]\), where \([\text{H}^+]\) is the hydrogen ion concentration in mol per litre (mol L\(^{-1}\)). pH itself has no unit.

Q1. Why do scientists write acidity as pH instead of simply writing the hydrogen ion concentration?

A. Because hydrogen ion concentrations in living systems are extremely small numbers spread over a very wide range — from about \(10^{-1}\) down to \(10^{-14}\) mol L\(^{-1}\). Taking the negative logarithm compresses that entire range onto a convenient scale running from 0 to 14, turns awkward decimals into single-digit whole numbers, and makes comparison instant. A gastric juice at pH 2 and blood at pH 7.4 are easier to compare as “2 and 7.4” than as “0.01 and 0.00000004”.

Because the scale is logarithmic, each step of one pH unit is a ten-fold change in hydrogen ion concentration, not a small arithmetical one. A solution at pH 4 is ten times more acidic than one at pH 5, and a hundred times more acidic than one at pH 6. This is the single most important thing to remember about pH, and the reason a “slight” drop of one or two units inside a fermenter is not slight at all.

The scale reads as follows: values from 0 to just below 7 indicate acidic solutions, a value of exactly 7 indicates a neutral solution (pure water at room temperature), and values above 7 up to 14 indicate basic or alkaline solutions. As the pH number goes down, acidity goes up — the scale runs backwards to intuition, which is a common source of error in examinations.

The hydrogen ion concentration of a culture medium is measured as \(1 \times 10^{-5}\) mol L\(^{-1}\). (a) Calculate its pH. (b) If contamination lowers the pH to 3, by how many times has the hydrogen ion concentration increased?

Why pH Matters So Much in Living Systems

The reasons pH is watched so closely in both the body and the biotechnology laboratory can be set out as follows.

  1. Enzymes are pH-specific. Every enzyme has an optimum pH at which its activity is highest, and activity falls away sharply on either side of it.
  2. pH alters the charge and shape of proteins. Changing the pH changes the charges on amino acid side chains, distorting the folded structure and eventually causing denaturation, with loss of activity.
  3. Microorganisms grow only within specific pH ranges. A production strain held outside its range grows slowly, or stops growing altogether.
  4. pH affects membrane permeability, nutrient uptake and metabolic pathways, so it controls not only how fast cells work but what they take in and what they make.
  5. pH governs the success of laboratory experiments, industrial fermentation and medical applications, from enzyme assays to the stability of an injectable drug.

The human digestive tract is the clearest demonstration of point (a). Pepsin works in the stomach, where gastric juice provides a strongly acidic pH of roughly 1.5 to 3.5, and it is inactive at neutral pH. Pancreatic amylase, acting a short distance further along in the small intestine, needs the neutral-to-alkaline conditions found there. Salivary amylase, working in the mouth at a pH of about 6.8, sits between the two. Three enzymes, all digesting food, all requiring different acidities — which is exactly why the body goes to the trouble of maintaining different pH values in different compartments.

You can see this relationship as a curve rather than a list in Fig. 3.3. Each enzyme’s activity is plotted against the pH of its surroundings; the peak of a curve marks that enzyme’s optimum pH, and the steepness of the fall on either side shows how quickly activity is lost when pH drifts.

Diagram

Fig. 3.3 – A graph with pH (0 to 14) on the x-axis and relative enzyme activity on the y-axis, showing three bell-shaped curves peaking at different points and labelled pepsin (~2), salivary amylase (~6.8) and pancreatic amylase (~7–8), with each peak marked “optimum pH”.

Measuring pH

Knowing that pH matters is of no use unless it can be measured quickly and repeatedly. Two methods are used, and both appear in the ISC Paper II practical in which tap water, tap water with a few drops of dilute HCl, and tap water with a few drops of NaOH are compared.

Measurement of pH Principle: Certain dyes change colour at particular hydrogen ion concentrations, and a glass electrode develops an electrical potential across its membrane that varies with the hydrogen ion concentration of the solution outside it. Steps: Using pH paper: dip the strip briefly in the solution → compare the colour developed against the printed colour chart → read off the approximate pH. Using a pH meter: calibrate the instrument with standard buffer solutions of known pH → rinse the electrode with distilled water → immerse it in the test solution → allow the reading to steady → record the pH → rinse the electrode again before the next sample. Used for: Checking and adjusting the pH of culture media, buffers and reagents before an experiment, and monitoring the pH of a fermentation continuously while it runs.

The calibration step is not a formality. A pH meter does not measure pH directly; it measures a voltage and converts it using a relationship fixed during calibration. If the meter is calibrated against standard buffers of known pH, every later reading is anchored to those known values. Skip the calibration, or allow the glass electrode to dry out between uses, and the readings drift — the instrument will still display a confident number, but the number will be wrong.

3.3 Buffer Solutions, Their Types and Buffering Capacity

Section 3.2 established that biological reactions run properly only within a narrow pH range, and that a drift of one unit is a ten-fold change in acidity. That leaves a practical problem. Reactions themselves generate acids and bases — a growing culture releases organic acids and carbon dioxide, an enzyme assay may liberate protons — so any reaction left to itself will alter the very conditions it needs. Something must hold the pH still while the chemistry goes on. That something is a buffer.

NoteDefinition

A buffer solution is an aqueous system that resists changes in its pH, maintaining a nearly constant pH when small quantities of an acid or a base are added to it.

What a Buffer Is Made Of

A buffer is not a single substance. It is a mixture of two species that between them can deal with whichever kind of trouble arrives:

  1. a weak acid together with its conjugate base — for example acetic acid, \(\text{CH}_3\text{COOH}\), together with sodium acetate, \(\text{CH}_3\text{COONa}\), which supplies acetate ions, \(\text{CH}_3\text{COO}^-\); or
  2. a weak base together with its conjugate acid — for example ammonia together with ammonium chloride.

This is why the conjugate-pair idea from Section 3.2 was worth the trouble. A buffer works because the two components are a conjugate pair: one of them is standing ready to absorb added \(\text{H}^+\) ions, and the other is standing ready to absorb added \(\text{OH}^-\) ions. The solution has, in effect, a defence prepared in both directions at once.

How a Buffer Resists a Change in pH

Take the acetate buffer as the working example.

When an acid is added, the concentration of \(\text{H}^+\) ions rises, which would normally send the pH down. In a buffer, the conjugate base intercepts them:

\[\text{CH}_3\text{COO}^- + \text{H}^+ \longrightarrow \text{CH}_3\text{COOH}\]

The added protons are not left free in solution; they are converted into molecules of a weak acid, which barely dissociates. Because the free \(\text{H}^+\) concentration hardly rises, the pH falls only slightly.

When a base is added, the concentration of \(\text{OH}^-\) ions rises, which would normally send the pH up. Now the weak acid component steps in:

\[\text{CH}_3\text{COOH} + \text{OH}^- \longrightarrow \text{CH}_3\text{COO}^- + \text{H}_2\text{O}\]

The hydroxide ions are converted into water, and the weak acid is turned into its conjugate base. The added alkali is neutralised and the pH stays nearly constant.

Flowchart

FC 3.2 – Buffer action in both directions: acid added → free \(\text{H}^+\) rises → conjugate base captures \(\text{H}^+\) → weak acid formed → pH falls only slightly | base added → free \(\text{OH}^-\) rises → weak acid donates \(\text{H}^+\) → water and conjugate base formed → pH rises only slightly

Three features make this two-way control possible:

  1. Weak acids do not dissociate completely, so they act as a reservoir that releases \(\text{H}^+\) only when it is needed.
  2. Conjugate bases can accept excess \(\text{H}^+\) ions and remove them from solution.
  3. Because both components are present at the same time, the system prevents sudden pH change in either direction.

Q2. What exactly is inside a buffer that lets it “absorb” added acid without its pH shooting up or down?

A. A conjugate acid–base pair. The conjugate base present in the solution reacts with added \(\text{H}^+\) ions and locks them up as undissociated weak acid molecules; the weak acid present reacts with added \(\text{OH}^-\) ions and converts them to water. Since the added ions are consumed almost as fast as they arrive, the concentration of free \(\text{H}^+\) — which is what pH actually measures — barely changes.

Q. Why must the acid in a buffer be a weak acid? Would a strong acid such as HCl not neutralise added alkali even better?

A. It would neutralise alkali, but it could do nothing about added acid, and it would not hold a steady pH in the first place. A strong acid dissociates completely, so all its protons are already free in solution and there is no undissociated reservoir left to draw on. A weak acid keeps most of its protons in molecular form, releasing them only as they are used up — and its conjugate base survives in solution to mop up incoming protons. Complete dissociation destroys both halves of the mechanism.

Properties of Buffers
  1. They maintain a nearly constant pH when small amounts of acid or base are added.
  2. They are composed either of a weak acid and its salt with a strong base, or of a weak base and its salt with a strong acid.
  3. They work effectively only within a limited pH range, approximately \(\text{p}K_a \pm 1\).
  4. Dilution does not significantly change the pH of a buffer, because dilution reduces the amounts of both components equally and leaves their ratio — which is what fixes the pH — unchanged.
  5. Buffers reduce pH changes; they do not abolish them completely.

Property (c) is the one that decides which buffer a scientist reaches for. A buffer is only useful near the pH at which its weak acid is half dissociated, and the reason for that will become clear in Section 3.4.

Types of Buffers and Their Uses

Three buffer systems cover most of the range met in a biotechnology laboratory, and all three appear in the ISC practical on preparation of buffers, which together span roughly pH 4 to pH 9.2.

Acetate vs Phosphate vs Borate buffer | Feature | Acetate buffer | Phosphate buffer | Borate buffer | |—|—|—|—| | Approximate pH range | 4–6 | 5.8–8.0 | 8–10 | | Major components | Acetic acid (\(\text{CH}_3\text{COOH}\)) and sodium acetate (\(\text{CH}_3\text{COONa}\)) | \(\text{H}_2\text{PO}_4^-\) / \(\text{HPO}_4^{2-}\) | Boric acid and borate salts | | Nature of the range | Acidic | Near-neutral, close to physiological pH | Alkaline | | Typical uses | Enzyme assays, biochemical experiments, protein studies at acidic pH | DNA and RNA extraction, cell culture media, protein purification, molecular biology work generally | Electrophoresis, enzyme assays under alkaline conditions, biochemical analysis |

Phosphate buffer is the workhorse of the three, and for a simple reason: its useful range straddles pH 7, which is where most cells, most enzymes and most nucleic acids are happiest. When you extract DNA, culture animal cells, or purify a protein without wanting to denature it, you are almost always working in phosphate buffer or something like it.

Buffering Capacity

A buffer is not inexhaustible. Each time it absorbs an added proton, one molecule of conjugate base is used up; each time it absorbs hydroxide, one molecule of weak acid is used up. Keep adding acid and eventually there is no conjugate base left to intercept it, and the next drop behaves as though it had been added to plain water.

NoteDefinition

Buffering capacity is the amount of acid or base a buffer can absorb before its pH changes appreciably — that is, the ability of a buffer to resist changes in pH.

Buffering capacity depends on two things:

  1. Concentration. The higher the concentration of the buffer components, the greater the buffering capacity, because there is simply more of the reservoir to draw on. A 1.0 M buffer resists far better than a 0.2 M buffer of the same pH.
  2. The ratio of the two components. Capacity is maximum when the concentration of the weak acid equals the concentration of its conjugate base, because at that point the buffer is equally well equipped to deal with added acid and added alkali.

Once the capacity is exceeded, the pH no longer changes gently — it changes rapidly. This is the limitation of buffer action, and it is visible as a shape rather than a number, as shown in Fig. 3.4.

Diagram

Fig. 3.4 – A graph with volume of acid added on the x-axis and pH on the y-axis, showing two curves: unbuffered distilled water falling steeply and immediately, and a buffer solution running almost flat over a long plateau before falling steeply once its capacity is exhausted. Label the flat region “buffering region” and the point where the curve turns down “capacity exceeded”.

Read Fig. 3.4 from left to right. Along the flat plateau, large additions of acid produce almost no change in pH — this is the buffer doing its job. The sharp bend marks the moment the conjugate base runs out. Beyond it, the buffered solution behaves no better than plain water, and the pH plunges.

Q3. If a buffer is so good at holding pH steady, why does it eventually fail when you keep adding acid to it?

A. Because the buffer works by consuming its own components. Every proton absorbed converts one conjugate base ion into a weak acid molecule, so the store of conjugate base steadily shrinks. Once it is used up, there is nothing left to intercept the next proton, which then stays free in solution and drops the pH sharply. Buffering capacity is a finite quantity, not a permanent property.

Two acetate buffers of the same pH are prepared, one at 0.10 M and the other at 0.50 M. (a) Which will resist pH change better, and why? (b) How many times greater is the buffering capacity of the second buffer compared with the first?

Buffers in the Human Body

The body faces exactly the problem a fermenter faces, on a larger scale and without the option of switching off. Respiring tissues pour carbon dioxide and organic acids into the blood every second of the day, yet blood pH is held at about 7.35 to 7.45. Three natural buffer systems make this possible.

  1. Bicarbonate buffer system — the carbonic acid–bicarbonate pair, which maintains blood pH at approximately 7.4 and is the body’s main extracellular buffer.
  2. Phosphate buffer system — functions mainly inside cells, where phosphate concentrations are high.
  3. Protein buffer system — the side chains of amino acids in proteins, including haemoglobin, can accept or donate protons and so act as buffers.

Even small deviations from the normal blood pH are dangerous, which is why these systems are supported by the lungs and the kidneys. The lungs adjust pH by controlling how much carbon dioxide is exhaled, since dissolved carbon dioxide forms carbonic acid; the kidneys adjust it by controlling what is excreted in the urine. Elsewhere in the body, buffering sets the stage for digestion: saliva at a pH of about 6.8 suits salivary amylase, and the pancreas secretes bicarbonate ions into the small intestine to neutralise the acid arriving from the stomach, protecting the intestinal lining and creating the near-neutral conditions that intestinal enzymes need.

Buffers in the Biotechnology Laboratory

Almost nothing in a molecular biology laboratory is done in plain water. Buffers are used because:

  1. Many reactions, especially enzyme-catalysed ones, require a constant pH to proceed at a predictable rate.
  2. Constant conditions make results comparable between tubes, between days and between laboratories.
  3. Buffers preserve biological samples — DNA, RNA, proteins and whole cells — by preventing the pH changes that would degrade or denature them.
  4. Buffers control the pH and therefore the net charge on molecules, which is essential for separation techniques such as gel electrophoresis and ion exchange chromatography.
  5. In analytical work, buffers hold the pH steady so that titrations give sharp end points.
  6. Pharmaceutical preparations, including injections, are buffered to keep the drug stable and to reduce irritation at the site of administration.
  7. Cell culture requires a physiological pH, which the medium’s buffer supplies and maintains as the cells metabolise.
  8. Fermentation and microbial growth are controlled by holding the medium at the specific pH the production strain requires.

Point (iv) is worth pausing on, because it explains a piece of laboratory practice that otherwise looks arbitrary. Charged molecules move in an electric field, and the charge a protein or a nucleic acid carries depends on the pH around it. Run electrophoresis in unbuffered solution and the pH will shift as the current passes, changing the charge on the very molecules being separated — the bands blur and the result becomes meaningless. The buffer is not a background detail of the experiment; it is what makes the separation reproducible.

3.4 The Henderson–Hasselbalch Equation and Preparation of Buffers

Section 3.3 explained why a buffer works. It did not answer the question a working scientist actually asks, which is more demanding: “I need a solution held at pH 7.2 — what exactly do I mix, and in what proportion?” Answering that requires a quantitative relationship between the composition of a buffer and the pH it settles at. The Henderson–Hasselbalch equation is that relationship.

The Equation

For an acidic buffer — a weak acid together with its salt, which supplies the conjugate base — the pH is given by:

\[\text{pH} = \text{p}K_a + \log\left[\frac{\text{Proton acceptor}}{\text{Proton donor}}\right]\]

Written in the form usually used for calculation, where the proton acceptor is the conjugate base supplied by the salt and the proton donor is the weak acid itself:

\[\text{pH} = \text{p}K_a + \log\frac{[\text{Salt}]}{[\text{Acid}]}\]

The symbols mean the following:

  1. \(\text{p}K_a = -\log K_a\), where \(K_a\) is the acid dissociation constant of the weak acid. \(\text{p}K_a\) is a fixed property of that particular acid at a given temperature, and like pH it carries no unit.
  2. \([\text{Salt}]\) is the concentration of the conjugate base, in mol L\(^{-1}\).
  3. \([\text{Acid}]\) is the concentration of the weak acid, in mol L\(^{-1}\).
  4. \(\text{pH}\) is the resulting pH of the buffer, again with no unit.

Because both concentrations appear inside the same ratio, their units cancel — which is the first useful thing the equation tells us.

Reading What the Equation Says

pH depends on a ratio, not on absolute amounts. Double the quantity of both components and the ratio is unchanged, so the pH is unchanged. This is precisely why diluting a buffer does not shift its pH, a property noted in Section 3.3 and now explained rather than merely stated. What dilution does reduce is the buffering capacity, since there is less of each component available to be consumed.

When the two concentrations are equal, pH equals \(\text{p}K_a\). If \([\text{Salt}] = [\text{Acid}]\), the ratio is 1, and the logarithm of 1 is zero, so the whole second term vanishes. This is also, from Section 3.3, the composition at which buffering capacity is maximum. The two facts sit together: the pH at which a buffer works best is the \(\text{p}K_a\) of its weak acid.

The useful range is \(\text{p}K_a \pm 1\). A ratio of 10 : 1 makes the log term \(+1\), and a ratio of 1 : 10 makes it \(-1\). Beyond these ratios, one component has become so scarce that it can absorb very little before running out. That single line of arithmetic is the whole justification for the “limited pH range” property listed earlier.

Q. Why does a scientist choose a buffer whose \(\text{p}K_a\) is close to the pH the experiment needs?

A. Because the closer the required pH is to the \(\text{p}K_a\), the closer the ratio of salt to acid is to 1 — and that is where the buffer has the largest reserve on both sides. Choose a buffer whose \(\text{p}K_a\) is far from the required pH and you are forced into a lopsided ratio, say 100 : 1, where one component is almost absent. The solution will show the correct pH on the meter but will collapse after absorbing very little acid or alkali. Correct pH with negligible capacity is a trap, because the failure only shows up once the experiment is already running.

Significance of the Henderson–Hasselbalch Equation
  1. It shows that the pH of a buffer depends on the ratio of salt to acid, not on their absolute quantities.
  2. It shows that when acid and salt concentrations are equal, pH equals \(\text{p}K_a\).
  3. It allows a buffer of any required pH to be prepared by calculating the ratio needed.
  4. It explains why buffers resist pH change on addition of small amounts of acid or base, since converting a little of one component into the other changes the ratio only slightly, and the logarithm dampens that change further.
  5. It is used routinely in biotechnology, biochemistry, medicine and physiology wherever pH has to be set or predicted.

Point (d) repays a moment’s thought, because it explains the degree of protection a buffer gives. Suppose a buffer holds equal amounts of acid and conjugate base, and enough acid is added to convert one-tenth of the conjugate base into the weak acid. The ratio shifts from 1 to about 0.82 — and since the logarithm of 0.82 is roughly \(-0.09\), the pH falls by less than a tenth of a unit. The same addition to unbuffered water would send the pH crashing. The logarithm is what converts a substantial chemical change into a trivial pH change.

You can see this behaviour as a curve in Fig. 3.5.

Diagram

Fig. 3.5 – Titration curve of a weak acid: volume of strong alkali added on the x-axis, pH on the y-axis. Mark the flat central region as the “buffering region”, the mid-point of that region as “half-neutralisation, where \([\text{Salt}] = [\text{Acid}]\) and \(\text{pH} = \text{p}K_a\)”, and show dashed lines at \(\text{p}K_a + 1\) and \(\text{p}K_a - 1\) bounding the effective range.

The x-axis of Fig. 3.5 records how much alkali has been added, and the y-axis the pH reached. The curve is steep at the start and at the end, but almost flat in the middle — and the middle is where the solution contains appreciable amounts of both the weak acid and its conjugate base. The exact centre of that flat stretch, where half the acid has been neutralised, is the \(\text{p}K_a\).

Preparing a Buffer of Required pH

Preparing buffers is a listed ISC practical, in which phosphate, acetate and borate buffers are made to cover the range of about pH 4 to pH 9.2. The calculation and the bench work follow directly from the equation above.

Preparation of a buffer of required pH Principle: The pH of a buffer is fixed by the ratio of conjugate base to weak acid, as given by the Henderson–Hasselbalch equation; so choosing a weak acid whose \(\text{p}K_a\) lies near the target pH, and then setting the correct ratio of its salt to itself, produces a solution of the required pH. Steps: 1. Select a buffer system whose \(\text{p}K_a\) is within about one unit of the required pH — acetate for acidic work, phosphate for near-neutral work, borate for alkaline work. 2. Use the Henderson–Hasselbalch equation to calculate the ratio \([\text{Salt}]/[\text{Acid}]\) that gives the required pH. 3. Decide the total molar concentration of the buffer, which fixes its buffering capacity, and calculate the mass or volume of each component needed. 4. Weigh or measure the components and dissolve them in distilled water in a clean beaker. 5. Calibrate the pH meter with standard buffers, then measure the pH of the prepared solution. 6. Adjust to the exact value by adding small amounts of the acid or the salt while stirring, then make up to the final volume. 7. Label with the composition, concentration and pH, and store as directed. Used for: Setting and holding the pH of enzyme assays, DNA and protein extractions, electrophoresis tanks, culture media and pharmaceutical preparations.

Two points of laboratory judgement are worth noticing in that list. Step 3 is a separate decision from step 2, because the ratio fixes the pH while the total concentration fixes the capacity — a 0.05 M and a 0.5 M phosphate buffer can read exactly the same pH but behave very differently in use. And step 6 is done before making up to the final volume, since adding the last of the water after adjusting would dilute the buffer to a concentration you did not intend. The approach recommended for school laboratories is to use easily available chemicals at reasonable cost, since the aim is to make the underlying calculation clear rather than to prepare a reagent of research-grade precision.

A buffer is prepared using a weak acid of \(\text{p}K_a = 4.75\). (a) What ratio of salt to acid is required to obtain a buffer of pH 4.75? (b) What ratio would be required for a buffer of pH 5.75?

During a DNA extraction the buffer must be held at pH 8.0. Two weak acids are available, one with \(\text{p}K_a = 7.2\) and one with \(\text{p}K_a = 4.8\). (a) Calculate the salt-to-acid ratio each would require. (b) State, with reasons based on your answers, which acid should be chosen.

Take the buffer you prepared in the laboratory and measure its pH. Now add distilled water to double its volume and measure the pH again. Record both readings, then predict — before you check — what would happen to the pH if you instead added a few drops of dilute HCl to each of the two solutions. Explain your prediction using the ratio idea from this section.

3.5 Physical Variables in Biochemical Processes

Getting the chemistry right is only half the job. A culture medium may be perfectly formulated and perfectly buffered, yet still give a poor yield — because the cells at the bottom of the vessel are starved of oxygen, or because the region near the heating jacket is several degrees warmer than the rest. Conditions that are correct on average are not the same as conditions that are correct everywhere, and it is the physical variables of a process that decide which of the two you get.

Temperature

Every organism and every enzyme has an optimum temperature, and the penalty for straying is asymmetric. Below the optimum, molecules move more slowly, reactions slow down and growth is sluggish — but nothing is destroyed, and the culture recovers when warmth is restored. Above the optimum, proteins begin to denature, and that damage is usually permanent.

A process is therefore designed to sit at, or slightly below, the optimum rather than above it. Holding that temperature evenly through a large volume is a genuine engineering problem, because the cells themselves release heat as they metabolise, and a densely growing culture will warm itself.

Fluid Flow

Fluid flow is the movement of liquids or gases. In a biochemical process it is the movement of the nutrient medium, of the gases sparged into it, and of the cells suspended in it. There are two kinds.

Laminar flow vs Turbulent flow | Feature | Laminar flow | Turbulent flow | |—|—|—| | Nature of movement | Smooth and orderly, in parallel layers | Chaotic and irregular, with eddies and cross-currents | | Typical speed | Low | High | | Mixing between layers | Very little | Extensive | | Use in biotechnology | Sterile air curtain in a laminar air flow cabinet; gentle handling of shear-sensitive cells | Vigorous mixing in fermenters, to distribute nutrients, gases and heat |

Diagram

Fig. 3.6 – Two horizontal pipes side by side. In the upper pipe, straight parallel arrows in neat layers labelled “laminar flow”; in the lower pipe, arrows curling into eddies and crossing one another, labelled “turbulent flow”. Show a dye stream entering each pipe — remaining as a thin line in the first, dispersing rapidly in the second.

Both kinds have their place, and the choice depends on what the flow is meant to achieve. Inside a fermenter, turbulent flow is wanted, because its whole purpose is to keep nutrients and dissolved oxygen evenly distributed, to keep the temperature uniform, and to stop cells from settling out at the bottom of the tank. In a laminar air flow cabinet, the opposite is wanted: a smooth, unbroken sheet of filtered air moving in one direction, because eddies would draw unfiltered room air — and the microorganisms it carries — back into the working area.

Q. If turbulence mixes so well, why not use the most vigorous stirring possible in every fermenter?

A. Because moving fluid exerts shear forces on whatever is suspended in it. Bacterial cells with tough walls tolerate this well, but animal cells, which have only a plasma membrane, and plant cell suspensions, which form fragile clumps, are damaged or torn apart by vigorous agitation. In such cultures the stirring rate is a compromise: fast enough to prevent settling and dead zones, gentle enough to leave the cells intact.

Newtonian and Non-Newtonian Fluids

How a fluid responds to being stirred depends on its viscosity — its resistance to flow. On this basis fluids fall into two groups.

Newtonian vs Non-Newtonian fluids | Feature | Newtonian fluid | Non-Newtonian fluid | |—|—|—| | Viscosity | Constant, whatever force (shear) is applied | Changes when force (shear) is applied | | Flow behaviour | Predictable and linear | Complex and non-linear | | Examples | Water, air, alcohol | Blood, yoghurt, toothpaste, slime, cell culture broth with high biomass | | Where met in biotechnology | Simple media and water-based solutions | Cell cultures and fermentation broths containing solids |

Non-Newtonian behaviour takes several forms. Some fluids thin out as they are stirred, as tomato ketchup does; some thicken, as a suspension of corn starch in water does; and some will not begin to flow at all until a minimum force is applied, as with toothpaste.

This matters more than it might appear. Many fermentation broths behave as non-Newtonian fluids, because they are crowded with cells, proteins and polysaccharides secreted during growth. A broth that was thin and easily stirred at the start of a batch can become thick and sluggish by the end, so the stirring that mixed it adequately on day one may leave dead zones on day three. Understanding fluid behaviour is what allows stirrers, pumps and bioreactors to be designed for efficient mixing and mass transfer throughout the run, not just at the beginning.

Fluid Mixing

Fluid mixing is the process of blending two or more fluids to form a uniform mixture. It is achieved in two ways:

  1. Mechanical agitation — using stirrers or impellers driven by a motor.
  2. Pneumatic mixing — using air bubbles, introduced through a sparger, to stir the liquid as they rise through it.

Good mixing ensures an even distribution of temperature, pH and nutrients, enhances microbial or cell growth during fermentation, and prevents the formation of dead zones — regions where no mixing occurs, where cells settle, and where the local conditions drift away from everything the process was designed for. Adding a concentrated glucose solution to a culture broth illustrates the point: without mixing, the cells nearest the inlet meet a sugar concentration high enough to inhibit them while the cells at the far side of the vessel remain starved.

Mass Transfer

Mass transfer is the movement of substances — gases or dissolved solutes — from one location to another, often across a boundary between two phases. It occurs by two mechanisms:

  1. Diffusion — movement from a region of higher concentration to a region of lower concentration.
  2. Convection — movement carried along by the bulk flow of the fluid itself.

Two kinds of mass transfer dominate a biochemical process. Oxygen transfer is the movement of oxygen from air bubbles into the liquid medium, and it is critical for aerobic fermentation. Nutrient transport ensures that all cells, not merely those near the feed point, receive the substances they need.

Oxygen transfer deserves particular attention because it is so often the step that limits the whole process. Oxygen is only sparingly soluble in water, so the medium can hold very little of it in reserve; a dense, actively respiring culture can consume the dissolved oxygen in a few seconds. Oxygen must therefore be supplied continuously, and the rate at which it crosses from bubble to liquid — which depends on how small the bubbles are, how long they stay in the liquid, and how well the vessel is stirred — often sets the maximum cell density a fermenter can support.

Heat Transfer

Heat transfer is the movement of heat from one place to another. It occurs in three ways: conduction through solids, convection through fluids, and radiation by electromagnetic waves.

In a biochemical process, heat transfer keeps the temperature within the optimal range for enzymes and cell growth, prevents overheating in bioreactors, and makes sterilisation processes possible — an autoclave works by transferring heat from steam into the material being sterilised. Heating or cooling jackets fitted around a fermenter are the usual arrangement: water at a controlled temperature is circulated through the jacket, conducting heat in or out through the vessel wall, while the stirring inside distributes it by convection.

Flowchart

FC 3.3 – How the physical variables act together in a stirred vessel: motor drives impeller → turbulent flow generated → bubbles broken into smaller bubbles and dispersed → oxygen and nutrients transferred evenly to all cells → metabolic heat carried to the vessel wall → heat removed through the cooling jacket → uniform temperature, pH and nutrient supply maintained throughout the vessel

Every one of these variables — flow, mixing, mass transfer and heat transfer — is controlled inside a single piece of equipment. That equipment is the bioreactor, and it is the subject of the next section.

3.6 Fermentation and Bioreactors

The word fermentation is used in two rather different senses, and confusing them is a common source of error in examinations.

In its strict biochemical sense, fermentation means the breakdown of a substrate such as glucose in the absence of oxygen, with the release of energy and the formation of products such as ethanol or lactic acid. Used this way, the word describes a particular kind of metabolism, and it is contrasted with aerobic respiration.

In its industrial sense — the sense in which biotechnologists use it every day — fermentation means any large-scale process in which microorganisms or their enzymes act on a substrate to produce a useful product. Used this way, the word covers processes that require plenty of oxygen just as readily as those that exclude it. The production of penicillin is called a fermentation even though the mould producing it must be supplied with air continuously.

NoteDefinition

Fermentation is the process in which microorganisms, or enzymes derived from them, act on a suitable substrate under controlled conditions to convert it into a useful product on a commercial scale.

What a Fermentation Produces

The range is wide, which is what makes the process so useful. Microbial fermentations yield alcohols and organic acids such as ethanol, lactic acid, acetic acid and citric acid; antibiotics such as penicillin; enzymes used in industry; vitamins; and, in modern recombinant processes, human proteins made by engineered cells. Traditional foods — curd, idli batter, bread, cheese, vinegar — are all fermentation products too, made by exactly the same biology at a domestic scale.

Whatever the product, the shape of the process is the same, and it has three stages.

Flowchart

FC 3.4 – The three stages of a fermentation process: upstream processing (select and maintain the production strain → prepare and formulate the medium → sterilise medium and equipment → raise the inoculum in stages) → fermentation (transfer the inoculum to the bioreactor → maintain pH, temperature, aeration and agitation → monitor growth and product formation until the batch is complete) → downstream processing (separate cells from broth by filtration or centrifugation → extract and purify the product → concentrate, formulate and package → quality control testing before release)

Notice how much of the work lies on either side of the fermentation itself. Preparing a sterile medium and building the inoculum up in stages, from a slant to a flask to a seed vessel, may take days; separating and purifying the product afterwards is frequently the most expensive part of the entire operation, because the desired substance is present in a dilute, messy mixture of cells, unused nutrients and by-products.

The Bioreactor

The vessel in which all this is done is the bioreactor, also called a fermenter.

NoteDefinition

A bioreactor is a closed vessel designed to carry out a biochemical process under controlled and sterile conditions, providing the optimum physical and chemical environment for microorganisms, or for plant and animal cells, to grow and form the desired product.

A bioreactor is far more than a large container. Everything discussed in Section 3.5 — flow, mixing, mass transfer and heat transfer — has to be delivered simultaneously and held steady for the length of the run, often several days, without a single contaminating organism getting in. The most widely used design is the stirred tank bioreactor, shown in Fig. 3.7.

Diagram

Fig. 3.7 – Vertical section of a stirred tank bioreactor, labelled: cylindrical stainless steel vessel; motor at the top; central shaft carrying two impellers; baffles fixed to the inner wall; sparger at the base with air bubbles rising; sterile air inlet with filter; exhaust gas outlet with filter; cooling jacket surrounding the vessel; pH probe, temperature probe and dissolved oxygen probe entering through ports; inoculum port; acid, alkali and antifoam addition ports; harvest valve at the bottom. Mark the liquid level, leaving a headspace above it.

Work through Fig. 3.7 part by part, asking what each part is for.

  1. Vessel body. Usually stainless steel, because it withstands repeated steam sterilisation, resists corrosion by acids and alkalis, and can be polished smooth so that no crevices remain where organisms could shelter. The vessel is filled only to about two-thirds or three-quarters of its volume, leaving a headspace for foam and rising gas.
  2. Impeller and shaft. Driven by a motor, the impeller agitates the contents. It keeps cells suspended, distributes nutrients, breaks large air bubbles into smaller ones and helps distribute heat.
  3. Baffles. Vertical strips fixed to the inner wall. Without them, a stirred liquid simply rotates as a body around the shaft and a vortex forms, which mixes badly; baffles interrupt that rotation and convert it into genuine turbulent mixing.
  4. Sparger. A perforated ring or plate at the base through which sterile air or oxygen is introduced as fine bubbles. The smaller the bubbles, the greater their total surface area, and the faster oxygen crosses into the liquid.
  5. Air filters. Fitted on both the incoming air line and the exhaust. The inlet filter keeps airborne contaminants out of a sterile vessel; the exhaust filter keeps the production organism from being released into the surroundings.
  6. Cooling jacket or coils. Water at a controlled temperature is circulated to remove the heat generated by metabolism and by stirring, or to supply heat when the culture must be warmed.
  7. Probes and control ports. A pH probe, a temperature probe and a dissolved oxygen probe report continuously on conditions inside. Their readings are used to trigger corrective action — adding acid or alkali to hold the pH at its set point, adjusting the jacket temperature, or increasing the air flow.
  8. Antifoam addition port. Vigorous aeration of a protein-rich broth produces foam, which can block the exhaust filter and carry cells out of the vessel. Small amounts of an antifoam agent are added to control it.
  9. Inoculum port and harvest valve. The culture is introduced through one and the finished broth withdrawn through the other, both designed so that the vessel is never opened to the room.

The sparged bioreactor is a simpler design in which there is no impeller at all: the rising stream of air bubbles alone does both the aerating and the mixing. Because there are no moving parts inside the vessel, it is mechanically simpler, cheaper to run and gentler on fragile cells — but the mixing is less vigorous, so it suits cultures that do not need intense agitation.

Q4. A fermenter is a sealed steel tank. How do the microbes inside it get oxygen, nutrients and an even temperature all the way through?

A. Nothing is left to chance inside the vessel. Sterile air is forced in through a sparger at the base, which breaks it into fine bubbles so that oxygen dissolves into the liquid across a large surface area. An impeller, working against fixed baffles, stirs the contents into turbulent flow, so that dissolved oxygen and nutrients are carried to every cell instead of remaining near the point where they entered. A jacket of circulating water removes the heat that the cells themselves generate, and the stirring spreads that cooling evenly. Probes for pH, temperature and dissolved oxygen report conditions continuously, and acid, alkali, air flow or coolant is adjusted automatically to hold each variable at its set point.

Q. Why is the whole vessel sterilised before every run, when the production organism is going to be added to it anyway?

A. Because the production organism has been chosen for the product it makes, not for its ability to win a fight. A stray contaminant that grows faster will consume the substrate, crowd out the production strain, and may release substances that spoil or poison the product — and in a vessel of several thousand litres, that means the entire batch, along with the days of work invested in it, is lost. Sterilising the medium and the equipment, and filtering every litre of air entering, ensures that the only organism growing is the one that was deliberately put there.

An ISC project on the organisation of a fermenter is one of the suggested project topics, and Fig. 3.7 is the natural starting point for it: for each labelled part, state what it does and what would go wrong if it were removed.

A stirred tank bioreactor of total capacity 2000 L is operated at 70% working volume. The fermentation yields 12 g of product per litre of broth. (a) Calculate the working volume. (b) Calculate the total mass of product formed in one batch. (c) If downstream processing recovers only 85% of the product, calculate the mass finally obtained.

3.7 Sampling Methods: Quadrat and Transect

Suppose you are asked how many dandelion plants grow in a ten-hectare meadow, or how many bacterial colonies are present in a litre of pond water. Counting every one of them is impossible, and even if it were possible it would be a waste of effort. The sensible approach is to examine a small, carefully chosen part of the whole and then reason outwards from it. That is sampling.

NoteDefinition

A population is the entire set of individuals, items or observations that a study is concerned with; a sample is the smaller subset actually examined; and sampling is the process of selecting that subset so that conclusions drawn from it can be applied to the whole population.

The entire value of a sample rests on one condition: it must genuinely represent the population. A biologist who places all her sampling squares on the green, damp patch of a field because it looks more interesting will conclude that the field is lush, and she will be wrong. Bias of this kind cannot be corrected afterwards by any amount of clever arithmetic, which is why sampling positions are chosen at random — for example by generating pairs of random numbers and using them as coordinates on the study area.

Q. Would it not be more accurate to place the sampling squares evenly across the whole area rather than randomly?

A. Regular spacing is sometimes used, but it carries a hidden risk. If the feature being studied happens to be arranged with a spacing of its own — crop rows, ridges, drainage lines — a regular grid may fall repeatedly on the same kind of position and miss everything in between. Random placement has no such pattern to collide with, so over enough samples it represents all parts of the area fairly.

The Quadrat Method

A quadrat is a square frame of known area that is placed on the ground, and everything of interest inside it is counted or recorded. Several quadrats are placed at random positions, and the results are pooled.

Quadrat sampling Principle: If sampling frames of known area are placed at random over a study area, the average number of individuals they contain reflects the number present in the area as a whole, so counts from a small measured area can be scaled up to the whole. Steps: 1. Choose a quadrat of suitable size for the organisms being studied. 2. Decide the number of quadrats to be laid, and fix their positions at random. 3. Place the quadrat at each chosen position. 4. Count the individuals of each species or category lying within the frame, following a fixed rule for individuals lying on the boundary. 5. Record the counts in a table, one row per quadrat. 6. Calculate density, frequency and abundance from the pooled data. Used for: Estimating the number and distribution of organisms that do not move about — plants, seeds, sessile animals, and colonies on a plate — over an area too large to count completely.

Two quantities are usually calculated from quadrat data. Density tells you how crowded the organisms are:

\[\text{Density} = \frac{\text{Total number of individuals of a species in all quadrats}}{\text{Total area sampled}}\]

Frequency tells you how widely they are spread, which is a different question altogether:

\[\text{Frequency (\%)} = \frac{\text{Number of quadrats in which the species occurs}}{\text{Total number of quadrats studied}} \times 100\]

A species may have high density but low frequency — many individuals, all packed into one corner — or low density with high frequency, meaning a few individuals scattered almost everywhere. Reporting only one of the two numbers hides half the story.

The ISC practical on sampling makes this concrete without needing a field at all. Yellow and green pea seeds are scattered at random on a table top, and a quadrat of 30 cm × 30 cm, divided into blocks of 6 cm × 6 cm, is laid over them. The seeds of each colour are counted block by block, and the frequency of each type is calculated. Working in a group, with the seeds scattered by someone who is not counting, keeps the exercise honest — and repeating it with a fresh scatter shows how much the answer varies from one sample to the next, which is exactly the variability that Section 3.10 will teach you to measure.

Diagram

Fig. 3.8 – A square quadrat frame of 30 cm × 30 cm divided by a grid into 25 smaller blocks of 6 cm × 6 cm, lying over a scatter of yellow and green pea seeds. Show seeds lying wholly inside blocks, and one or two lying across a grid line to raise the boundary rule.

A student lays 20 quadrats, each of area \(1\ \text{m}^2\), in a field. Species A is found in 14 of the quadrats, with a total count of 84 individuals. (a) Calculate the density of species A per square metre. (b) Calculate its percentage frequency. (c) Species B has a density of 4.2 per square metre but a frequency of only 15%. Describe, in one sentence, how the distribution of species B differs from that of species A.

The Transect Method

A quadrat answers the question “how much is there?”. It is much less good at answering “how does it change from here to there?” — and in ecology that second question is often the interesting one. Vegetation changes as you walk from the water’s edge up a river bank, or from open ground into the shade of a wood. A sampling method that follows the direction of change is needed, and that is a transect.

A transect is a line laid across the study area, usually along the direction in which conditions are changing, and observations are made along it.

Transect sampling Principle: Organisms are distributed along environmental gradients such as moisture, light, altitude or salinity, so recording what occurs at intervals along a line running across the gradient reveals how the community changes with position, not merely how much of it is present. Steps: 1. Identify the gradient to be studied and lay a measuring tape or rope across it in a straight line. 2. Decide whether to record every organism touching the line (line transect) or every organism within a fixed width on either side of it, usually by laying quadrats at intervals (belt transect). 3. Work along the line from one end to the other, recording each organism together with its distance along the line. 4. Note the physical conditions at intervals as well, so that the biological change can be related to the environmental change. 5. Plot the results against distance to show the pattern of change. Used for: Studying zonation and the change of communities along a gradient — up a sea shore, across a pond margin, along a hillside, or from a polluted outfall to clean water downstream.

Quadrat vs Transect | Feature | Quadrat | Transect | |—|—|—| | Shape of sampling unit | A square frame of known area | A line laid across the area, sampled along its length | | Placement | Random positions across the whole area | Deliberately along the direction of an environmental gradient | | Question it answers | How many are there, and how widely spread? | How does the community change from one place to another? | | Data obtained | Density, frequency, abundance | Zonation and change in composition with distance | | Best suited to | Areas that are reasonably uniform | Areas where conditions change steadily across the site |

The two methods are not rivals. A belt transect is, in effect, a line of quadrats, so it produces density and frequency figures and shows how they shift with position. Which method is chosen depends entirely on the question being asked — and choosing correctly is itself part of designing a sound study.

Whichever method is used, the outcome is the same in form: a set of numbers standing in for a much larger reality. Those numbers now have to be summarised and judged, and that is what the remainder of this chapter is about.

3.8 Data Collection: Primary and Secondary Data

Statistical analysis begins long before any calculation is done. It begins with the question of where the numbers came from, because no amount of careful arithmetic can rescue a conclusion drawn from unsuitable data. In research, the methods of collecting information fall into two broad categories: primary data and secondary data.

Primary Data
NoteDefinition

Primary data is data originated for the first time by the researcher, through direct effort and experience, specifically to address the research problem at hand. It is also called first-hand or raw data.

The defining features follow from that definition. Primary data is factual, original and collected with the immediate aim of solving the problem in front of the investigator. Its collection is under the investigator’s direct control and supervision, which means the investigator decides what is measured, how it is measured and how carefully — and therefore knows exactly how trustworthy the result is.

In biotechnology, primary data means the readings a researcher generates first-hand through experiments, observations, surveys or trials for a particular study or project. For example:

  1. Data obtained from performing a gene expression analysis in a laboratory.
  2. Results from a clinical trial testing a new vaccine, or a field trial of a genetically modified crop.
  3. The colony counts, optical density readings and pH measurements recorded during a fermentation run.

Such data can be gathered by many methods — surveys, direct observation, physical testing, mailed questionnaires, questionnaires filled in and returned by enumerators, personal interviews, telephonic interviews, focus groups and case studies — and the choice of method is itself part of designing the study.

The price of all this control is cost. Collecting primary data is expensive and slow, because the work is carried out by the organisation or agency itself and requires investment, equipment, trained manpower and time. A clinical trial may run for years before it yields its first usable table.

Secondary Data
NoteDefinition

Secondary data is second-hand data obtained from information that has already been collected and published by other researchers, institutions or organisations, and is used for reference, analysis or comparison in a new study.

Secondary data relates to the past rather than to the present moment, and it was gathered for somebody else’s purpose. It is often the refined form of somebody’s primary data — the tables and summaries produced after statistical methods were applied to raw observations. Its sources include scientific journals, public databases such as NCBI, patent records, government publications and health reports, censuses, internal records of an organisation, reports, books and websites. A study of microbial strains or fermentation techniques that draws on previously published research papers is working with secondary data.

Its great advantage is convenience: it is easily available, and it saves both the time and the cost that first-hand collection would demand. But convenience comes with two standing warnings:

  1. The data was gathered for a purpose other than the present problem, so its usefulness may be limited in several ways — relevance, accuracy, the exact conditions under which it was recorded, or how up to date it is.
  2. The objective and the method adopted by the original collectors may not suit the current situation at all. These factors must be examined before the data is used, not after the conclusions have been drawn.

Q. If primary data is more reliable, why does any researcher bother with secondary data?

A. Because collecting everything first-hand is often impossible, wasteful, or both. No single laboratory can sequence a genome that has already been sequenced and published, or repeat a national health survey covering millions of people. Secondary data also supplies the background against which new findings are judged — you cannot claim your strain gives a higher yield without published figures to compare against. The rule is not to avoid secondary data, but to know its origin and its limits before leaning on it.

Comparing the Two

Primary data vs Secondary data | Feature | Primary data | Secondary data | |—|—|—| | Origin | Originated by the researcher for the first time | Already existing, collected earlier by other investigators, agencies or organisations | | Time reference | Real-time data | Relates to the past | | Purpose of collection | Collected to address the problem at hand | Collected for purposes other than the present problem | | Collection process | Involved, lengthy and demanding | Rapid and easy | | Sources | Surveys, observations, experiments, questionnaires, personal interviews | Government publications, websites, books, journal articles, internal records | | Resources required | Large amounts of time, cost and manpower | Relatively inexpensive and quickly available | | Control over quality | Specific to the researcher’s needs; the researcher controls the quality | Neither specific to the researcher’s needs, nor under the researcher’s control | | Form of the data | Available in raw form | The refined form of primary data, obtained after statistical methods have been applied to it | | Reliability | More reliable and accurate, being collected directly and under supervision | Less reliable for the present purpose, since its conditions of collection are not controlled or always known |

Many good studies use both. A team investigating an improved fermentation process might take yield figures from published literature as secondary data to decide which strains are worth testing, and then generate its own primary data by running those strains under identical conditions in its own bioreactors. The published figures narrow the search; only the team’s own measurements can settle the answer.

using other people’s data Secondary data is somebody else’s work, and using it carries obligations. Published results must be cited so that credit goes where it is due and readers can trace the original; presenting another group’s figures as one’s own is plagiarism, whatever the field. Where the data concerns people — clinical records, genetic profiles, health survey responses — a second obligation arises: the individuals concerned consented to their data being used for a particular purpose, and reusing it for a different purpose without fresh consent or proper anonymisation breaches that agreement. Those who argue for freer sharing point out that pooled data has accelerated genetics and public health enormously, and that data locked away helps nobody. Those who argue for tighter control point out that genetic data cannot be truly anonymised, since it identifies not only the donor but their relatives. Most institutions resolve this through ethics committees, which examine each proposed use of existing human data before it is permitted.

The ISC practical on data collection asks you to gather both kinds — some primary data of your own and some secondary data from published sources — tabulate them and draw a conclusion. The tabulation matters as much as the collection. Raw observations recorded loosely on the back of a page are nearly worthless a week later; observations entered into a properly headed table, with units stated and the date and conditions noted, can still be analysed years afterwards. Once tabulated, the data is ready to be summarised, and that is the subject of the next section.

3.9 Measures of Central Tendency

A set of data, once collected, is usually too long to be understood by looking at it. Fifty colony counts, or the marks of a whole class, or the enzyme activities recorded in thirty tubes, tell you very little in raw form. What you want first is a single number that stands for the whole set — a value around which the observations cluster. That number is a measure of central tendency.

NoteDefinition

A measure of central tendency is a single value that describes a set of data by identifying the central position within that set.

Measures of central tendency are also called measures of central location, and, because they condense many observations into one, they belong to the family of summary statistics. Three of them are used routinely: the mean, the median and the mode. All three are valid, but they are not interchangeable, and a good part of biostatistics consists of knowing which one to trust in a given situation.

Mean

The mean is the ordinary average. It is calculated by adding all the values and dividing by the total number of values.

\[\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}\]

\[\bar{x} = \frac{x_1 + x_2 + x_3 + \cdots + x_n}{n}\]

Here \(x_1, x_2, \ldots, x_n\) are the individual observations, \(n\) is the number of observations, and \(\bar{x}\) (read “x bar”) is the mean. The mean carries the same unit as the observations themselves — a mean of five DNA concentrations measured in µg/µL is itself in µg/µL.

Suppose the expression levels of a gene in five different samples are 10, 12, 8, 11 and 9. Then:

\[\bar{x} = \frac{10 + 12 + 8 + 11 + 9}{5} = \frac{50}{5} = 10\]

The mean of 10 describes this set well. Every observation lies close to it, so a single number genuinely stands for all five.

When the Mean Misleads

Now consider the marks obtained by ten students of a class:

Student A B C D E F G H I J
Marks 15 18 16 14 15 15 12 17 90 95

The mean of these ten marks is 30.7. Yet inspecting the raw data suggests that 30.7 is a poor description of a typical student, since eight of the ten scored between 12 and 18, and not one student scored anything close to 30.7. The mean has been pulled upwards by the two unusually high marks. Values of this kind, lying far away from the bulk of the data, are called outliers, and the mean is highly sensitive to them, because every observation enters the sum with its full weight.

This is not a defect to be hidden; it is a property to be understood. In some situations the pull of extreme values is exactly what you want — a mean yield across many fermentation batches should take the failed batches into account. In others, as here, it produces a number that represents nobody. When that happens, a different measure of central tendency is needed.

Q5. Ten students score 15, 18, 16, 14, 15, 15, 12, 17, 90 and 95. Their average is 30.7 — yet no student scored anywhere near 30.7. Is the average lying, and what should we use instead?

A. The average is not lying; it is answering a different question from the one being asked. The mean divides the total equally among all members, so two very high scores lift it far above the bulk of the data. It is arithmetically correct but unrepresentative. When a data set contains outliers like this, the median is the better measure of central tendency, because it depends only on the position of the middle value and not on how extreme the highest and lowest values happen to be. The median of these ten marks is 15.5 — a number that describes a typical student in this class far more honestly than 30.7 does.

Median

The median is the middle value in a data set when the observations are arranged in ascending or descending order.

  1. If the number of values is odd, the median is the middle number.
  2. If the number of values is even, the median is the average of the two middle numbers.

\[\text{For odd } n: \quad \text{Median} = x_{\left(\frac{n+1}{2}\right)}\]

\[\text{For even } n: \quad \text{Median} = \frac{x_{\left(\frac{n}{2}\right)} + x_{\left(\frac{n}{2}+1\right)}}{2}\]

where \(x_{(k)}\) denotes the value occupying the \(k\)th position after the data has been arranged in order.

For the data 8, 9, 10, 11, 12, there are five values, so the median is the third value: 10.

For the data 8, 9, 10, 11, there are four values, so the median is the average of the second and third:

\[\text{Median} = \frac{9 + 10}{2} = 9.5\]

Arranging the data in order is not an optional preliminary — it is the whole method. Take the following set of readings:

06 09 13 12 15 12 11 02 07 14 17

Rearranged in order of magnitude, smallest first:

02 06 07 09 11 12 12 13 14 15 17

There are eleven values, so the median is the sixth: 12. It is the middle mark because five scores lie before it and five after it.

For an even number of observations the two middle scores are averaged. Take:

62 37 52 47 42 67 32 47

Rearranged in order:

32 37 42 47 47 52 62 67

There are eight values, so we take the fourth and fifth scores — both 47 — and average them, giving a median of 47.

Notice what the median ignores. It does not matter how large the largest value is; if the 95 in the class data above were 950, the median would still be 15.5. This insensitivity to extremes is precisely why the median is preferred whenever a data set contains outliers or is strongly skewed.

Mode

The mode is the value that occurs most frequently in a data set.

A data set can have one mode, in which case it is called unimodal; more than one mode, when it is bimodal or multimodal; or no mode at all, if every value occurs exactly once and none is more frequent than the rest.

In the data 5, 7, 9, 7, 6, 8, 7, the value 7 appears three times and no other value appears more than once, so the mode is 7.

Consider the number of bacterial colonies counted in seven Petri dishes: 10, 12, 10, 15, 14, 10, 12. Counting the frequency of occurrence of each value:

  1. 10 occurs 3 times
  2. 12 occurs 2 times
  3. 15 and 14 occur 1 time each

The mode is therefore 10 colonies.

On a bar chart or histogram, the mode is the tallest bar — so it can be thought of as the most popular or most commonly observed option. Fig. 3.9 shows this.

Diagram

Fig. 3.9 – A histogram with the measured value on the x-axis and frequency on the y-axis, showing bars of varying height. Arrow pointing to the tallest bar, labelled “Mode”. Show a second, smaller peak elsewhere on the axis to illustrate what a bimodal distribution would look like.

Read Fig. 3.9 by looking at height, not position: the x-axis tells you which value, and the y-axis tells you how often it occurred. The mode is found by scanning across for the greatest height and then reading down to the value beneath it.

The mode is the only one of the three measures that can be used for data which is not numerical at all. If you record the colour of pea seeds, or the Gram reaction of isolates from a soil sample, there is no mean and no median to calculate — but there is certainly a most frequent category.

Q. Which measure would you report for the number of colonies on a set of plates — mean, median or mode?

A. The mean, in most cases, because colony counts from replicate plates cluster around a central value and every plate carries information about the true concentration in the sample. But check the raw counts first. If one plate has been contaminated and shows an enormous count, that outlier will drag the mean up, and either the median should be reported instead or the spoiled plate excluded with the reason recorded. What you must never do is choose the measure after seeing which one gives the answer you were hoping for.

Mean vs Median vs Mode | Feature | Mean | Median | Mode | |—|—|—|—| | What it is | The arithmetic average of all values | The middle value when data is arranged in order | The most frequently occurring value | | Uses all observations? | Yes, every value contributes | No, only the position of the middle value matters | No, only the frequencies matter | | Effect of outliers | Strongly affected | Largely unaffected | Unaffected | | Suitable for non-numerical data | No | No | Yes | | Best used when | Data is fairly symmetrical, with no extreme values | Data is skewed or contains outliers | The most common category or value is what matters |

The number of colonies counted on nine replicate plates is: 24, 27, 25, 26, 24, 28, 25, 24, 96. (a) Calculate the mean, the median and the mode. (b) State which of the three best represents a typical plate, and justify your choice. (c) One plate was found to be contaminated. Recalculate the mean after excluding it, and comment on the difference.

3.10 Standard Deviation and Standard Error

A mean on its own can be dangerously incomplete. Consider two sets of DNA concentration readings that both average 5.2 µg/µL. In the first, every reading lies between 5.1 and 5.3; in the second, the readings range from 1.0 to 9.5. The two sets have the same central value but tell entirely different stories about the reliability of the measurement. The first describes a method under control; the second describes something badly wrong.

To capture that difference, a second number is needed alongside the mean — one that measures how widely the observations are scattered. That is the standard deviation.

Diagram

Fig. 3.10 – Two curves drawn on the same axes, with the measured value on the x-axis and frequency on the y-axis. Both peak at the same point on the x-axis, marked “mean”. One curve is tall and narrow, labelled “low standard deviation — consistent results”; the other is low and broad, labelled “high standard deviation — variable results”.

Standard Deviation
NoteDefinition

Standard deviation (SD) is a measure of how spread out or dispersed the data is around the mean. It shows how much variation or inconsistency exists in a data set.

For a sample, it is calculated as:

\[\text{SD} = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}\]

where \(x_i\) is each individual observation, \(\bar{x}\) is the mean of all the observations, \(n\) is the total number of observations, and \(\Sigma\) means “the sum of”. The standard deviation carries the same unit as the original measurements, so an SD calculated from concentrations in µg/µL is itself in µg/µL — which is what makes it directly comparable with the mean.

The logic of the formula is worth following, because each step exists for a reason. The quantity \((x_i - \bar{x})\) is how far one observation lies from the mean. These deviations are then squared, for two reasons: squaring removes the negative signs, which would otherwise cancel the positive ones and always sum to zero, and it gives extra weight to observations that lie far out. The squared deviations are added, divided to give an average, and finally the square root is taken to return the answer to the original units.

The division is by \((n-1)\) rather than by \(n\). This is because a sample tends to be slightly more tightly clustered than the population it came from — the extremes of a population are rare and so are easily missed by a small sample. Dividing by the smaller number \((n-1)\) makes the answer slightly larger, correcting for that tendency to underestimate.

Worked Example

A biotechnology laboratory measures the DNA concentration, in µg/µL, in five samples:

4.8 5.0 5.2 5.4 5.6

Step 1 — Calculate the mean:

\[\bar{x} = \frac{4.8 + 5.0 + 5.2 + 5.4 + 5.6}{5} = \frac{26}{5} = 5.2\]

Step 2 — Find each deviation from the mean, square it, and add the squares:

\[\sum (x_i - \bar{x})^2 = 0.16 + 0.04 + 0 + 0.04 + 0.16 = 0.4\]

Step 3 — Divide by \((n-1) = 4\):

\[\frac{0.4}{4} = 0.1\]

Step 4 — Take the square root:

\[\text{SD} = \sqrt{0.1} = 0.32\]

The result is read as follows. A low SD means the data points lie close to the mean, so the results are consistent. A high SD means the data points are spread out, so the results are more variable. Here an SD of 0.32 µg/µL against a mean of 5.2 µg/µL indicates a set of measurements in reasonable agreement with one another.

Standard Error

Standard deviation describes the scatter of the individual observations. But in most experiments the number you actually want to report is the mean — and the mean of one sample would come out a little differently if the experiment were repeated with a fresh sample. How far can the mean you obtained be trusted as an estimate of the true value? That is a different question, and it is answered by the standard error.

NoteDefinition

Standard error (SE) is a statistical measure that estimates the variability of the sample mean from the true population mean — that is, how precisely the sample mean estimates the real value.

\[\text{SE} = \frac{\text{SD}}{\sqrt{n}}\]

where SD is the standard deviation of the sample and \(n\) is the number of observations, or sample size. Like the standard deviation, standard error carries the unit of the original measurement.

Suppose a biotechnologist measures enzyme activity in four bacterial strains and records, in units/mL: 20, 22, 21, 23.

  1. Mean \(= \dfrac{20 + 22 + 21 + 23}{4} = 21.5\)
  2. Standard deviation \(\approx 1.29\), calculated by the four steps shown above.
  3. Standard error \(= \dfrac{1.29}{\sqrt{4}} = \dfrac{1.29}{2} = 0.645\)

In biotechnology, standard error is used to assess the reliability of sample data — for instance when measuring gene expression, enzyme activity or the response to a drug. A lower standard error indicates that the mean is more reliable, and it matters most in experiments where only a small number of samples can be examined, which is the usual situation in a school or research laboratory.

Q. If more replicates are performed, does the standard deviation get smaller?

A. No — and this is the distinction most often missed. Standard deviation measures how variable the material and the method really are, and that variability does not change simply because you measured it more often; with more readings your estimate of the SD becomes better, not smaller. Standard error, however, does fall as \(n\) increases, because \(n\) appears under a square root in the denominator. More replicates cannot make an inconsistent assay consistent, but they can pin down the average of that inconsistent assay more precisely. Note also that because of the square root, four times as many replicates are needed to halve the standard error.

Standard Deviation vs Standard Error | Feature | Standard Deviation (SD) | Standard Error (SE) | |—|—|—| | What it shows | The spread of individual data points around the mean | How accurately the sample mean estimates the true population mean | | Effect of increasing sample size | Does not change systematically | Decreases as sample size increases | | Used to describe | Variability | Precision | | Formula | \(\sqrt{\dfrac{\sum (x_i - \bar{x})^2}{n-1}}\) | \(\dfrac{\text{SD}}{\sqrt{n}}\) |

These two quantities are exactly what turn a set of laboratory readings into a defensible result. When you record replicate optical density values in Bradford’s assay for protein estimation, or replicate colony counts, or the quadrat counts from Section 3.7, the mean tells you the value obtained and the standard deviation tells you how much the replicates disagreed. If your replicates scatter widely, the honest conclusion is not that the mean is wrong but that the method needs tightening before the mean is worth quoting at all. In published work this is why results are shown as “mean ± SD” or “mean ± SE”, with the vertical error bars on a graph representing that spread — a mean quoted with no measure of its variability is an incomplete result.

The protein concentration of a sample, measured in mg/mL in five replicate tubes, is: 2.4, 2.6, 2.5, 2.7 and 2.3. (a) Calculate the mean. (b) Calculate the standard deviation. (c) Calculate the standard error. (d) State how many replicates would be needed to reduce the standard error to half its present value, assuming the standard deviation stays the same.

3.11 Concept of Probability

Biology deals constantly in uncertainty. Will this batch become contaminated? Will this seed germinate? Will this colony carry the recombinant plasmid? None of these questions has a yes-or-no answer in advance, but none of them is a matter of pure ignorance either — each has a chance attached to it that can be estimated, quoted and acted upon. Probability is the branch of mathematics that puts a number on that chance.

The word itself comes from the Latin probabilitas, carrying the senses of credibility, likelihood, provability, worthiness of approval and likeliness. It was applied to matters that were not easily accessible or understandable — that is, to situations where certainty was out of reach and a reasoned estimate was the best that could be had.

NoteDefinition

Probability is the ratio of the number of favourable outcomes to the total number of all possible outcomes.

\[P(E) = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}\]

where \(P(E)\) is the probability of the event \(E\). Because the favourable outcomes are always a part of the total, the value of \(P(E)\) lies between 0 and 1: a probability of 0 means the event cannot occur, a probability of 1 means it is certain, and everything of biological interest lies somewhere in between. Probability has no unit, and it is often expressed as a percentage by multiplying by 100.

Probability entered mathematics through a dispute about gambling. In 1654 a French nobleman put to Blaise Pascal a puzzle about how the stakes should be fairly divided if a game of chance had to be abandoned before it finished. Pascal took the problem up in a series of letters with Pierre de Fermat, and between them the two worked out a systematic way of reasoning about uncertain outcomes. Neither man was interested in gambling for its own sake; what they had produced, almost incidentally, was the foundation of probability theory as a branch of mathematics in the seventeenth century. Today theories of probability and probabilistic modelling are used in almost every field — business, industry and the sciences, including telephone exchanges, computer processing and genetics.

Theoretical and Experimental Probability

Probability tells us about the chances of occurrence of a particular phenomenon, and it is classified into two kinds according to how that chance is arrived at.

NoteDefinition

Theoretical probability is based on mathematical calculation of the occurrence of an event or phenomenon under ideal conditions.

NoteDefinition

Experimental probability is based on the actual results obtained by repeated testing and observation.

The difference is one of source. Theoretical probability is worked out by reasoning about the situation before any trial is performed: a fair coin has two equally likely faces, so the theoretical probability of a head is \(\frac{1}{2}\), or 0.5. Experimental probability is worked out after the event, by counting what actually happened and dividing by the number of trials attempted.

These two numbers are rarely identical, and understanding why is the point of the whole distinction. Toss a coin ten times and you may well get seven heads, giving an experimental probability of 0.7 — the coin has not become unfair, and the theory has not been disproved. Random variation simply produces uneven results in small runs.

Q. If experimental probability keeps disagreeing with theoretical probability, which one is right?

A. Both, but they are answering different questions. Theoretical probability states what would happen in the long run under ideal conditions; experimental probability states what did happen in the particular trials you performed. A large disagreement is worth investigating, because it may mean the ideal assumptions do not hold — the coin may be weighted, the seeds may not all be viable, the pipette may be miscalibrated. A small disagreement, especially in a short run of trials, is expected and needs no explanation beyond chance itself.

The way to reconcile them is to increase the number of trials. As trials accumulate, the experimental probability settles closer and closer to the theoretical value, because unusual runs in one direction are increasingly balanced by unusual runs in the other. This behaviour is shown in Fig. 3.11.

Diagram

Fig. 3.11 – A line graph with the number of trials on the x-axis (running from 10 to 1000) and the experimental probability of obtaining a head on the y-axis (0 to 1). The line swings widely at the left-hand side and gradually settles towards a horizontal dashed line drawn at 0.5, which is labelled “theoretical probability”.

Read Fig. 3.11 from left to right. Early in the run, each new toss changes the fraction substantially, so the line lurches. Further along, each additional toss is one among hundreds and can shift the fraction only slightly, so the line steadies. The lesson for laboratory work is direct: conclusions drawn from a handful of observations are unreliable not because the observer was careless, but because small numbers are inherently noisy.

Why Probability Matters in Biotechnology

Every sampling exercise in Section 3.7 rests on probability, since the reason for placing quadrats at random is to give every part of the study area an equal chance of being examined. The variability described by standard deviation and standard error in Section 3.10 is variability of exactly this random kind. And when a plate of transformed bacteria is screened, or a batch is tested for contamination, or a diagnostic test is interpreted, the question being asked is a probability question: given what was observed, how likely is each explanation?

Deviations between what is expected and what is observed also have practical causes that a good scientist looks for before blaming chance. Seeds may fail to germinate, a few may not have been viable, a colony may have been miscounted, a pipette may deliver slightly less than its nominal volume. Random variation explains small deviations; systematic errors of this kind explain large ones.

A student tosses a coin 50 times and records 32 heads. (a) Calculate the experimental probability of obtaining a head. (b) State the theoretical probability. (c) Suggest one reason for the difference, and state what the student could do to bring the two values closer together.

Roll a die 30 times and record how often a six appears; calculate the experimental probability and compare it with the theoretical value. Now pool your results with those of the whole class, recalculate the experimental probability from the combined data, and comment on what happens to the difference.

3.12 Summary

  1. Biochemical technology applies the chemical and metabolic activities of living cells, or of components obtained from them, to manufacture useful products on a commercial scale under controlled conditions.

  2. Because these processes depend on living cells, they require a suitable nutrient substrate, optimum pH and temperature, freedom from contamination, and a substrate that is cheap and where possible reusable, so that production remains economical and sustainable.

  3. An acid is a proton donor and a base is a proton acceptor. When an acid donates its proton the species left behind is its conjugate base, and when a base accepts a proton the species formed is its conjugate acid; the members of a conjugate pair differ by exactly one proton.

  4. pH is the negative logarithm to base 10 of the hydrogen ion concentration, \(\text{pH} = -\log_{10}[\text{H}^+]\). The scale runs from 0 to 14: below 7 is acidic, exactly 7 is neutral, above 7 is basic. Because the scale is logarithmic, a change of one pH unit is a ten-fold change in acidity.

  5. pH is critical in biological systems because enzymes are pH-specific, because pH alters the charge and structure of proteins and can denature them, because microorganisms grow only within specific pH ranges, and because pH affects membrane permeability, nutrient uptake and metabolic pathways. pH is measured with pH paper or with a pH meter, which must be calibrated against standard buffers.

  6. A buffer solution resists changes in pH when small quantities of acid or base are added. It contains either a weak acid with its conjugate base, or a weak base with its conjugate acid. Added \(\text{H}^+\) is removed by the conjugate base and added \(\text{OH}^-\) is removed by the weak acid, so free hydrogen ion concentration barely changes.

  7. Buffers maintain a nearly constant pH, work effectively only within about \(\text{p}K_a \pm 1\), are not significantly altered in pH by dilution, and reduce but do not abolish pH change. The common systems are acetate buffer (about pH 4–6), phosphate buffer (about pH 5.8–8.0) and borate buffer (about pH 8–10).

  8. Buffering capacity is the ability of a buffer to resist pH change. It increases with buffer concentration and is maximum when the weak acid and conjugate base concentrations are equal. Once the capacity is exceeded, further addition of acid or base changes the pH rapidly.

  9. The Henderson–Hasselbalch equation, \(\text{pH} = \text{p}K_a + \log\frac{[\text{Salt}]}{[\text{Acid}]}\), relates the pH of a buffer to the ratio of conjugate base to weak acid. When the two concentrations are equal the log term is zero and \(\text{pH} = \text{p}K_a\). A buffer of required pH is prepared by selecting a system whose \(\text{p}K_a\) lies near the target pH, calculating the ratio, deciding the total concentration, and adjusting the final pH on a calibrated meter.

  10. The human body is buffered by the bicarbonate system, which holds blood pH at about 7.4, by the phosphate system inside cells, and by the protein buffer system. In the laboratory, buffers keep enzyme reactions reproducible, preserve DNA, RNA, proteins and cells, control charge during electrophoresis and chromatography, stabilise pharmaceutical preparations, and maintain cell culture and fermentation at their required pH.

  11. Physical variables decide whether correct conditions exist everywhere in a vessel rather than merely on average. Fluid flow may be laminar (smooth, orderly, low speed) or turbulent (chaotic, mixed, high speed). Fluids are Newtonian if their viscosity is constant, and non-Newtonian if viscosity changes with applied shear; many fermentation broths are non-Newtonian because of their cells, proteins and polysaccharides.

  12. Fluid mixing is achieved by mechanical agitation using stirrers or impellers, or by pneumatic mixing using sparged air. Mass transfer moves gases and solutes by diffusion or convection, with oxygen transfer being critical for aerobic fermentation. Heat transfer occurs by conduction, convection and radiation, and keeps the vessel within the optimum temperature range.

  13. Fermentation, in its industrial sense, is any large-scale process in which microorganisms or their enzymes convert a substrate into a useful product under controlled conditions. A complete process has three stages: upstream processing, fermentation, and downstream processing.

  14. A bioreactor is a closed vessel that provides controlled, sterile conditions for cells to grow and form a product. In the stirred tank type, the vessel body, impeller, baffles, sparger, sterile air filters, cooling jacket, probes for pH, temperature and dissolved oxygen, antifoam port, inoculum port and harvest valve each perform a specific function. The sparged type has no impeller and is mixed by rising air bubbles alone.

  15. Sampling examines a representative subset of a population because examining the whole is impractical. Sampling positions are chosen at random to avoid bias. A quadrat is a frame of known area placed at random positions, from which density and percentage frequency are calculated; a transect is a line laid along an environmental gradient, sampled as a line transect or a belt transect, and it reveals how a community changes with position.

  16. Primary data is originated first-hand by the researcher for the problem at hand; it is raw, real-time, under direct supervision, more reliable, but expensive and slow to collect. Secondary data is second-hand data already collected and published by others; it is quick and inexpensive but was gathered for a different purpose and may lack relevance, accuracy or specificity.

  17. Measures of central tendency describe the centre of a data set. The mean is the arithmetic average and uses every observation, but is strongly affected by outliers. The median is the middle value of ordered data and is largely unaffected by outliers. The mode is the most frequent value and is the only measure usable for non-numerical categories.

  18. Standard deviation measures how widely individual observations are spread around the mean, \(\text{SD} = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}\). A low SD means consistent results and a high SD means variable results. Standard error, \(\text{SE} = \frac{\text{SD}}{\sqrt{n}}\), measures how precisely the sample mean estimates the true population mean; it falls as sample size increases, and a lower SE means a more reliable mean.

  19. Probability is the ratio of favourable outcomes to the total number of possible outcomes, with values between 0 and 1. Theoretical probability is calculated for ideal conditions; experimental probability is obtained from actual repeated trials. The two differ in short runs because of random variation, and converge as the number of trials increases.

Flowchart

FC 3.5 – The logic of the chapter: living cells need controlled conditions → pH identified as the most critical chemical variable → pH held steady by buffers → buffer composition calculated using the Henderson–Hasselbalch equation → physical variables (flow, mixing, mass transfer, heat transfer) controlled so conditions are uniform → all of this delivered inside a bioreactor during fermentation → the process generates observations → observations obtained by sampling and recorded as primary or secondary data → data summarised by mean, median and mode → variability judged by standard deviation and standard error → chance quantified by theoretical and experimental probability

3.13 Key Terms

  1. Biochemical technology the branch of biotechnology that uses the chemical and metabolic activities of living cells, or components obtained from them, to manufacture useful products on a commercial scale under controlled conditions.

  2. Nutrient substrate the raw material supplied to a culture, which provides the carbon, nitrogen, minerals and energy that the cells need for growth and product formation.

  3. Acid a substance that donates a proton (\(\text{H}^+\)) to another substance during a chemical reaction.

  4. Base a substance that accepts a proton (\(\text{H}^+\)) from another substance during a chemical reaction.

  5. Conjugate base the species that remains after an acid has donated its proton, and which can accept that proton back.

  6. Conjugate acid the species formed when a base accepts a proton, and which can donate that proton again.

  7. Conjugate acid–base pair two species that differ from each other by exactly one proton, such as \(\text{CH}_3\text{COOH}\) and \(\text{CH}_3\text{COO}^-\).

  8. pH the negative logarithm to base 10 of the hydrogen ion concentration of a solution, \(\text{pH} = -\log_{10}[\text{H}^+]\); it has no unit.

  9. Optimum pH the pH at which a particular enzyme or organism shows its highest activity or growth rate.

  10. Buffer solution an aqueous system containing a conjugate acid–base pair, which maintains a nearly constant pH when small quantities of acid or base are added to it.

  11. Weak acid an acid that does not dissociate completely in solution, so that most of its protons remain held in molecular form as a reservoir.

  12. Buffering capacity the amount of acid or base a buffer can absorb before its pH changes appreciably.

  13. Acetate buffer a buffer of acetic acid and sodium acetate, effective at approximately pH 4–6 and used for enzyme assays and protein work at acidic pH.

  14. Phosphate buffer a buffer of \(\text{H}_2\text{PO}_4^-\) and \(\text{HPO}_4^{2-}\), effective at approximately pH 5.8–8.0 and used for nucleic acid extraction, protein purification and cell culture media.

  15. Borate buffer a buffer of boric acid and borate salts, effective at approximately pH 8–10 and used for electrophoresis and alkaline enzyme assays.

  16. \(K_a\) (acid dissociation constant) the equilibrium constant that measures the extent to which a weak acid dissociates in solution.

  17. \(\text{p}K_a\) the negative logarithm to base 10 of the acid dissociation constant, \(\text{p}K_a = -\log K_a\); it equals the pH at which a weak acid is half dissociated.

  18. Henderson–Hasselbalch equation the relation \(\text{pH} = \text{p}K_a + \log\frac{[\text{Salt}]}{[\text{Acid}]}\), used to calculate the pH of a buffer and to prepare a buffer of required pH.

  19. Viscosity the resistance of a fluid to flow.

  20. Laminar flow smooth, orderly flow in parallel layers, occurring at low speed and with little mixing between the layers.

  21. Turbulent flow chaotic, irregular flow with eddies and cross-currents, occurring at high speed and producing extensive mixing.

  22. Newtonian fluid a fluid whose viscosity remains constant regardless of the force or shear applied to it, such as water or alcohol.

  23. Non-Newtonian fluid a fluid whose viscosity changes when force or shear is applied, such as blood, yoghurt or a dense fermentation broth.

  24. Fluid mixing the blending of two or more fluids to form a uniform mixture, achieved by mechanical agitation or pneumatic mixing.

  25. Mechanical agitation mixing brought about by stirrers or impellers driven by a motor.

  26. Pneumatic mixing mixing brought about by bubbles of air introduced by sparging.

  27. Dead zone a region within a vessel where no effective mixing occurs, so that cells settle and local conditions drift away from the set values.

  28. Mass transfer the movement of gases or dissolved solutes from one location to another, often across a phase boundary, by diffusion or by convection.

  29. Diffusion movement of a substance from a region of higher concentration to a region of lower concentration.

  30. Convection movement of a substance carried along by the bulk flow of the fluid containing it.

  31. Heat transfer the movement of heat from one place to another, by conduction through solids, convection through fluids, or radiation as electromagnetic waves.

  32. Fermentation the process in which microorganisms, or enzymes derived from them, act on a substrate under controlled conditions to produce a useful product on a commercial scale.

  33. Upstream processing all the preparatory work preceding a fermentation, including strain selection, medium formulation, sterilisation and building up the inoculum.

  34. Downstream processing the separation, extraction, purification and formulation of the product after the fermentation is complete.

  35. Bioreactor a closed vessel designed to carry out a biochemical process under controlled and sterile conditions, providing the optimum environment for cells to grow and form a product; also called a fermenter.

  36. Impeller the motor-driven blade assembly inside a stirred tank bioreactor that agitates the contents, keeps cells suspended and breaks up air bubbles.

  37. Baffles vertical strips fixed to the inner wall of a bioreactor that interrupt the rotation of the liquid and convert it into genuine turbulent mixing.

  38. Sparger a perforated ring or plate at the base of a bioreactor through which sterile air or oxygen is introduced as fine bubbles.

  39. Stirred tank bioreactor the common bioreactor design in which mixing and aeration are achieved by a motor-driven impeller working with a sparger and baffles.

  40. Sparged bioreactor a bioreactor design with no impeller, in which the rising stream of air bubbles alone provides both aeration and mixing.

  41. Population the entire set of individuals, items or observations with which a study is concerned.

  42. Sample the smaller subset of a population that is actually examined.

  43. Sampling the process of selecting a subset of a population so that conclusions drawn from it can be applied to the whole.

  44. Random sampling sampling in which every part of the study area or population has an equal chance of being selected, so that bias is avoided.

  45. Quadrat a square frame of known area placed at random positions on the ground, within which organisms are counted.

  46. Density the total number of individuals of a species recorded, divided by the total area sampled.

  47. Frequency the percentage of sampling units in which a species occurs, indicating how widely it is spread.

  48. Transect a line laid across a study area, usually along an environmental gradient, along which observations are recorded to show how a community changes with position.

  49. Line transect a transect in which only the organisms actually touching the line are recorded.

  50. Belt transect a transect in which organisms within a fixed width on either side of the line are recorded, usually by laying quadrats at intervals.

  51. Primary data data originated for the first time by the researcher through direct effort, specifically to address the problem at hand; also called first-hand or raw data.

  52. Secondary data second-hand data already collected and published by other researchers, institutions or organisations, and used for reference, analysis or comparison.

  53. Measure of central tendency a single value that describes a data set by identifying the central position within it.

  54. Mean the arithmetic average, obtained by adding all the values and dividing by the number of values.

  55. Median the middle value of a data set when the observations are arranged in ascending or descending order.

  56. Mode the value that occurs most frequently in a data set.

  57. Outlier an observation lying far away from the bulk of the data, which strongly affects the mean but not the median.

  58. Standard deviation (SD) a measure of how spread out the individual observations are around the mean, showing how much variation exists in a data set.

  59. Standard error (SE) a measure of how precisely the sample mean estimates the true population mean, calculated as the standard deviation divided by the square root of the sample size.

  60. Probability the ratio of the number of favourable outcomes to the total number of possible outcomes, taking values between 0 and 1.

  61. Theoretical probability probability calculated mathematically for the occurrence of an event under ideal conditions.

  62. Experimental probability probability obtained from the actual results of repeated testing and observation.

3.14 Points to Ponder

  1. The pH scale was invented not by a theoretical chemist but by someone trying to solve a brewery’s practical problem. How often, do you think, does a measuring tool matter more to the progress of a science than a new discovery does? Can you think of another quantity whose way of being expressed changed the way people were able to think about it?

  2. A buffer holds pH steady by consuming itself. Are there other systems in biology that maintain stability by being used up in the process, and what does that suggest about the cost of keeping anything constant?

  3. A solution can show exactly the correct pH on a meter and still be almost useless as a buffer. What does this tell you about the difference between a measurement and a property? Where else in laboratory work might a correct-looking reading conceal an unfit reagent?

  4. Sterilising a fermenter, filtering every litre of air entering it, and discarding a contaminated batch are all expensive. If a company found that contamination spoiled only one batch in fifty, on what grounds should it decide how much to spend on prevention?

  5. The exhaust air leaving a bioreactor is filtered before it is released. Whose interest does that filter protect — the manufacturer’s, or the public’s? Should the standard for what may be released depend on the organism being grown inside?

  6. A biotechnologist choosing between primary and secondary data is really choosing between control and cost. Suppose a decision about the safety of a product must be taken quickly and no budget exists for fresh trials. Is it responsible to rely on published data collected under different conditions? What would you require before doing so?

  7. Genetic and clinical data, once published, can be reused by anyone. Does a person who consented to their data being used in one study have a continuing right to say how it is used later? Consider what is lost if the answer is yes, and what is risked if the answer is no.

  8. Ten students’ marks averaged 30.7 when eight of them scored below 19. The average was arithmetically correct and practically misleading. Where in public life do you see averages quoted in ways that hide more than they reveal — incomes, examination results, rainfall, recovery rates?

  9. A researcher who calculates the mean, the median and the mode and then reports whichever best supports her hypothesis has broken no rule of arithmetic. What rule has she broken, and how could a reader of her paper ever detect it?

  10. Increasing the number of replicates lowers the standard error but does not lower the standard deviation. Why might a laboratory be tempted to add replicates rather than to improve its method, and what would be lost by taking that route?

  11. A result reported as “mean ± SE” always looks more precise than the same result reported as “mean ± SD”. Is choosing the one that looks better a form of dishonesty, or simply a choice of emphasis? What should decide which is reported?

  12. If experimental probability approaches theoretical probability only over many trials, how many trials are enough? Consider how you would answer that question for a coin toss, for a germination test on seeds, and for a clinical trial of a vaccine — and why the answer differs.

  13. A quadrat placed where the vegetation looks interesting produces data that no statistical treatment can repair. Why is bias introduced at the moment of collection so much more dangerous than an error made during calculation?

  14. Many industrial fermentations run on molasses, whey and other by-products that another industry would have had to dispose of. Should processes that consume waste be judged by different economic or environmental standards from those that consume purpose-grown crops?

3.15 Objective Questions

  1. In the reaction \(\text{NH}_3 + \text{H}_2\text{O} \rightarrow \text{NH}_4^+ + \text{OH}^-\), water behaves as:
  1. A Brønsted–Lowry base
  2. A Brønsted–Lowry acid
  3. Neither an acid nor a base
  4. A conjugate base of ammonia
  1. A solution at pH 4 compared with a solution at pH 6 is:
  1. 2 times more acidic
  2. 20 times more acidic
  3. 100 times more acidic
  4. 1000 times more acidic
  1. The conjugate base of \(\text{H}_2\text{PO}_4^-\) is:
  1. \(\text{H}_3\text{PO}_4\)
  2. \(\text{HPO}_4^{2-}\)
  3. \(\text{PO}_4^{3-}\)
  4. \(\text{H}_3\text{O}^+\)
  1. A buffer is most effective within a pH range of approximately:
  1. \(\text{p}K_a \pm 0.1\)
  2. \(\text{p}K_a \pm 1\)
  3. \(\text{p}K_a \pm 5\)
  4. Any pH from 0 to 14
  1. A buffer solution is diluted with an equal volume of distilled water. Its pH will:
  1. Fall by one unit
  2. Rise by one unit
  3. Remain almost unchanged, though its buffering capacity falls
  4. Become exactly 7
  1. Which buffer would be the most suitable choice for a procedure that must be carried out at pH 9?
  1. Acetate buffer
  2. Phosphate buffer
  3. Borate buffer
  4. Any of the three, since all buffers work at all pH values
  1. Baffles are fitted to the inner wall of a stirred tank bioreactor in order to:
  1. Increase the working volume of the vessel
  2. Prevent the liquid from simply rotating around the shaft and forming a vortex
  3. Filter the incoming air
  4. Remove metabolic heat from the broth
  1. A sparger delivers air as very fine bubbles rather than as a few large ones because fine bubbles:
  1. Rise more quickly to the surface
  2. Contain more oxygen per bubble
  3. Provide a greater total surface area for oxygen transfer
  4. Reduce the need for a cooling jacket
  1. A fermentation broth becomes thicker and harder to stir as biomass accumulates during the run. This broth is behaving as:
  1. A Newtonian fluid
  2. A non-Newtonian fluid
  3. A gas
  4. A laminar fluid
  1. Laminar flow, rather than turbulent flow, is deliberately used in:
  1. A stirred tank bioreactor
  2. A laminar air flow cabinet
  3. A sparged bioreactor
  4. An autoclave
  1. Which of the following is an example of upstream processing?
  1. Centrifuging the broth to remove cells
  2. Sterilising the medium and raising the inoculum
  3. Purifying the product by chromatography
  4. Packaging the final product
  1. A species is recorded in 6 out of 25 quadrats laid at random. Its percentage frequency is:
  1. 6%
  2. 24%
  3. 25%
  4. 60%
  1. A transect, rather than a set of scattered quadrats, would be the better method for studying:
  1. The total number of weeds in a uniform field
  2. How vegetation changes from the edge of a pond to dry land
  3. The number of colonies on a Petri plate
  4. The mode of a set of colony counts
  1. Which of the following would be classified as primary data?
  1. A sequence downloaded from a public database
  2. A figure quoted from a government health report
  3. Optical density readings recorded by the student during her own experiment
  4. A table reproduced from a published research paper
  1. In the data set 4, 4, 5, 6, 7, 7, 7, 40, the measure of central tendency that would be least affected by the value 40 is the:
  1. Mean
  2. Median
  3. Standard deviation
  4. Standard error
  1. The only measure of central tendency that can be applied to non-numerical data, such as the colour of pea seeds, is the:
  1. Mean
  2. Median
  3. Mode
  4. Standard deviation
  1. In the formula for standard deviation, the squared deviations are divided by \((n-1)\) rather than by \(n\) because:
  1. One observation is always discarded as an outlier
  2. A sample tends to underestimate the spread of the population it came from
  3. It makes the arithmetic simpler
  4. The mean is not known
  1. A set of replicate readings has a standard deviation of 2.4 and a sample size of 9. The standard error is:
  1. 0.27
  2. 0.80
  3. 2.40
  4. 7.20
  1. If the number of replicate measurements is increased fourfold, the standard error will:
  1. Double
  2. Be halved
  3. Be reduced to one-quarter
  4. Remain unchanged
  1. The probability of an event that is certain to occur is:
  1. 0
  2. 0.5
  3. 1
  4. 100

  1. Assertion (A): A buffer must contain a weak acid rather than a strong acid. Reason (R): A strong acid dissociates completely, leaving no undissociated reservoir of protons and no conjugate base in solution.

  2. Assertion (A): Diluting a buffer does not significantly change its pH. Reason (R): Dilution reduces the concentrations of both components equally, so the ratio of salt to acid is unchanged.

  3. Assertion (A): The mean is not always the best measure of central tendency. Reason (R): The mean uses every observation, so it is pulled towards extreme values or outliers.

  4. Assertion (A): Performing more replicates reduces the standard deviation of an experiment. Reason (R): Standard error decreases as the sample size increases.

  5. Assertion (A): Sampling positions in a field study are chosen at random. Reason (R): Random placement gives every part of the study area an equal chance of being examined, so bias is avoided.

  6. Assertion (A): Oxygen must be supplied continuously to an aerobic fermentation. Reason (R): Oxygen is only sparingly soluble in water, so the medium can hold very little of it in reserve.

  7. Assertion (A): A pH meter should be calibrated with standard buffers before use. Reason (R): The meter measures a voltage and converts it to pH using a relationship fixed during calibration.

  8. Assertion (A): Secondary data should be examined carefully before being used in a new study. Reason (R): Secondary data was collected for a different purpose, under conditions the present researcher did not control.

  9. pH is a measure of:

  1. Acidity or basicity
  2. Temperature
  3. Salinity
  4. Pressure
  1. Which of the following is a weak acid used in biological buffers?
  1. HCl
  2. \(\text{H}_2\text{SO}_4\)
  3. \(\text{CH}_3\text{COOH}\)
  4. NaOH
  1. A buffer resists change in:
  1. Volume
  2. pH
  3. Temperature
  4. Pressure
  1. Which instrument is used to measure pH?
  1. Spectrophotometer
  2. pH meter
  3. Thermometer
  4. Barometer
  1. What does a pH of 7 indicate?
  1. Acidic
  2. Basic
  3. Neutral
  4. Alkaline
  1. Which of the following can act as a buffer system?
  1. NaOH and NaCl
  2. HCl and NaCl
  3. \(\text{CH}_3\text{COOH}\) and \(\text{CH}_3\text{COONa}\)
  4. \(\text{HNO}_3\) and \(\text{KNO}_4\)
  1. The unit of buffer capacity is:
  1. mol/pH
  2. pH/mol
  3. L/mol
  4. atm
  1. A solution with pH = 2 is:
  1. Basic
  2. Acidic
  3. Neutral
  4. Amphoteric
  1. Which component is not part of a buffer system?
  1. Weak acid
  2. Strong acid
  3. Conjugate base
  4. Weak base
  1. Which of these values represents the most acidic solution?
  1. pH 1
  2. pH 5
  3. pH 7
  4. pH 10
  1. Which of the following buffer systems is used in human blood?
  1. Acetate buffer
  2. Phosphate buffer
  3. Carbonic acid–bicarbonate buffer
  4. Ammonia buffer
  1. Buffer capacity increases with:
  1. Lower temperature
  2. Lower concentration
  3. Higher concentration
  4. Presence of strong acid
  1. Which of the following is a quantitative method of data collection?
  1. Survey
  2. Interview
  3. Numerical measurements
  4. Open-ended questions
  1. In which range does a phosphate buffer work effectively?
  1. pH 3–5
  2. pH 5–7
  3. pH 6.8–7.4
  4. pH 8–9
  1. What happens to pH when a strong acid is added to a buffer?
  1. Increases sharply
  2. Decreases slightly
  3. Remains unchanged
  4. Increases greatly
  1. Which of the following is a conjugate acid–base pair?
  1. HCl and \(\text{Cl}^-\)
  2. \(\text{H}_2\text{O}\) and \(\text{H}_2\text{SO}_4\)
  3. \(\text{NH}_4^+\) and \(\text{NH}_3\)
  4. NaOH and NaCl
  1. The Henderson–Hasselbalch equation relates pH with:
  1. Acid strength
  2. Temperature
  3. \(\text{p}K_a\) and buffer components
  4. Osmotic pressure
  1. If the \(\text{p}K_a\) of an acid is 4.76, what will be the pH of its buffer when acid and conjugate base are equimolar?
  1. 3.76
  2. 4.00
  3. 4.76
  4. 5.76
  1. In a phosphate buffer system, \(\text{H}_2\text{PO}_4^-\) acts as:
  1. A strong base
  2. A weak acid
  3. A strong acid
  4. A neutral molecule
  1. Buffer capacity is lowest when:
  1. \(\text{pH} = \text{p}K_a\)
  2. [Acid] = [Base]
  3. [Acid] \(\neq\) [Base]
  4. Strong base is added
  1. If the \([\text{A}^-]/[\text{HA}]\) ratio is 10 and \(\text{p}K_a = 5\), what is the pH of the buffer?
  1. 6
  2. 4
  3. 5
  4. 7
  1. Which type of buffer system is used in electrophoresis experiments?
  1. Acetate
  2. Tris-HCl
  3. Phosphate
  4. Carbonate
  1. The accuracy of pH measurement decreases if:
  1. The glass electrode is dry
  2. The buffer is freshly prepared
  3. The temperature is constant
  4. The solution is dilute
  1. Which of the following is secondary data?
  1. Results from your own lab experiment
  2. Data from a government health report
  3. Results from a class test
  4. Bacterial colonies you counted yourself
  1. Primary data is typically:
  1. Already interpreted
  2. Past data
  3. Collected by someone else
  4. Raw and first-hand
  1. Which of the following statements is true about secondary data?
  1. It is always less reliable
  2. It is collected for current research only
  3. It may lack specificity but is easy to obtain
  4. It requires lab-based surveys

  1. Assertion (A): Enzyme activity decreases when there is a slight change in pH. Reason (R): pH changes alter the structure and charge of proteins.

  2. Assertion (A): Buffer solutions resist changes in pH when small amounts of acid are added. Reason (R): The conjugate base present in the buffer reacts with added \(\text{H}^+\) ions.

  3. Assertion (A): When the concentration of acid equals its conjugate base, \(\text{pH} = \text{p}K_a\). Reason (R): The logarithmic term in the Henderson–Hasselbalch equation becomes zero.

  4. Assertion (A): Buffer solutions can resist pH change indefinitely. Reason (R): Buffers have unlimited capacity to neutralise acids and bases.

  5. Assertion (A): Primary data is generally more reliable than secondary data. Reason (R): Primary data is collected directly for a specific research purpose.

  6. Assertion (A): Experimental probability always equals theoretical probability. Reason (R): Experimental probability is based on actual observations.

  7. The term pH stands for __________ of hydrogen.

  8. According to Brønsted–Lowry theory, acids are __________ donors.

  9. A substance that accepts a proton is called a __________.

  10. Buffer solutions resist changes in __________.

  11. Probability is the ratio of favourable outcomes to __________ outcomes.

  12. Data collected for the first time is called __________ data.

  13. In buffer systems, weak acids do not completely __________.

  14. The Henderson–Hasselbalch equation shows that pH depends on the __________ of salt to acid.

  15. Enzymes function best at their __________ pH.

  16. Secondary data is also known as __________ data.

  17. Experimental probability is based on __________ and observation.

  18. Buffers maintain pH due to the presence of a weak acid and its __________.

  19. If \([\text{H}^+]\) increases, the pH value will __________.

  20. When acid and salt concentrations are equal, pH equals __________.

  21. In acetate buffer, \(\text{CH}_3\text{COOH}\) acts as the __________ acid.

  22. The pH range of phosphate buffer is approximately __________.

  23. Primary data collection requires more __________ and manpower.

  24. Probability theory was first applied in games of __________.

  25. Enzyme denaturation occurs due to change in __________ and structure of proteins.

  26. Buffer capacity is maximum when acid equals __________ base.

  27. Secondary data may lack __________ for current research problems.

  28. Primary data is under the direct __________ of the investigator.

  29. Buffers are essential because biological reactions require __________ pH.

  30. Secondary data saves time but may lack __________.

  31. Primary data is more reliable because it is __________ and controlled.

  32. The Henderson–Hasselbalch equation is useful for preparing __________ solutions.

  33. For fermentation processes, controlling __________ ensures maximum yield.

  34. A research study combining survey and observation generates __________ data.

  35. A well-designed buffer must operate within its __________ range.

  36. Data analysis requires both __________ and secondary data sources.

3.16 Check Your Concepts

  1. Why is it more accurate to say that a substance “acts as an acid in this reaction” than to say that it “is an acid”? Support your answer using the behaviour of water in two different reactions.

  2. A student writes that \(\text{H}_2\text{O}\) accepts a proton and becomes \(\text{H}_2\text{O}^+\). Identify the error, write the correct product, and name the conjugate pair involved.

  3. Why does a change of only one pH unit matter so much more in a fermenter than a change of one degree of temperature usually does? Justify your answer using the nature of the pH scale.

  4. Pepsin, salivary amylase and pancreatic amylase all digest food, yet each works at a different pH. Explain why the body maintains different pH values in different parts of the digestive tract instead of one convenient value throughout.

  5. A technician measures the pH of a solution, obtains a reading of 7.2, and reports it. Later the result is found to be wrong. Suggest two things the technician might have failed to do, and explain how each would produce an incorrect reading.

  6. Explain, with equations, what happens inside an acetate buffer when (i) a small amount of HCl is added and (ii) a small amount of NaOH is added.

  7. What would happen if someone attempted to make a “buffer” from a strong acid and its salt? Explain why such a mixture would fail to hold a steady pH.

  8. Two phosphate buffers are prepared, one at 0.05 M and one at 0.5 M, both adjusted to exactly pH 7.0. A meter reads the same value for both. In what way are they not equivalent, and how would the difference reveal itself during an experiment?

  9. Compare and contrast buffering capacity and buffer range. Which one is altered by changing the concentration of the buffer, and which by changing the choice of weak acid?

  10. Explain why the pH of a buffer equals the \(\text{p}K_a\) of its weak acid when the salt and acid concentrations are equal, and why buffering capacity is greatest at this point.

  11. A scientist needs a buffer at pH 8.0 and has two weak acids available, with \(\text{p}K_a\) values of 7.2 and 4.8. She selects the first. Justify her decision in terms of the ratio each would require, and predict what would go wrong if she had chosen the second.

  12. Arrange the following steps of buffer preparation in the correct order, and explain why the adjustment step must come before making up to the final volume: adjust to the exact pH with stirring; calculate the salt-to-acid ratio; dissolve the components in distilled water; select a buffer system with a suitable \(\text{p}K_a\); calibrate the pH meter; make up to the final volume.

  13. Look at Fig. 3.4 in this chapter. Describe what the flat region of the buffer curve represents, what the sharp bend represents, and why the unbuffered curve has no flat region at all.

  14. Blood pH must stay near 7.4, yet respiring tissues pour acids into it continuously. Explain how the bicarbonate buffer system, the lungs and the kidneys share this task between them.

  15. Why would running a gel electrophoresis in distilled water rather than in a buffer produce blurred, unreliable bands? Answer in terms of what determines the movement of a charged molecule.

  16. Turbulent flow is wanted inside a fermenter but avoided inside a laminar air flow cabinet. Explain the reason in each case.

  17. A plant cell suspension culture is stirred at the same speed as a bacterial culture and gives a very poor yield. Suggest what has gone wrong and how the process should be modified.

  18. Compare Newtonian and non-Newtonian fluids under the headings: viscosity, predictability of flow, examples, and relevance to fermenter design.

  19. What is a “dead zone” in a bioreactor? Explain how it forms and what happens to the cells trapped in it.

  20. Distinguish between the strict biochemical meaning of the word “fermentation” and its industrial meaning. Give one example of a process that qualifies under the second meaning but not the first.

  21. Redraw Fig. 3.7 from memory and label the following parts: impeller, baffles, sparger, cooling jacket, air inlet filter, exhaust filter, pH probe, harvest valve. Against each label, write in one line what would go wrong if that part were removed.

  22. Arrange the following in the correct sequence and name the three stages of the process they belong to: purify the product; sterilise the medium; monitor pH and dissolved oxygen; select the production strain; separate cells from the broth; raise the inoculum in stages; transfer the inoculum to the bioreactor.

  23. A sparged bioreactor has no impeller at all. State one advantage and one disadvantage of this design compared with a stirred tank bioreactor.

  24. Why is a fermenter never filled to its full capacity? What is left in the space above the liquid, and why is it needed?

  25. A contaminating organism enters a 5000 L fermenter on the second day of a seven-day run. Trace the likely consequences for the culture and for the product, and explain why sterilisation of the vessel and filtration of the incoming air are treated as non-negotiable.

  26. A student places all ten of her quadrats along the shaded edge of a field because it is easier to reach. Explain why her results cannot be corrected afterwards, however carefully she counts.

  27. Species P has high density but low frequency; species Q has low density but high frequency. Describe the distribution of each species in the field, and explain why reporting only one of the two measures would be misleading.

  28. Compare the quadrat method and the transect method under the headings: shape of the sampling unit, how positions are chosen, the question each answers, and the kind of data obtained.

  29. In the pea-seed quadrat practical, the seeds are scattered by a student who is not doing the counting. Explain the purpose of this arrangement.

  30. A team gathers yield figures from published papers to decide which microbial strains are worth testing, then measures the yield of those strains in its own laboratory. Identify the type of data used at each stage and justify why both were needed.

  31. “Secondary data is quick and cheap, so it should always be preferred.” Give two arguments against this statement and one situation in which secondary data is nevertheless the correct choice.

  32. Ten students score 15, 18, 16, 14, 15, 15, 12, 17, 90 and 95. Calculate the mean and the median. Explain why the two values differ so widely, and state which one you would report and why.

  33. Match the following: | Column A | Column B | |—|—| | a. Mean | i. Unaffected by extreme values, uses only the middle position | | b. Median | ii. The only measure usable for categories such as seed colour | | c. Mode | iii. Measures the spread of individual observations | | d. Standard deviation | iv. Uses every observation and is pulled by outliers | | e. Standard error | v. Measures how precisely the sample mean estimates the true value |

  34. Why are the deviations from the mean squared before they are added in the standard deviation formula? What would happen if they were simply added as they are?

  35. Two laboratories report the same mean enzyme activity. Laboratory A reports a standard deviation of 0.2 units/mL; laboratory B reports 4.5 units/mL. What does this tell you about the two sets of results, and which laboratory’s mean would you trust more?

  36. Explain why increasing the number of replicates reduces the standard error but not the standard deviation. What does this imply about a researcher who repeats a poorly controlled assay many times and then reports a very small standard error?

  37. Why is a mean reported without any measure of variability considered an incomplete result? What do the vertical error bars on a published graph represent?

  38. A student tosses a coin ten times and obtains seven heads. He concludes that the coin is biased. Evaluate his conclusion, and describe what he should do before making such a claim.

  39. Explain why experimental probability approaches theoretical probability as the number of trials increases. Relate your answer to why conclusions drawn from very few observations are unreliable.

  40. A large and persistent gap between experimental and theoretical probability is treated differently from a small one. Explain why, and give two possible systematic causes a scientist should investigate.

  41. Define pH.

  42. What is a buffer solution?

  43. Who proposed the Brønsted–Lowry theory?

  44. What is primary data?

  45. What is the pH of pure water?

  46. Why is pH important for enzyme activity?

  47. What happens when an acid is added to a buffer?

  48. Why are buffers necessary in biological systems?

  49. What is meant by a conjugate acid–base pair?

  50. Differentiate between primary and secondary data (one point).

  51. Identify the acid in the reaction: \(\text{HCl} + \text{H}_2\text{O} \rightarrow \text{H}_3\text{O}^+ + \text{Cl}^-\).

  52. What happens to pH when \([\text{H}^+]\) concentration increases?

  53. State the pH range of acetate buffer.

  54. Give one use of phosphate buffer in biotechnology.

  55. Why does a small change in pH affect biological systems significantly?

  56. Why is maintaining sterility important in culture media?

  57. How does buffer capacity depend on concentration?

  58. Why may secondary data be unreliable?

  59. Why is maintaining optimum pH essential in fermentation?

  60. Why is primary data considered more reliable than secondary data?

  61. Why are buffers important in laboratory experiments?

  62. Why must nutrient media be free from impurities?

  63. Why is buffering capacity limited?

  64. Suggest a suitable buffer for enzyme activity at acidic pH.

  65. Design one method to collect primary data.

  66. Propose a reason for using reusable substrates in biotechnology.

  67. Suggest a condition required for maximum microbial growth.

  68. Propose one application of buffers in medicine.

  69. What is pH and why is it important in enzyme activity?

  70. What is a buffer and what is its role in biochemical reactions?

  71. What is meant by theoretical and experimental probability?

  72. A lab technician notices that an enzyme stops working when the pH changes slightly from its optimum. Explain why enzymes are sensitive to pH changes.

  73. A student prepares a solution with very low pH. Predict its effect on protein structure and justify your answer.

  74. Why should secondary data be used carefully in research? Give two reasons.

  75. A biologist needs a pH of 7.5–9.2 in an experiment. Which buffer should they use?

  76. During an experiment, the pH of a solution is found to be 3. What does this indicate about the solution’s nature and its hydrogen ion concentration?

  77. You are calibrating a pH meter before an enzyme experiment. Why is it important to use a buffer during calibration?

  78. During fermentation, the pH drops from 6 to 4. Explain how this change affects microbial metabolism and product yield.

  79. In a biotechnology lab, a buffer is added during an enzyme reaction. Explain how the buffer maintains pH stability.

  80. A study uses both surveys and published reports. Identify the types of data used and justify your answer.

  81. A buffer solution fails to resist pH change after repeated addition of acid. Explain the reason, based on buffering capacity.

  82. A slight change in pH in a bioreactor affects product yield. Analyse the role of pH in this process.

  83. A researcher prefers primary data over secondary data despite the higher cost. Justify this choice.

  84. Why are buffers essential in DNA extraction and protein purification experiments? Give two reasons.

  85. Why is maintaining optimum temperature and pH necessary for microbial growth? Give two reasons.

  86. A buffer is used in a pharmaceutical preparation to maintain drug stability. Explain how pH affects drug effectiveness.

  87. Students from different backgrounds such as Rohan, Zainab, Smith and Su Lin are learning about primary and secondary data. They are asked to compare both types of data in terms of reliability, cost and usefulness. Explain the differences between primary and secondary data with suitable examples.

  88. A research group consisting of Tanaka, Priya and Fatou Diop conducts repeated experiments to study enzyme activity. They notice that the actual results differ slightly from the expected theoretical outcomes. Explain the difference between theoretical probability and experimental probability with reference to this case.

  89. In a biotechnology lab, researchers like Pierre, Kavya and Omar Sheikh use secondary data from previous studies. Later, they realise that the data does not match their current research conditions. Analyse the limitations of secondary data and explain why it must be used carefully.

  90. A research team including Elena Petrova, Rahman ul Haq and Neha Guleria is designing a study to collect data on enzyme efficiency. Propose a method to collect primary data and explain why it is more reliable than secondary data.

3.17 Case-Based Questions

  1. A technician is asked to prepare 500 mL of buffer at pH 7.2 for a DNA extraction. The stock chemicals available are acetic acid with sodium acetate (\(\text{p}K_a\) 4.76), a phosphate pair (\(\text{p}K_a\) 7.2), and boric acid with borate salts (\(\text{p}K_a\) 9.2). She selects the phosphate pair, dissolves the two components in distilled water, checks the pH on a meter she has just switched on, finds it reads 7.2, and makes the solution up to 500 mL. The extraction later fails, and the pH of the used buffer is found to have fallen to 5.9.
  1. Justify her choice of the phosphate pair over the other two systems.
  2. Identify one step she omitted before taking her pH reading, and explain how this could produce a false value.
  3. Suggest a second reason, unrelated to the meter, why the pH of the buffer could have fallen so far during the extraction.
  4. State what she should change when preparing the buffer for the next attempt, and why.
  1. A small unit produces an organic acid by fermentation in a 500 L stirred tank bioreactor. The medium is inoculated at pH 6.8, which is the optimum for the production strain. Over the first eighteen hours the culture grows well, but the pH probe records a steady fall to 4.9, after which growth slows sharply and the product yield for the batch is less than half of what was expected.
  1. Explain why the pH fell during the run, even though the medium was correctly adjusted at the start.
  2. Calculate how many times the hydrogen ion concentration increased between pH 6.8 and pH 4.9, to the nearest order of magnitude.
  3. Explain, in terms of enzyme structure, why growth slowed as the pH fell.
  4. Suggest two measures the unit could take to prevent this happening in the next batch.
  1. An engineering student is shown two bioreactors in a pilot plant. Vessel A is a stirred tank with a motor, a shaft carrying two impellers, baffles on the inner wall and a sparger at the base. Vessel B has no moving parts inside at all; air is introduced at the bottom and the rising bubbles alone mix the contents. The plant manager explains that Vessel A is used for a bacterial fermentation and Vessel B for an animal cell culture.
  1. Name the type of bioreactor represented by Vessel B.
  2. Explain why the animal cell culture has been assigned to Vessel B rather than Vessel A.
  3. State the function of the baffles in Vessel A, and predict what would happen to mixing if they were removed.
  4. Both vessels have a filter on the incoming air line and another on the exhaust. Explain the purpose of each filter.
  1. Three groups of students carry out the pea-seed quadrat practical. Group I scatters the seeds and then places the quadrat over the densest patch, because “that is where the seeds are”. Group II lays the quadrat at positions decided by a pair of random numbers. Group III lays five quadrats in a straight line running from the edge of the table to the centre. All three then count the yellow and green seeds in each block and calculate the frequency of each colour.
  1. Identify which group’s data can be used to estimate the frequency of each seed colour on the whole table, and justify your answer.
  2. Explain what is wrong with Group I’s method, and why careful counting cannot repair the problem.
  3. Group III’s arrangement resembles a sampling method described in this chapter. Name it, and state the situation in which it would be the appropriate choice.
  4. Suggest one change to the practical that would let the class see how much the answer varies from one sample to the next.
  1. A team measures the activity of an enzyme in eight replicate tubes and records, in units/mL: 41, 43, 42, 44, 41, 43, 42 and 89. They calculate a mean of 48.1 units/mL and report it in their project file. When their teacher asks about the spread, they calculate a large standard deviation. One student suggests running the assay twenty more times so that the standard error becomes small and the result “looks better”.
  1. Identify the outlier in the data set and calculate the median.
  2. Explain why the mean of 48.1 units/mL is a poor description of a typical tube.
  3. Evaluate the student’s suggestion. State what more replicates would and would not achieve.
  4. Describe what the team should do about the eighth tube, and what they must record if they exclude it.
  1. A research student is investigating whether a newly isolated bacterial strain produces more amylase than strains already described. She downloads enzyme activity figures for twelve published strains from journal articles and a public sequence database, then grows her own isolate in her laboratory and measures its activity in triplicate. Her supervisor warns her against comparing the two sets of numbers directly.
  1. Classify each of the two sets of figures as primary or secondary data, giving one reason in each case.
  2. Explain the supervisor’s warning. Give two specific reasons why the published figures might not be directly comparable with her own.
  3. Describe how she should modify her study so that a fair comparison becomes possible.
  4. State one obligation she has towards the authors of the published figures when she writes up her work.
  1. A hospital laboratory holds records of genetic test results from several thousand patients, collected over ten years with each patient’s consent for the purpose of diagnosis. A university group now asks for access to the records, anonymised, in order to study how common a particular variant is in the region. The hospital’s ethics committee is asked to decide.
  1. State one strong argument in favour of granting access.
  2. State one strong argument against granting access, referring to the special nature of genetic data.
  3. Explain why removing names from the records may not fully protect the individuals concerned.
  4. Suggest one condition the committee might attach if it decides to permit the study.
  1. A class simulates random sampling by tossing coins. Working alone, one student tosses her coin 20 times and records 14 heads, and concludes that her coin is weighted. The class then pools every student’s tosses, giving a combined total of 1000 tosses with 508 heads.
  1. Calculate the experimental probability of a head from the individual student’s data and from the pooled class data.
  2. State the theoretical probability of obtaining a head with a fair coin.
  3. Evaluate the student’s conclusion in the light of the pooled result.
  4. Explain what this exercise demonstrates about drawing conclusions from a small number of observations in a laboratory experiment.
  1. A biotechnology student is preparing notes on pH and buffers for lab work.
  1. Define pH and write its mathematical expression.
  2. Define a buffer solution.
  3. State any two properties of buffers.
  1. A research intern is studying the types of data used in experiments.
  1. Define primary data and secondary data.
  2. Give one example of each.
  3. State one difference between them.
  1. During an enzyme experiment, slight variation in pH caused loss of activity.
  1. Explain why enzymes are pH-sensitive.
  2. Describe how pH affects protein structure.
  3. Give one example of an enzyme functioning at a specific pH.
  1. A buffer is used in a biochemical reaction to maintain stability.
  1. Explain how a buffer resists changes in pH when an acid is added.
  2. Explain what happens when a base is added.
  3. Why are buffers essential in biological systems?
  1. A laboratory prepares a buffer where \(\text{p}K_a = 5\) and \([\text{Salt}]/[\text{Acid}] = 10\).
  1. Calculate the pH using the Henderson–Hasselbalch equation.
  2. Identify whether the solution is acidic or basic.
  3. State one application of such a buffer.
  1. A buffer solution fails to maintain pH after repeated addition of acid.
  1. Explain the concept of buffering capacity.
  2. Analyse why the buffer fails beyond a certain limit.
  3. Suggest how buffer capacity can be increased.
  1. A biotech company must choose between primary and secondary data for research.
  1. Compare primary and secondary data (any two points).
  2. Evaluate why primary data is more reliable.
  3. Justify a situation where secondary data may still be preferred.
  1. A scientist selects a buffer with \(\text{p}K_a\) close to the required pH in an experiment.
  1. Explain the Henderson–Hasselbalch principle involved.
  2. Justify why this choice is ideal.
  3. Evaluate what would happen if the \(\text{p}K_a\) were far from the required pH.
  1. You are designing a biotechnology experiment involving enzyme reactions.
  1. Propose suitable conditions of pH and temperature.
  2. Suggest a buffer system and justify your choice.
  3. Explain how you will ensure the stability of the reaction.
  1. A pharmaceutical company is developing a drug that must remain stable in the human body.
  1. Explain the role of buffers in maintaining drug stability.
  2. Propose one buffer system suitable for physiological pH.
  3. Design a brief strategy to ensure drug effectiveness.
  1. A fermentation unit producing antibiotics requires an optimum pH of 3. During monitoring, the hydrogen ion concentration is found to be \(1 \times 10^{-2}\) M.
  1. Calculate the pH of the medium.
  2. Predict whether microbial yield will increase or decrease. Justify your answer on the basis of pH.
  1. An enzyme used in food processing works best at pH 5. Due to contamination, the pH drops to 3.
  1. Calculate how many times the acidity has increased.
  2. Explain how this affects enzyme structure and activity.
  1. A pharmaceutical lab is preparing a buffer for a drug that must remain stable at pH 6. The acid used has \(\text{p}K_a = 6\).
  1. Calculate the ratio of the concentration of salt to that of acid required.
  2. Explain why this ratio gives maximum buffering capacity.
  1. In a bioreactor, a buffer solution has \(\text{p}K_a = 5\). The technician sets the ratio of the concentration of salt to the concentration of acid at 0.1.
  1. Calculate the pH of the solution.
  2. Predict whether the medium is suitable for organisms requiring slightly acidic conditions.
  1. During DNA extraction, the buffer pH must be maintained at 7. If \(\text{p}K_a = 6\):
  1. Calculate the required ratio of salt to acid.
  2. Explain why deviation from this pH can damage DNA.
  1. In a lab test, a coin is used to simulate random sampling. Out of 200 trials, heads appear 120 times.
  1. Calculate the experimental probability.
  2. Compare it with the theoretical probability and comment on the variation.
  1. A biotech company is choosing between two data collection methods. The primary data cost is ₹80,000 while the secondary data cost is ₹20,000.
  1. Calculate how many times more expensive primary data is.
  2. Justify why the company may still choose primary data.
  1. In a clinical dataset of 500 samples, 350 are primary data and 150 are secondary data.
  1. Calculate the percentage of primary data.
  2. Evaluate which dataset contributes more to reliability, and why.
  1. Two buffers are used in a fermentation setup. The concentration of buffer X is 0.2 M, while that of buffer Y is 1.0 M.
  1. Which buffer will resist pH change better?
  2. Calculate how many times greater its buffering capacity is, and justify your answer.
  1. During an international biotechnology workshop, participants such as Anil, Aisha, Kim and Fernandez observe that enzyme activity changes drastically with slight variation in pH. The instructor explains that biological systems function only within a narrow pH range, and that even minor changes can disrupt biochemical processes.
  1. Define pH and give its mathematical expression.
  2. Why are enzymes sensitive to pH changes?
  3. What happens to protein structure when pH changes significantly?
  4. Give one example of an enzyme functioning at a specific pH.
  1. In a pharmaceutical laboratory, scientists like Dr Emily, Dr Patel and Dr Hassan are developing a drug that must remain stable at a pH of 7. Due to experimental error, the pH drops to 5, affecting the drug’s effectiveness. The team decides to use a buffer system to maintain stability.
  1. Which type of buffer system can be used near pH 7?
  2. Explain how the buffer maintains pH stability.
  3. What will happen if pH is not maintained?
  4. Why are buffers important in pharmaceutical preparations?
  1. In a biotechnology company, a fermentation process shows inconsistent results. Scientists such as Naledi Khumalo, Arjun Singh and Mohammed Farouq find that the pH fluctuates frequently despite adding a buffer. This leads to reduced product yield and enzyme inactivation.
  1. What is buffering capacity?
  2. Why did the buffer fail to maintain pH?
  3. How can buffering capacity be increased?
  4. Analyse the effect of pH fluctuation on enzyme activity.
  1. You are part of a biotechnology team including Dr Ananya, Dr Sara and Dr Lucas. The team is designing a fermentation experiment where maintaining constant pH is critical for maximum yield. You are asked to propose a suitable experimental setup.
  1. Suggest a suitable buffer system and justify your choice.
  2. Mention two conditions required for optimal microbial growth.
  3. How will you maintain pH throughout the process?
  4. Design one method to monitor pH changes during the experiment.

3.18 Practice With Data

  1. The hydrogen ion concentration of a culture medium is \(2.5 \times 10^{-6}\) mol L\(^{-1}\). (a) Calculate its pH to two decimal places. (b) State whether the medium is acidic, neutral or alkaline. (c) The medium must be adjusted to pH 7.0. State whether acid or alkali must be added, and by what factor the hydrogen ion concentration must change.

  2. During a fermentation the pH is monitored and found to fall from 7.0 to 4.0 over twelve hours. (a) Calculate how many times the hydrogen ion concentration has increased. (b) The production strain has an optimum pH of 6.5 and tolerates a range of \(\pm 0.5\) units. Calculate the hydrogen ion concentration at the two limits of its tolerance. (c) State at approximately what point in the fall the culture would have passed outside its tolerated range.

  3. A buffer is prepared from a weak acid of \(\text{p}K_a = 6.10\). (a) Calculate the pH obtained when the salt-to-acid ratio is 1 : 1. (b) Calculate the pH when the ratio is 4 : 1. (c) Calculate the ratio required to obtain a buffer of pH 5.10. (d) State whether the buffer in part (c) lies within the effective working range of this acid, and justify your answer.

  4. A laboratory must prepare a buffer at pH 7.40. Three weak acids are available, with \(\text{p}K_a\) values of 4.76, 7.20 and 9.20. (a) Calculate the salt-to-acid ratio each acid would require to give pH 7.40. (b) State which acid should be selected, giving a reason based on your calculated ratios. (c) Explain what would happen to the buffering capacity if the acid with \(\text{p}K_a\) 4.76 were used instead.

  5. Two acetate buffers of identical pH are prepared, one at 0.05 M and the other at 0.40 M. (a) Calculate how many times greater the buffering capacity of the second buffer is. (b) Predict what the pH meter would read for each buffer immediately after preparation. (c) Explain why identical meter readings do not mean the two buffers will behave identically during an experiment.

  6. A student lays 25 quadrats, each of area \(0.25\ \text{m}^2\), across a grassland plot. Species A occurs in 20 quadrats with a total count of 150 individuals; species B occurs in 5 quadrats with a total count of 145 individuals. (a) Calculate the total area sampled. (b) Calculate the density of each species per square metre. (c) Calculate the percentage frequency of each species. (d) Describe how the distribution of the two species differs, using both sets of figures.

  7. In the pea-seed practical, a quadrat of 30 cm × 30 cm is divided into 25 blocks of 6 cm × 6 cm. A group counts 180 yellow seeds and 120 green seeds within the quadrat, with yellow seeds present in 22 blocks and green seeds in 15 blocks. (a) Calculate the frequency of each seed colour as a percentage of the total seeds counted. (b) Calculate the percentage of blocks in which each colour occurs. (c) Calculate the average number of seeds of each colour per block. (d) Explain why the answers to (a) and (b) are different quantities and should not be confused.

  8. The number of bacterial colonies counted on nine replicate plates is: 62, 58, 65, 60, 59, 63, 61, 57 and 64. (a) Calculate the mean. (b) Calculate the median. (c) State whether the data set has a mode, and justify your answer. (d) Comment on which measure of central tendency best represents this data set and why.

  9. A tenth plate in the experiment above is later included, with a count of 148 colonies. (a) Recalculate the mean for all ten plates. (b) Recalculate the median. (c) State which of the two measures has changed more, and explain why. (d) The tenth plate is found to have been contaminated. State what the analyst should do, and what must be recorded.

  10. The DNA concentration of five samples, in µg/µL, is: 6.0, 6.4, 6.2, 6.6 and 5.8. (a) Calculate the mean. (b) Calculate the sum of the squared deviations from the mean. (c) Calculate the standard deviation. (d) Calculate the standard error. (e) State the unit attached to each of your answers to (a), (c) and (d).

  11. Two teams measure the same enzyme preparation. Team X records, in units/mL: 30, 31, 29, 30, 30. Team Y records: 24, 36, 28, 34, 28. (a) Calculate the mean for each team. (b) Calculate the standard deviation for each team. (c) Calculate the standard error for each team. (d) State which team’s mean is the more reliable estimate, and justify your answer using your calculated values.

  12. An assay has a standard deviation of 3.6 units/mL. (a) Calculate the standard error obtained with 4 replicates, with 16 replicates and with 36 replicates. (b) Plot or describe the trend in your three answers. (c) Calculate how many replicates would be needed to bring the standard error below 0.5 units/mL. (d) State what happens to the standard deviation as the number of replicates is increased, and explain why.

  13. A stirred tank bioreactor of total capacity 5000 L is operated at 75% working volume. The fermentation gives a product concentration of 8.5 g L\(^{-1}\) of broth. (a) Calculate the working volume. (b) Calculate the total mass of product formed in one batch. (c) Downstream processing recovers 78% of the product. Calculate the mass finally obtained. (d) If the plant runs 40 such batches in a year, calculate the annual output in kilograms.

  14. During a fermentation, the dissolved oxygen concentration in the broth is recorded as follows.

Time (hours) 0 4 8 12 16 20
Dissolved oxygen (mg L\(^{-1}\)) 7.8 6.9 4.2 1.1 1.0 1.0
  1. Describe the trend shown by the data.
  2. Suggest what is happening in the vessel between 8 and 12 hours.
  3. Explain why the value does not fall to zero after 12 hours.
  4. Suggest two adjustments to the operating conditions that would raise the dissolved oxygen level during the middle of the run.

  1. A student rolls a die 60 times and records the number of sixes as 14. (a) Calculate the experimental probability of obtaining a six. (b) State the theoretical probability. (c) Calculate the difference between the two values. (d) The whole class pools its results, giving 1200 rolls with 207 sixes. Calculate the pooled experimental probability and comment on how the difference from the theoretical value has changed.

  2. In a screening experiment, 480 bacterial colonies are tested and 36 are found to carry the desired recombinant plasmid. (a) Calculate the experimental probability that a randomly picked colony is a recombinant. (b) Express this as a percentage. (c) Estimate how many colonies would need to be screened to obtain 15 recombinants, assuming the same probability holds. (d) State one reason why the actual number needed might differ from your estimate.