Reversible binding between biomolecules—for example, between a cell-surface receptor such as the insulin receptor and its corresponding natural ligand such as insulin—is central to innumerable physiological transactions. Binding of the dye HABA to egg-white avidin is a simple, reliable, and colorful laboratory model for introducing beginning biology students to the principles underlying reversible binding. They can probe the reaction quantitatively with a spectrophotometer, and model it mathematically using only high-school algebra and a spreadsheet program such as Microsoft Excel.

## Introduction

For six years, we offered an alternative freshman biology lab called Mathematics in Life Sciences (MLS) that emphasized investigation while integrating simple mathematics more intimately into the curriculum. One of its modules has previously been described in these pages (Smith et al., 2015). Here I describe another module, in which students explored a reversible molecular binding reaction.

Reversible, non-covalent binding interactions are a ubiquitous component of physiological processes, both inside and outside cells. A familiar example is the interaction between a cellular receptor and its natural *ligand* (the hormone or other biomolecule that naturally binds that receptor). For instance, insulin is the natural ligand for the insulin receptor on muscle, liver, and adipose cells. When insulin engages the receptor, the cell is stimulated to import glucose from the surrounding fluid, thus lowering blood glucose level.

Physiological systems like the insulin response are typically complex and difficult to study directly in detail. Progress often comes from model systems that preserve essential features of the complex phenomena of interest, but that are much easier to study experimentally. In this investigation students explored such a model: the reversible binding of the artificial ligand HABA to the protein avidin. In the module's first hour, they followed detailed instructions to quantify the reaction. These measurements were followed by guided student investigation, which spanned five hours of class time spread out over several weeks and interspersed with other modules, but which can be shortened as needed. In the course of this investigation, students came to understand the reaction in increasing depth, ultimately modeling it mathematically using only high-school algebra. Detailed instructor's guides and teaching materials are available on a companion website https://mls.missouri.edu/equilibrium-binding/.

## Avidin, Biotin, and HABA

Avidin is a biotin-binding protein in the oviducts of birds, reptiles, and amphibians that is deposited in the whites of their eggs. Its adaptive function is suspected to be antimicrobial: by sequestering biotin, it slows the growth of microbes that require that vitamin (White & Whitehead, 1987). Because the binding interaction is non-covalent, it's reversible; avidin and biotin can *associate* to form the avidin-biotin complex (the *forward* reaction), and at the same time the complex can *dissociate* to release the two individual molecules again (the *reverse* reaction). Such systems come to a natural *equilibrium* state, in which the forward and reverse reactions exactly balance, so that the net concentrations of the reactants remain constant even though individual molecules continue to associate and dissociate. In the case of the avidin-biotin interaction, dissociation is so slow (half-life 6–7 weeks; Green, 1975) and association so fast that actually measuring them is technically challenging. Super-slow dissociation plus super-fast association = super-super-high *affinity* (binding strength).

Avidin consists of four identical subunits. Each subunit (molecular mass 16,500 Da) is a polypeptide of 128 amino acids with carbohydrate attached to one of its amino acid side chains. The subunits bind biotin independently of one another; as far as binding equilibrium is concerned, therefore, each tetramer acts the same as would four separate monomers. Consequently, avidin concentration is expressed in terms of individual monomer subunits. Biotin bound to avidin is buried in a deep pocket in the protein's three-dimensional structure (Figure 1).

HABA is a dye that binds reversibly to avidin with weak affinity, inhabiting the same binding pocket as does biotin (Green, 1965; Livnah et al., 1993a). Avidin-HABA binding results in a dramatic change in HABA's spectral properties that's easy to measure with a spectrophotometer. Whereas HABA dissolved in aqueous solution absorbs light maximally at a wavelength λ_{max} = 348 nm and has a pale yellow color, the avidin-HABA complex absorbs light maximally at λ_{max} = 500 nm and has a bright red color. The avidin-HABA system is an attractive experimental model for introducing students to binding interactions (e.g., Ninfa et al., 2010).

## Performance and Results

Before the first session (100 min.), students read a handout covering the information in the first two sections of this article. This prepared them to participate in discussion during the session.

Each student was supplied with four 1.5-mL microtubes containing exactly 1.2 mL of HABA at a particular concentration in buffer (see Appendix); the HABA concentrations for different students ranged from 1.1 to 100 µM in logarithmic progression. Each student was also supplied with a 500-µL microtube containing 100 µL of an avidin solution in water ( Appendix; the same avidin concentration for all students); a 500-µL microtube with 100 µL water; a 500-µL microtube with 100 µL biotin at the same concentration as avidin in the avidin microtube ( Appendix); four disposable 1.5-mL polystyrene spectrophotometer cuvettes (e.g., Fisher 14-955-127); pipetters and tips for pipetting 40- and 1000-µL volumes; access to a vortex mixer and to a beaker for discarding used disposable labware. All supplies were at room temperature. A single Jenway model 6705 spectrophotometer adjusted to 500 nm served the entire class; any comparable instrument can be substituted.

Working as accurately as possible, each student pipetted exactly 40 µL water into two of his/her 1.5-mL microtubes (the reference tubes), and exactly 40 µL avidin into the other two 1.5-mL microtubes (the sample tubes). The microtubes were vortexed (equilibrium is attained almost instantly), and exactly 1 mL from each was pipetted into a disposable cuvette (two reference cuvettes, two sample cuvettes). The color in the reference and sample cuvettes were compared visually; the latter should be red compared to the former, the contrast being more and more pronounced the higher the HABA concentration. To quantify the color change, the student brought the four cuvettes to the spectrophotometer, rapidly zeroing with one of the reference cuvettes, reading and recording A_{500} (absorbance at 500 nm) with one of the sample cuvettes, zeroing with the other reference cuvette, and reading and recording A_{500} with the other sample cuvette. One reference and one sample cuvette were discarded; the other two cuvettes were brought back to the workbench, where the student pipetted 40 µL from the biotin microtube into each (a 24 percent molar excess of biotin over the avidin in the sample cuvette) and stirred the contents with the pipette tip; students were forewarned to note any visible color change (the red color should disappear in all sample cuvettes, regardless of HABA concentration) in anticipation of discussion later in the session. The two cuvettes (now containing biotin) were brought back to the spectrophotometer, zeroing with the reference cuvette and reading and recording A_{500} with the sample cuvette as before. Each student thus generated three A_{500} measurements for a single HABA concentration, the class as a whole generating data for the entire range of concentrations.

The foregoing steps consumed ~70 minutes of lab time; the time could be reduced by skipping the duplicate reads (i.e., only one reference and one sample), and by the instructor doing the post-biotin spectrophotometric reads after the lab. The module's investigative phase began immediately in the remaining ~30 minutes, when students were asked to explain why the red color disappeared after adding biotin, even when the amount of biotin added was much less than the amount of HABA already in the cuvette. (The reason is that biotin binds avidin far more strongly than does HABA, and thus completely displaces HABA as long as it's in molar excess over avidin.) The class as a whole was always able to arrive at a cogent answer, setting the tone for the entire investigation.

If we think of avidin as a model for a cellular receptor (e.g., a hormone receptor on the surface of a cell), HABA as a model for that receptor's natural ligand (e.g., the hormone that binds to and activates the receptor), and the color change as a model for the physiological effect triggered by receptor engagement; then biotin could be thought of as a model for a drug that antagonizes the natural ligand, thus blocking its physiological effect. The antagonist drug occupies the receptor's binding site, preventing the natural ligand from binding. The antagonist drug and the natural ligand thus compete for the same binding site on the receptor. Biotin thought of in this way is an extraordinarily potent antagonist drug because of its super-high affinity for the target receptor. Competitive inhibitors are a major category of drugs, though there are many other categories. (Of course, we're distorting reality to make our analogy here, since the biotin “antagonist” is actually avidin's presumed natural ligand.)

After the lab, the instructor entered the students’ spectrophotometric data (duplicate pre-biotin readings, a single post-biotin reading for each student) into a spreadsheet such as Microsoft Excel; the spreadsheet also included the HABA and avidin concentrations during the pre-biotin readings (these are the total input concentrations of the respective molecules, regardless of whether or not they are bound to a partner). The spreadsheet document was distributed to the students so they had access to the entire class's results. Students were assigned the homework task of graphing the data-points as in Figure 2 (open circles, closed triangles, and open squares only; other features were created later), using a logarithmic scale for the *x* axis to match the logarithmic distribution of HABA concentrations. This exercise required prior experience with the spreadsheet program, which in the MLS course occurred in computer labs in previous modules. An exemplar student worksheet is provided on the companion website.

Reversible binding studies like that in Figure 2 are ubiquitous in biochemistry, physiology, and pharmacology. In each case, the concentration of a ligand analogous to HABA is varied, and some binding measurement analogous to A_{500} is made at each ligand concentration.

## Mathematical Model

The 100-minute lab described in the previous section can stand on its own, but in the MLS course it was followed by an investigative phase (~300 min. of classroom time) devoted to understanding the reaction and modeling it mathematically. The instructor's guides and teaching materials on the companion website detail the steps by which this was achieved.

Letting A stand for free avidin, B for free HABA, and AB for the avidin-HABA complex (“free” avidin and HABA are molecules that aren't part of AB complexes), the reversible binding reaction can be diagrammed as in Figure 3.

### Graphing and Defining Variables

To begin their investigation, students were asked to sketch graphically how the concentration of avidin-HABA complex will change with time in the course of one of the binding reactions, starting at its initial value of 0. With some guidance from the instructor, they were able to come up with something qualitatively like the ascending curve in Figure 4 (the variable definitions in that graph emerge only later). Why does the curve flatten out with time, asymptotically approaching an upper boundary? Part of the reason is that as more and more avidin-HABA complex accumulates, the rate at which complexes dissociate increases in concert, slowing the overall rate at which the complex accumulates. Concomitantly, accumulation of avidin-HABA complex also reduces the concentrations of free avidin and free HABA, slowing the rate at which they can associate to make more complex. Eventually, dissociation and association reactions come to balance each other; that's the equilibrium state. These effects are a qualitative expression of the mass action law (next subsection).

Once a satisfactory graph of complex concentration had been developed, students were asked to sketch how the free avidin concentration evolves with time on the same graph. The purpose of this task was to reveal a key stoichiometric constraint: that at any point in time, the amount by which the complex concentration has increased from its initial value of 0 must equal the amount by which the free avidin and free HABA concentrations have decreased from their non-zero initial values. The descending avidin and HABA concentration curves are thus the same as the ascending complex concentration curve flipped upside down, as depicted in Figure 4.

Students found it much easier to define the algebraic variables of their mathematical model, and to express the stoichiometric constraint in the previous paragraph, with the curves sketched in Figure 4 in hand than without them. Their notation systems were equivalent to the one in this paragraph and in Figure 4, though the particular choice of algebraic symbols differed. We let *a*, *b*, and *c* stand for the concentrations of free A, free B, and AB complex, respectively, at any moment in time (all concentrations in the lab are expressed in µM units). We call the initial concentrations of free A and free B (before any binding reactions have occurred) *a*_{0} and *b*_{0}, which equal the known total input concentrations of avidin and HABA, respectively; the initial concentration of the avidin-HABA complex is 0. The momentary concentrations *a*, *b*, and *c* change with time from their initial values at time 0, ultimately attaining their final equilibrium values, which we'll call *a*_{eq}, *b*_{eq}, and *c*_{eq}, respectively. The stoichiometric constraint in the previous paragraph implies that at any moment during the reaction, *a*_{0} − *a* = *b*_{0} − *b* = *c*. The time course for *c* thus completely determines the time course for *a* and *b*: *a* = *a*_{0} − *c* and *b* = *b*_{0} − *c*. As students develop their own systems of algebraic symbols, the instructor should be ready to help them avoid the confusions that arise from using the same symbol for two different quantities (e.g. using *b* for both *b*_{0} and *b*_{eq}).

### Mass Action Law

Applied to the bimolecular forward (association) reaction, mass action implies that the forward reaction rate at any moment in time is proportional to both *a* and *b* at that moment. Letting *k*_{1} stand for the proportionality constant, the forward reaction rate = *k*_{1}*ab*; *k*_{1} is called the *association rate constant*. Applied to the unimolecular reverse (dissociation) reaction, mass action implies that the reverse reaction rate at any moment in time is proportional to *c* at that moment. Letting *k*_{–1} stand for the proportionality constant, the reverse reaction rate = *k*_{–1}*c*; *k*_{–1} is called the *dissociation rate constant*. The two rate constants are independent of each other; neither is specified by the mass action law. From the two rate equations, the entire time course of a reaction (as graphed in Figure 4) can be modeled with the aid of calculus. In the MLS module, however, we considered only the final equilibrium state, which can be modeled using high-school algebra.

Equilibrium is attained when the concentrations *a*, *b*, and *c* have changed from their initial values (*a*_{0}, *b*_{0}, and 0) to final equilibrium values (*a*_{eq}, *b*_{eq}, and *c*_{eq}) that bring the forward and reverse reaction rates into balance: *k*_{1}*a*_{eq}*b*_{eq} = *k*_{–1}*c*_{eq}. Rearranging, *a*_{eq}*b*_{eq}/*c*_{eq} = *k*_{–1}/*k*_{1} = *K*_{D}, where the constant *K*_{D} defined in the second part of the equation is called the *dissociation equilibrium constant*, and has units of concentration (here µM). This equation shows that the equilibrium state doesn't depend on the magnitudes of the *k*_{–1} and *k*_{1} rate constants individually, but only on their ratio *K*_{D}.

The stoichiometric constraints *a*_{0} − *a*_{eq} = *b*_{0} − *b*_{eq} = *c*_{eq} (Figure 4) allow the unknown variables *a*_{eq} and *b*_{eq} to be eliminated from the equation above, leaving a single unknown variable, *c*_{eq}, on the left-hand side:

When students were asked in class to solve Equation 1 for *c*_{eq}, they found that it is quadratic with solution:

(The expression inside the radical is non-negative; the quadratic solution with a plus sign preceding the radical is physically impossible.)

### Beer's Law and the Absorption Coefficient

Students did not measure the complex concentration *c*_{eq} directly. Instead, they measured A_{500} (absorbance at 500 nm), which according to Beer's law (the subject of a previous module in the MLS course) is proportional to *c*_{eq}, the proportionality constant being the complex's *absorption coefficient ε*:

Equation 3 gives predicted values of the dependent variable A_{500} in terms of a known constant (the total input avidin concentration *a*_{0}), an independent variable with known values (the total input HABA concentrations *b*_{0}), and two constants with unknown values (the dissociation equilibrium constant *K*_{D} and the absorption coefficient *ε*). Unknown constants such as *K*_{D} and *ε* are called *parameters*; the next section focuses on using the available data to estimate their values.

## Parameter Estimation

If the mathematical model in Equation 3 accurately describes binding equilibrium, there should be values of its two parameters *K*_{D} and *ε* that bring the predicted values of the dependent variable A_{500} into agreement with the observed values of that dependent variable. Conversely, parameter values that bring the model's predictions into agreement with the observations can be considered estimates of those parameters. The final session in the MLS binding equilibrium module was a 100-minute computer lab centered on estimating *K*_{D} and *ε*. Students had already encountered a simpler estimation problem involving a single parameter, in the Beer's law module mentioned earlier.

Using the spreadsheet with the observed absorbances A_{500} for each total input HABA concentration *b*_{0} (the open circle and filled triangle data-points in Figure 2; an exemplar spreadsheet, ParameterEstimationExemplar.xlsx, is provided on the companion website), students wrote a formula that gives the mathematical model's predicted absorbance for each HABA concentration. The formulae's predictions were graphed (smooth solid curve in Figure 2) to compare them visually to the observed measurements. The formulae obtained values of the parameters *K*_{D} and *ε* from two of the spreadsheet's cells. Initially, the parameter values in those cells were just guesses (e.g., *K*_{D} = 10 µM, *ε* = 0.1 µM^{−1}), and the predicted absorbances in the solid curve consequently diverged greatly from the corresponding observed absorbances. The students were given the task of finding values of the parameters that brought the predictions into as close agreement as possible with the observations. To supplement their visual assessment of agreement, they entered a formula for a conventional badness-of-fit criterion in one of the spreadsheet's cells: *S* = the sum of the squares of the deviations of the predictions from the corresponding measurements. Their goal was thus more clearly defined: to find the optimal pair of parameter values according to the least-squares criterion (i.e., the pair of values that minimized *S*). They soon learned that optimizing two parameters simultaneously by trial and error is a challenging task. In the Beer's law module, they had been introduced to least-squares optimization, and learned to use Solver, a Microsoft Excel add-in utility that automates this task for any number of parameters. The solid curve in Figure 2 is the optimal fit of the mathematical model to the data (the two outliers at 11 µM HABA were excluded from the optimization); the optimal values of *K*_{D} and *ε* were 5.9 µM and 0.0308 µM^{−1}, respectively.

The foregoing parameter values are estimates, of course, not exact measurements. Suboptimal parameter values result in systematic deviations from the data, but small deviations may plausibly be attributed to systematic errors in the data and thus do not convincingly exclude the corresponding parameter values. Indeed, even the optimal fit deviates systematically from the data to a slight degree. Unsurprisingly, somewhat different optimal parameter values emerged from the students’ data in different years (optimal *K*_{D} 4.9–10.6 µM; optimal *ε* 0.0308–0.0352 µM^{−1}). The companion website includes suggestions for an optional in-class discussion of uncertainty in parameter values.

The fit of the solid curve to the data in Figure 2 seems excellent, its systematic deviations very slight. That accuracy, in conjunction with the well-established physical principles from which the mathematical model was developed (mass action and Beer's law), give us considerable confidence in Equation 3 as an explanation of the hidden physical processes underlying the observed results. By the same token, we can be relatively confident of predictions based on that equation. Those predictions are extensive. The model allows the equilibrium state to be predicted for different concentrations of avidin and HABA than those actually investigated. Moreover, it is the end-point of an even deeper mathematical model describing the entire time-course along which the system attains equilibrium (Figure 4).

## Pedagogical Strategy

The pedagogical approach of the equilibrium binding lab module is detailed on the companion website, and will be summarized only briefly here. It aligns well with the K-12 Next Generation Science Standards (National Research Council et al., 2013), whose general principles project naturally to the college level.

The module is an example of “inquiry learning” (e.g., Hmelo-Silver et al., 2007), but the focus of inquiry was not the physical process of acquiring data; the spectrophotometric measurements were acquired via step-by-step instructions, as in traditional “cookbook” labs. Inquiry was focused instead on the mental process of developing scientific understanding from the data, from relevant scientific principles presented in reading and lectures (e.g., mass action) or expounded in previous modules (e.g., Beer's law), and from students’ background knowledge (especially high-school algebra).

The inquiry was highly guided or “scaffolded” (Hmelo-Silver et al., 2007; Kirschner et al., 2006) in important ways. First, the inquiry evolved in planned stages, each class session and homework task focusing on a restricted part of the overall inquiry that students could master for themselves. Second, short lectures and readings introduced specific scientific principles (e.g., the mass action law) just as needed in the inquiry. Third, prior modules provided worked examples that could be applied to the problem at hand. In particular, the Beer's law module not only introduced students to spectrophotometry and to the law itself, but also included a computer lab in which least-squares optimization was used to estimate an absorption coefficient parameter; at the same time, that computer lab served to enhance their mastery of the spreadsheet program.

In spite of extensive guidance, there was a great deal of intellectual distance for students themselves to traverse, both individually and as a group. For instance, they had to recognize that their prior module on Beer's law taught them not just about the absorption coefficient for the particular chromophore in that module, but about the absorption coefficients for chromophores in general, including the avidin-HABA complex; and not just about estimating absorption coefficients, but about estimating parameters in general, of any kind and in any number. Such generalizations, which come so naturally to experienced scientists, and which are so central to scientific practice, are a revelation to beginning science students.

As central as mathematics is to the practice of science, and was to MLS's ambitions, the MLS course did not emphasize new mathematical techniques, as in traditional modeling courses. We sought instead to show students how even the elementary mathematics they had already learned in high school illuminates a great diversity of biological phenomena. At this early stage in their education, we hoped, students would come to understand the scientific enterprise as a great web of deeply interconnected ideas rather than a collection of separate “disciplines” with distinct “professional skills.”

My colleagues Carmen Chicone, Michael Henzl, Miriam Golomb, Ricardo Holdo, and Jeni Hart, as well as Ravit Golan Duncan of Rutgers University, helped critique the manuscript. The University of Missouri Mathematics in Life Sciences Program was supported by U.S. National Science Foundation grant DMS 0928053, Dix H. Pettey Principal Investigator.

## References

### Appendix

#### Advance preparation

Detailed instructions are available in the BindingEquilibriumPreparation document on the companion website.

**HABA stock solution:**

Into a 16 × 100 mm glass test tube, dissolve 140 mg of HABA [2-(4′-hydroxyphenylazo)benzoic acid; e.g., Sigma H5126-5G] in 7 ml ethanol; the nominal HABA concentration is 20 mg/mL (82.6 mM).

Transfer the 20 mg/mL ethanolic solution to a 15-mL polypropylene screw-cap centrifuge tube (or other tightly sealed polypropylene vessel).

Measure 24.75 mL Dulbecco's phosphate-buffered saline without calcium or magnesium (DPBS; e.g., Fisher Scientific SH30028FS) into a 50-mL polypropylene screw-cap centrifuge tube.

Start the tube vortexing with the cap off, and pipette 250 µL of the 20-mg/mL HABA solution directly into the vortexing buffer. (A fine yellow precipitate forms temporarily but disperses immediately.) Continue vortexing until the precipitate is fully dissolved.

Measure A

_{348}(absorbance at 348 nm) of a 1/40 dilution, and calculate the true HABA concentrations in the 15- and 50-mL tubes assuming a molar absorption coefficient of 20,700 M^{−1}. (Should be about 75 mM and 750 µM, respectively.) Store the tubes with caps on securely in the dark at −20ºC.Upon thawing and before opening a tube, mix to ensure homogeneity and centrifuge briefly at low speed to drive all liquid to the bottom.

**Biotin stock solution**: The 10-mM biotin stock solution is used to titrate avidin (second bullet in the next section); therefore biotin solid of high purity should be used (e.g., Fisher BP2321), and the 10-mM stock solution in DPBS should be prepared as accurately as practicable. The stock solution is stored at −20ºC, and thawed and refrozen as needed. Upon thawing and before opening a tube, mix to ensure homogeneity and centrifuge briefly at low speed to drive all liquid to the bottom.

**Avidin stock solution.** Avidin can be purchased in 10- or 100-mg quantities (Merck-Millipore 189725), enough for one and ten 20-student labs, respectively; the smaller amount is about twice as expensive per mg as the larger amount. Here I describe the stock solution for the smaller amount, to be used within a few days without freezing. (The BindingEquilibriumPreparation document on the companion website describes preparing frozen aliquots of the larger amount, to be used over a period of years.)

The entire contents of the vial (nominally 10 mg) is dissolved in 1.2 mL water (all water is distilled or deionized), transferred to a 1.5-mL microtube, and microfuged for a few minutes to collapse bubbles and pellet particulates.

The supernatant is transferred to a larger vessel (capacity at least 3 mL) and diluted with 1 mL additional water (total volume 2.2 mL, enough for 20 students). The nominal avidin subunit concentration is 275.5 µM, but it is advisable to determine the concentration of biotin-binding sites empirically by titration with biotin (Green, 1965); a detailed titration protocol is included in the BindingEquilibriumPreparation document on the companion website.

The stock solution can be stored at 4ºC for a few days before use in the student lab.

**Preparations just before lab**: The BindingEquilibriumPreparationWorksheet Microsoft Excel spreadsheet on the companion website automates the calculations below.

Starting with the 10-mM biotin stock above, make at least 2.2 mL of a dilution in DPBS with a biotin concentration equal to the avidin concentration in the avidin stock solution above.

Dispense 100 µL water, 100 µL of the avidin stock solution (previous section), and 100 µL of the biotin dilution (previous bullet) into 500-µL microtubes for each student.

Calculate the volume of 100-µM HABA dilution needed = $0.2+5.5\xd7(1\u2212fn)/(1\u2212f)$ mL, where

*n*is the number of students and*f*is defined in the next bullet. Thaw the 50-mL tube of HABA stock solution above, vortex the tube vigorously, and centrifuge it briefly to drive solution to bottom. In a 50-mL polypropylene conical centrifuge tube, mix HABA stock solution and DPBS diluent in appropriate volumes to give the calculated final volume with a final HABA concentration of 100 µM. (It is significantly more accurate to measure the DPBS diluent into the tube by weight rather than by volume.) Screw the cap on the 50-mL stock solution tube securely and return that tube to the −20ºC freezer.In

*n*− 1 vessels make a logarithmic (geometric) serial dilution series of 5.5-mL HABA solutions by mixing DPBS diluent and the 100-µM HABA solution (previous bullet) in appropriate volumes. The HABA concentration in each dilution in the series is a fraction*f*of the HABA concentration in the previous dilution, where the serial dilution fraction $f=11/1000n\u22121$; the final dilution in the series has a HABA concentration of 1.1 µM. The actual HABA concentrations can differ slightly from the target concentrations if convenient, and the volumes can differ slightly from 5.5 mL; but it is important that the actual HABA concentrations be known as accurately as practicable. Dispense exactly 1.2 mL of the 100-µM solution and of each dilution into four 1.5-mL microtubes, which will be used by one of the students (4 ×*n*microtubes altogether).