ACTAUNIVERSITATISUPSALIENSISUPPSALA2006

Digital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 184

Sensitivity, Noise and Detectionof Enzyme Inhibition in ProgressCurves

OMAR GUTIÉRREZ ARENAS

ISSN 1651-6214ISBN 91-554-6569-2urn:nbn:se:uu:diva-6886

Roberticooooooooo!Deja algo yunta.

List of papers

This thesis is based on the following papers, which will be referred to in the test by the Roman numerals.

I Gutiérrez, O.A.; Salas, E.; Hernández, J.; Lissi, E.; Castrillo, G.; Reyes, O.; Garay, H.; Aguilar, A.; García, B.; Otero, A.; Chavez, M.; and Duarte, C. A. (2002) An Immunoenzymatic Solid-Phase As-say for Quantitative Determination of HIV-1 Protease Activity. Anal Biochem 307(1): 18-24.

II Gutierrez, O.A.; Chavez, M. and Lissi, E. (2004) A theoretical ap-proach to some analytical properties of heterogeneous enzymatic as-says. Anal Chem 76: 2664-2668

III Gutierrez, O.A., and Danielson, U.H. Sensitivity analysis and error structure of progress curves. In progress at Analytical Bio-chemistry.

IV Gutierrez, O.A., and Danielson, U.H. Identification of substrate conversion levels for sensitive and robust detection of com-petitive enzyme inhibition with end point assays. In progressat Analytical Biochemistry.

Reprints of the papers were made with permission from the publisher

Contents

Introduction.....................................................................................................9The problem: when to measure? ................................................................9Enzymatic assays in primary screening......................................................9

Drug discovery ......................................................................................9Common assay platforms ....................................................................11Assay optimisation. Z’-factor ..............................................................11

Current views ...........................................................................................13Follow the tradition: Initial rate measurements ...................................14Non-initial rate region as an acceptable compromise ..........................14Non-initial rate region as the best choice.............................................15

Present Investigation.....................................................................................17Prelude......................................................................................................17

Two assumptions and one definition ...................................................17Quantification with heterogeneous enzymatic assays (HtEA).............18Sensitivity analysis of HtEA................................................................20

Sensitivity and noise in progress curves from homogeneous assays........21Sensitivity analysis of homogeneous assays........................................21Fitting the sensitivity model to experimental data...............................22The error structure of progress curve data ...........................................26A model for the standard deviation of progress curve data .................28

Detection of inhibition in progress curves ...............................................33IDL-factor: Inhibition Detection Limit................................................33Definition of assay populations ...........................................................35Detection limit and robustness in assay populations ...........................37IDL-factor vs. Z’-factor .......................................................................40

An answer. Its limitations.........................................................................41

Summary in Swedish ....................................................................................42

Acknowledgements.......................................................................................44

References.....................................................................................................46

Abbreviations

p Substrate conversion level µB Signal at the beginning of the reaction µ Signal at the end of the reaction R = µ /µB. Direction of signal change KM Michaelis constant Ki Inhibition constant KMs Michaelis constant for the substrate KMp Michaelis constant for the product K = [S]o/KMs. Initial saturation levelKps = KMs/KMp. Substrate-Product competition coefficient CV Coefficient of variation

c CV representing the constant error component p CV in the proportional error component t CV in the transient error component

Sn Sensitivity of p to variations in [E]osd Standard deviation IDL Inhibition Detection Limit HIV-PR Human immunodeficiency virus protease GR Glutathione reductase GST Glutathione sulfotransferase

9

Introduction

The problem: when to measure? Suppose you want to determine if compound X is an inhibitor of enzyme E. In order to pursue that goal, you have a typical enzymatic assay that allows you to test the effect of X on the activity of the enzyme E. As in most enzy-matic assays, the activity of the enzyme E is inferred by the intensity with which it speeds up the conversion of the substrate to product. Due to techni-cal and economical issues, the concentration of X that you can have in the assay is limited and you can just afford a single measurement point in the course of the reaction catalyzed by E. In order to maximize the probability to detect a potential inhibition by X, how much substrate should have been converted into product to make that single measurement?

The problem above is a real one; the goal and constrains that lead to the final question are not artificially imposed. The ultimate intention of the pre-sent thesis is to provide a refutable answer to this question.

Enzymatic assays in primary screening Drug discovery The target-centered approach is the prevailing paradigm for the development of drugs for the treatment of diseases or some other conditions affecting the quality of human life in its broadest sense (1, 2). First, the biochemical com-ponents of a given disease are identified. Then, the effort is placed on devel-oping a compound that targets one or more of these components to restore the system to its normal function. A basic requirement for a compound to be used as a drug is its ability to interact and modify the activity of the targeted component in the ailing system.

Enzymes have historically made good targets for drug action (3). Such a relevant position derives from their protagonical role in living systems. Moreover, enzymes are particularly druggable, i.e. rather large molecules with featurely, specific and functionally relevant surface spots that support the binding of small compounds (< 500 Da) (4, 5). The active site of en-

10

zymes, a cavity for substrate binding and catalysis, is a paradigmatic exam-ple of such spots. This explains why many drugs that target enzymes are compounds that bind to the active center, producing a pattern of total com-petitive inhibition (e.g. gleevec, tamiflu, viagra and penicillin). Approxi-mately 30-50% of the current and potential protein targets of human non-infectious diseases are enzymes (2, 5, 6). Among enzyme targets, protein kinases are predominant (5-7), followed by proteases (6, 8), protein phos-phatases (9), phosphodiesterases (10), ATPases (11), cytochrome P450 (5, 6) and dehydrogenases (5, 12, 13). All these enzymes catalyse practically irre-versible reactions. The proportion and diversity of enzymes among protein targets of infectious diseases may be similar or even higher.

The first step to identify compounds that interact with a target enzyme and provide the starting point for the development of a drug, is generally to perform a primary screening of compound collections using a biochemical assay. With this assay, the effect of the screened compounds on the isolated target is assessed. The number of compounds, of natural or synthetic origin, included in a single primary screening usually exceeds the thousand. There has been an intense development of procedures for biasing the compound collection to be screened towards desired properties. These properties in-clude higher potency (e.g. virtual docking (14)), more room for chemical modifications (e.g. fragment-based discovery (15, 16)), higher chemical diversity/novelty and better pharmacokinetic profile (e.g. Lipinsky’s rules (17)). Still, the number of structures to be screened remains high since these pharmacologically relevant regions of the chemical space are big (18, 19). The screening of thousands of compounds in short periods is called high-throughput screening (HTS). The pursuit of HTS has prompted the develop-ment of very characteristic technological (e.g. automatization, miniaturiza-tion) and methodological (e.g. austerity in data collection) infrastructure that is also suitable for medium and low throughput screening efforts (15, 20-22).

The number and structural diversity of active compounds (“hits”) result-ing from the primary screening depend on the characteristics of the enzy-matic assay used for the screening. Success is often restricted by difficulties in detecting those inhibitors that are present at low concentrations and/or have intrinsically low affinities for the target. For example, this is often the case in the primary screening of natural product and synthetic fragment col-lections, two popular types of libraries screened for small and structurally diverse inhibitors (23, 24). Inhibitors rarely occur at high concentrations in natural product extracts (24), and in fragment libraries they tend to have rather low affinities for the target due to their small size (23). Thus, the in-hibitory effect of compounds from both types of libraries is generally ex-pected to be low. Improving the detection limit of assays used for primary screening of these and other libraries would consequently improve the detec-tion of potentially useful compounds.

11

Common assay platforms The rate of the catalytic transformation of substrates by enzymes is used as a measure of enzyme activity. By extension, the interaction of a compound with the enzyme is commonly measured as the effect it has on that rate. The experimental setting for making these measurements is the enzymatic assay. There is a multitude of enzymatic assays to monitor the effect of compounds on validated target enzymes. They can be homogeneous or with separation steps, end-point or continuous multiple points, labeled or label free. The pressure for high throughput has favored homogeneous, end-point and la-beled assays. This is called the “mix and read” format.

Just one or two measurements at a single compound concentration and asingle reaction time are done after stopping the reaction. These end-point measurements do not just exist as a mean of increasing the throughput but they are also a result of the limitations of many assay designs. That is the case of some widely used assay platforms, i.e. versatile experimental ar-rangements that can be applied to a variety of targets and reactions by intro-ducing slight modifications. These assay platforms have been used for pro-tein kinases (25), proteases (26), phosphatases (27), phosphodiesterases (28), dehydrogenases (29), DNA polymerase (30), etc.

The most common assay platforms are scintillation proximity assay, chemiluminescent proximity assay (e.g. AlphaScreen), fluorescence reso-nance energy transfer and fluorescence polarization (31-33). The first three of them are proximity-based assays where energy donors and acceptors have to be in close proximity to interact. This close proximity is achieved by the specific non-covalent binding of the substrate or product of the reaction to some partner. In fluorescence polarization there are also many instances where a binding event precedes the development of the signal.

The non-covalent binding of the substrate or the product to a partner(s) in order to develop the signal is usually much slower than the catalytic reaction. In many instances the optimal conditions for this step do not overlap those for enzyme action. Generally, these are the reasons to stop the reaction and to make a single end-point measurement.

Although these four assay platforms dominate the landscape of primary screening assays, there are many others also used for this purpose (34-36). Something common to almost all these enzymatic assays is that they are performed in microplates of 96, 384 or 1536 wells.

Assay optimisation. Z’-factor Every enzymatic assay used for primary screening needs to be optimised. Assay optimisation is a budget-observant procedure (e.g. via miniaturiza-tion) intended to get as close as possible to the proper conditions for the en-zyme catalytic activity (i.e. pH, ionic strength, temperature and additives),

12

and to tune the assay so that its analytic performance is maximized. Since primary screening assays are intended for the reproducible discrimination between active and inactive compounds, the most relevant analytical per-formance parameters are limit of detection and robustness. Briefly, the limit of detection is the compromise threshold demarking, with certain confi-dence, the active from the inactive compounds. In a restricted sense, the robustness is a measure of the stability of that threshold to variations in the assay conditions.

In practice, the most used indicator to monitor the quality of primary screening assays and the effect of assay modifications is the Z’-factor (37). The relation of Z’-factor with the limit of detection and the robustness of an assay is far from clear. Z’-factor is computed with Eq.1. The elements it takes into account are shown in Figure 1, where sd and µ are the standard deviation and the mean of the control of null enzyme activity (B), i.e. no enzyme, and the enzymatic activity control (C), respectively.

BCCB sdsdZ 31' [1]

Figure 1: Schematic representation of the elements used for computing the Z’-factor. The Z’-factor is empirical since in its formulation there is not clues about the process generating the signal; there is no mechanistic under-pinning.

These controls are also known as the negative (µB) and the positive (µC)control of enzyme activity. Most enzymatic assays are designed in a way that the signal increases as the reaction progresses, so that µB < µC. This is not an accident but something done deliberately, albeit not based on a defined for-malism. In practice, the sd and µ are estimated from a number of replicates of both controls.

The larger the difference between the mean signals of the controls, and the lower their dispersion, the better the assay. The Z’-factor is always below 1. An assay with Z’ > 0.5 is considered an excellent assay. The exact limit for this qualification may vary (33). The signal obtained with an inhibitory compound acting on the enzyme will lie between µB and µC, in a position that results from the potency of the inhibition and the random error inherent to any measurement. Compounds with no effect on the enzyme activity will lie at both sides of the enzymatic control, µC, in a random fashion. There

13

could be also measurements out of the [µB, µC] range due to enzyme activa-tors in the screened library or compounds that interfere with the measured signal for physical reasons rather than for effects on the enzyme activity.

The success of the Z’-factor lies on its simplicity, and on the fact that it combines the otherwise incomplete information contained in older indicators such as signal/noise ratio and signal/background ratio (33, 37). However, it must be noted that Z’-factor is strictly phenomenological, i.e. it has no estab-lished mechanistic underpinning. The expression of Z’ is the same regardless the mechanism of the process being screened.

This is the reason why, even though the Z’-factor is preferentially used to evaluate the assay quality and to drive the assay optimization, system-specific and other complementary considerations are also taken into account (32, 33, 38).

One very general consideration is linearity between the signal and the concentration of the analyte, i.e. a linear calibration relationship. In the case of enzymatic assays the analyte is the substrate and/or the product of the reaction. A linear calibration relationship is the simplest and most amenable for calculations and modeling, but this is not always achieved. Inner filter effects in fluorescence, Lambert-Beer’s Law deviations in absorbance or hyperbolic saturation behavior when the signal is mediated by the binding of the substrate/product to some partner (e.g. proximity assays) usually underlie non linear relationships (33). In the first two instances, the solution is to work at lower analyte concentrations, and in the third, to use a binding part-ner of higher affinity and/or at higher concentration.

The enzyme concentration is usually set low, below 100 nM, and in a re-gion where it has a linear relationship with the signal. The concentration within this linear range is set according to productivity issues, i.e. reaction time and enzyme cost. This will depend on how efficiently the enzyme trans-forms the substrate. Other constrains such as enzyme inactivation by adsorp-tion could play a role (38). A necessary condition for the existence of linear-ity between the signal and the enzyme concentration is to make the end-point measurements in the initial rate region of the reaction, where less than 10% of substrate has been converted to product. Again, a linear relationship is the simplest and most amenable for calculations and modeling. However, what certainly matters in primary screening assays is detection limit and robust-ness. Here is our problem restated: are they optimized with initial rate meas-urements?

Current views The current views about the best position in a reaction progress curve to make end-point measurements can be divided in three groups with an in-creasing level of heterodoxy.

14

Follow the tradition: Initial rate measurements It is advisable to make end-point measurements in the initial velocity region (38-40). This is the settled rule after almost 100 year of success in enzyme kinetics modeling. The extent of the initial velocity region varies among systems, experimental conditions and, certainly, authors (41, 42). It is com-mon to set 10% of substrate conversion as the upper limit of the initial veloc-ity range and this seems to be the standard in end-point primary screening assays (33, 42, 43). Well-known estimations of inhibition potency are estab-lished with the use of initial velocity measurements (see below). Further-more, measurements in the initial segment of the reaction are the most time saving.

Non-initial rate region as an acceptable compromiseThis position is not frontally opposed to the traditional one. Measurements in the non-initial velocity region are just advocated for assays where high a proportion of transformed substrate is required to obtain a significant change in the signal and a high Z’-factor (33, 43). It is proposed that the cost of this strategy is not particularly high in terms of overlooked potency. However, the formalization of this argument by Wu and colleagues (43) has an impor-tant problem: the extrapolation of the degree of inhibition ( ) from a variable based on initial velocity measurements, to one based on measurements made at any substrate conversion level is misleading.

Let’s go to the basics. Eq.2 represents the original definition of (44),

co

io

vv1 [2]

where voi and vo

c are the initial velocity of the reactions with and without (enzymatic controls) the inhibitor, respectively. As defined in Eq.2, is, in the case of total inhibition, the fraction of inhibited enzyme in relation to the total enzyme concentration. For end-point measurements made in the initial velocity range, i.e. µo

c and µoi, Eq.2 equals Eq.3.

Bco

Bio1 [3]

In this case the initial rate formalism can be used to further developments such as the hyperbolic inhibition curve (Eq.4) for a total competitive inhibi-tor),

15

IICI

50

if [E]o /Ki << 1 [4]

where, [I] and [E]o are the concentrations of inhibitor and enzyme, respec-tively, Ki is the dissociation constant of the enzyme-inhibitor complex. IC50is a very popular measure of potency that represents the concentration of inhibitor which reduces the enzyme activity to a half, i.e. = 0.5,

M

oi K

SKIC 150 [5]

where [S]o is the initial substrate concentration and KM is the Michaelis con-stant.

Wu and colleagues extended Eq.3 to measurements done at any substrate conversion level and this is misleading. For example, it is clear that under the assumption [S]o/KM <<1, does not depend on the substrate conversion level. Thus, the potency expressed as the inhibitor concentration that pro-duces certain value of is constant all over the reaction. However, their ex-tended definition of and the whole argument implies that it does change (43). Furthermore, they stated that this extended has a hyperbolic relation-ship with the inhibitor concentration and that is not true. Nevertheless, this is not particularly relevant given that the rigorous expression behaves almost identically up to 70% of substrate conversion. The latter may explain the good fit of Wu’s formalism to experimental data (43).

Thus, such an argument, based on potency, should not be cautious at all, but encouraging, given that apparently nothing is lost by making end-point measurements in the non-initial rate region.

Non-initial rate region as the best choiceThe point where the signal from the reaction of enzyme activity controls diverges the most from the signal of the reaction with an inhibitory sample is not in the initial velocity region. Thus the optimal point for measuring is not in the initial velocity region but in the maximal divergence region. To the best of our knowledge the first, and lonely, advocate of this view is John C. Owicki. The argument is simple (then, beautiful); it seems likely that some-one else dealt with it before or after. In fact, what our proposal has in com-mon with this one was developed independently. The visibility of Owicki’s proposal is very low, besides, it is pretty succinct and with little formaliza-tion (45). He also argued that the higher the degree of inhibition, the higher the substrate conversion level of the reference where the difference between progresses curves is maximal. Mild product inhibition wouldn’t affect sig-

16

nificantly this basic conclusion. This is probably the work most related to the present study.

None of the three groups described above connected their arguments to the notion of limit of detection in a meaningful way. All of them defended their proposals based on mean values properties, i.e. linearity, potency and differ-ence between progress curves. None of them took into account the disper-sion of the data, which is an essential part of any definition of limit of detec-tion. For example, it could be the case that the difference between the control and the inhibited progress curve is maximal at a point where the dispersion (i.e. standard deviation) of the measurements is also maximal. In that case, the conclusions drawn from the analysis of mean value properties would be jeopardized.

Here I present, in essence, a formalization of the interplay of mean values properties and standard deviation to model the limit of detection of competi-tive inhibitors for a typical end-point enzymatic assay. For reaching that point some developments were needed, such as a model for the standard deviation of progress curve data. We provided them here.

17

Present Investigation

Prelude(Papers I and II)

Two assumptions and one definition In this study, certain assumptions have been made. Some of them were justi-fied in the introductory section above. Among these assumptions the follow-ing two play a ubiquitous role.

Assumption 1: reactions are practically irreversible ( G << 0). This im-plies that at the equilibrium almost all the substrate has been converted to product, i.e. [S]o ~ [P] .

Assumption 2: the measured signal (M) is linearly related to the analyte concentration, which can be the substrate, [S], the product, [P], or both, so that

M = s [S] + p [P] [6]

where s and p are proportionality coefficients. M has been subtracted from any existent signal at 0 analyte concentration.

All the experimental systems used in this work, and most of the enzyme targets and assays used in primary screening, fulfill these two conditions. These assumptions simplify the modeling considerably and are needed to derive the consequences of the following definition.

Substrate conversion level (p): the proportion of substrate that has been con-verted to product at a given reaction time.

oo SP

SSp 1 [7]

p is a useful indicator of the reaction progress. Combining this definition with the two assumptions given above, results in

18

p varies from 0 at the beginning of the reaction where the signal is µB, to 1 at the end of the reaction where the signal is µ . The linear relationship be-tween the signal and the substrate conversion level (Eq.8) allows the straightforward conversion of values, results and conclusions obtained from one into the other. In many instances p is used interchangeably with the sig-nal, M.

The following parameter,

BR [9]

describes whether the signal increases (R > 1) or decreases (R < 1) with the reaction. This will prove to be a highly relevant variable.

All the routines used in the fittings and the simulations of this work were programmed in MatLab 7.1.

Quantification with heterogeneous enzymatic assays (HtEA) The starting point of what is being presented here was the development of an assay platform for endopeptidases (e.g. HIV-1 protease and Plasmepsin II). This was an end-point HtEA, uncommon but not novel in its approach (46, 47). As represented in the following scheme, a solid-phase (SP) substrate is hydrolyzed by an endopeptidase in solution (E)

Scheme 1

Scheme 1 represents an irreversible reaction with neither substrate/product inhibition nor enzyme inactivation. The substrate was a biotinylated peptide bound to streptavidin-coated microplates. The remaining substrate after in-cubation with the endopeptidase was measured using a monoclonal antibody against SSP followed by a peroxidase-antibody conjugate in an ELISA-like fashion. Experimental details of the assay are described in Paper I. The dis-tinctive feature of HtEAs is that the limited binding capacity of the solid-

osB S

B

BMp [8] Pp

19

phase and the area/volume ratio of the system (~10 cm-1) generally render substrate concentrations below the KM. Thus, the system is not saturable by substrate but by the enzyme, and this implies that [ES] is negligible with respect to [E] but not with respect to [S]. The main scientific contribution was to introduce a model for the time course of the enzymatic reaction in this unusual format,

oM

o

EKE

kep

2

1 [10]

Eq.10 was used for the estimation of kinetics parameters (k2 and KM) and the proper quantification of enzymatic and inhibitory activity. This implied measuring whole progress curves instead of just initial velocity rates. Al-though the monitoring of whole progress curves has been a common practice in HtEA, a quantitative treatment like the one proposed was missing. Ex-pressions like Eq.10 that model the mean value of the monitored signal are also known as mean models. Complementarily, expressions that model the standard deviation of the monitored signal are called standard deviation functions. This is the nomenclature used in this work.

Eq.10 is the integrated form of

oM

ooo EK

Ekdt

dpv 2 [11]

Eq.11 had been around for a while to model the hydrolysis of solid poly-meric substrates (e.g. cellulose), systems that bear some resemblance with HtEA (48). However, in the case of solid polymeric substrates the reaction scenario is further complicated by the overlapping of binding sites in the solid substrate (49, 50). The similarity of Eq.11 with the Michaelis-Menten equation (Eq.12) is evi-dent,

oM

oooo SK

SEkdtSdv 2 [12]

Both describe the initial reaction rate of saturable systems, but with opposing saturating species. However, the different roles of the substrate and the en-zyme in the reaction have deep consequences on the limits of validity of the quasi-steady state approximation, or more precisely, on the meaning of the approximation d[ES]/dt ~ 0. Here the analogy ends (51).

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Sensitivity analysis of HtEA As with any saturable system, working with the saturating species below and above the KM is essential to estimate the kinetic parameters of an HtEA. However, since these assays are intended to measure enzymatic activity it was not clear whether or not the same domain of [E]o was optimal for ana-lytical purposes. That was the main issue in Paper II where we made use of a tool known as sensitivity analysis (52). The sensitivity of the variable F to changes in y is nothing but the first derivative of F respects y. This is the core of sensitivity analysis.

Results obtained from analysing the sensitivity of the substrate conversion level are easily extrapolated to the signal by multiplying for the constant (µ- µB) which is the total signal window of the assay. From Eq.10 it follows that the sensitivity is

ppEK

Ked

dpoM

M

o

1ln1 [13]

where

o

oo E

Eded

With the change from d[E]o to d[e]o the focus is changed from absolute to relative variations in the enzyme concentration. This transformation will prove to be simplifying. Eq.13 can be split in two components,

Time-independent component Transient component

oM

M

EKK

[14] pp 1ln1 [15]

The time-independent component indicates that the higher the ratio [E]o/KMthe lower the sensitivity of the signal to variations in [e]o. This is the ex-pected result from the curvature of the hyperbolic function that describes many saturable systems (e.g. Langmuir and Michaelis-Menten models). The same expression was obtained for the sensitivity of the relative initial veloc-ity to variations in the relative enzyme concentration. Thus, if the maximal sensitivity of the signal to variations in the enzyme activity is desired, then the region [E]o << KM is optimal in this sense.

The transient component of the sensitivity (Eq.15) is a non-monotonous continuous function with a single maximum at p = 0.63 and equal to 0 at the beginning (p = 0) and at the end (p = 1) of the reaction. That is why we

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named this sensitivity component as transient. This behavior suggests that measuring at substrate conversion levels around p = 0.63 would maximize the ability of HtEA to detect differences in enzymatic activity caused, for example, by a competitive inhibitor. The realization of this simple but over-looked property and of its analytical potential constituted a turning point in this work.

Sensitivity and noise in progress curves from homogeneous assays (Paper III)

Sensitivity analysis of homogeneous assays Despite their potential, HtEAs are far exceeded in variety and abundance by their homogeneous counterparts. Most enzymatic assays used in primary screening of enzyme inhibitors are homogeneous, with both enzyme and substrate in solution. Thus, we turned our attention to homogeneous assays in an attempt to unveil the significance of the transient component of the sensitivity in a more relevant field. For homogeneous assays the time-independent component of the sensitivity is not at stake since the condition [E]o << KM is characteristic to all of them (Eq.19).

In the absence of cooperativity, the integrated form of a Michaelis-Menten equation describes the pro-gress of a reaction catalysed by an enzyme in a ho-mogeneous scenario. For monosubstrate irreversible reactions with neither enzyme inactivation nor sub-strate inhibition, like the one represented in Scheme 2, the rate equation and its integrated form are repre-sented by Eq.16 and Eq.17 , respectively, Scheme 2

pKK

pekdtdp

ps

o

1111 [16]

tekpKK

pK opsps 1ln11 [17]

22

where

o

ocat

SEkk [18]

[e]o is the enzyme concentration relative to an arbitrary value [E]o. K is the ratio [S]o/KMs and Kps is the ratio KMs/KMp. Kps accounts for the possibility of competitive product inhibition. The sensitivity of the substrate conversion level to variations in the enzyme concentration is obtained from Eq.17 as,

BpK1KK1

pK1Kp1lnKK11ped

dp

ps

psps

o

[19]

B is an empirical parameter whose role is explained later. As said before, in this case (Eq.19) there is no time-independent component of the sensitivity since we have restricted the analysis to the first order region, i.e. [E]o << KM. Nevertheless, the transient component of the sensitivity persists and shares some properties with the one already described for HtEA (Eq.15). Equation 19 is a non-monotonous positive function that equals 0 at the be-ginning (p = 0) and at the end (p = 1) of the reaction (Fig. 2). The position and amplitude of the maximum depend on K and Kps. The substrate conver-sion level at the maximum is higher than 0.5 for any K and Kps.

Fitting the sensitivity model to experimental data Experimental values of dp/d[e]o are not obtainable. A differential, i.e. d[e]o,is as unimaginable small as infinity is unimaginable huge. Then, to test Eq.19 we used the ratio of finite differences, p/ [e]o, as an experimental proxy of dp/d[e]o. The experimental sensitivity, p/ [e]o, is denoted here as Sn.

In order to obtain experimental data to test the sensitivity model (Eq.19), three real systems were analysed: HIV-1 protease (HIV-PR), yeast glu-tathione reductase (GR) and glutathione sulfotransferase (GST). Some rele-vant features of these systems are shown in Table I, and further details are described in Paper III.

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Table I: Homogeneous enzymatic assays used in this work. F stands for fluorescence and O.D. for optical density.

The initial concentration of the substrates underlined in Table I was set more than 25 times lower than the accompanying substrate concentration so that the reactions could be considered as monosubstrate. However, for GR and GST the parameters kcat, KM and Kps were apparent, i.e. they still depended on the concentration of the non-underlined substrate.

The enzymatic assays were all continuous and the reactions were moni-tored until the signal stabilized. Experimental substrate conversion levels were computed from the monitored signal (Eq.8) at 8 different enzyme con-centrations ([e]o) and varying reaction times for the three different systems (Fig. 2). Each relative enzyme concentration differed only 3.5% (i.e. [e]o = 0.035) from the previous one. Four replicates of progress curves were ob-tained for each of them. The experimental values of the sensitivity, Sn, were computed as the slope of the plot “p vs. [e]o” at different reaction times. This slope corresponded to p/ [e]o (Fig. 2).

The mean model of progress curves (Eq.17) was fitted to (t, [e]o, p) data sets from GR and GST. A modified version of Eq.17 was used in the case of HIV-PR. This version accounts for enzyme inactivation as a first order proc-ess that affects free (E) and bound (ES & EP) species with the same intrinsic rate. The significant inactivation of the HIV-PR in the time scale of the ex-periments (~ 2 hours) has been reported (53) and it was indicated here by non overlapping curves in a Selwyn plot (not shown). The estimated sub-strate conversion level, ep , for the mean enzyme concentration (i.e. 1.14) is obtained with the fitted mean model and used as a regressor for the sensitiv-ity analysis (Fig. 2). The model for sensitivity was fitted to ( ep , Sn) data sets.

If Eq.17 successfully describes an experimental system, then Eq.19 should also do because the latter is a derivation of the former. Thus, once Eq.17 has proved its mettle, evaluating the fit of Eq.19 to experimental data is more a confirmatory exercise than an original trial. However, this paved the way for the later development of a related concept, the transient compo-nent of the noise.

E Reaction Signal R |µ - µB|HIV-PR Flr-(aa)9-Qn + H20 Flr-(aa)4 + (aa)5-Qn + H+ F 355/500 nm 9.37 0.278

GR NADPH + H+ + GSSG NADP+ + 2GSH O.D. 340 nm 0.03 0.119 GST CDNB + GSH GSDNB + Cl– + H+ O.D. 340 nm 14.9 1.417

24

Figure 2: I, Progress curve data for GR. The mean of four replicates (•) is repre-sented for each enzyme concentration at different reaction times. The red curve is the predicted value of the substrate conversion level, ep , obtained with the fitted mean model (Eq.17) for the mean enzyme concentration (i.e. 1.14). II, Illustration of the computation of the experimental sensitivity, Sn, at a time point marked in I with a dashed line. III, “Sn vs. ep ” plot. The sensitivity model (Eq.19) with and without the parameter B fitted to the experimental Sn (•) is represented with solid and dashed lines, respectively.

The fitting of both the mean model and the sensitivity model was performed by a non-linear least squares procedure where 2 was minimized. The 2

minimum was used for a 2 test of goodness of fit (Box 1) (54). In both cases, the experimental 2 were computed assuming homoscedastic data, i.e.the standard deviations of p and Sn were constant at any point in the reaction (55). The goodness of fit and the parameter estimates of both the mean model and the sensitivity model are shown in Table II.

25

Box 1: 2 test for goodness of fit The 2 distribution with a number of degrees of freedom (DoF) is com-monly used to assess the quality, or goodness, of fit of a given model to experimental data. This test is based on two assumptions: the data is nor-mally distributed and statistically independent. The experimental value of the statistic 2 for a model for the variable “y” is computed as,

n

i i

ii yyDoF 1

2

22 ˆ1 [B1.1]

DoF = n – par [B1.2] n: number of data points

y: experimental value y : predicted value (from

model fit) : standard deviation of y

par: parameters of the model

The mean of the theoretical 2 distribution is 1. The critical value, or sig-nificance level (sl), is dependent on the DoF. A model-independent refer-ence of data dispersion, i.e. , is required for computing the experimental 2

and it is generally obtained from replicate measurements. A successful fit means that the dispersion of residuals from the fitted model is similar to this

(54). The lower the sl for an experimental 2 > 1, the lower the probability that the tested model is a good descriptor of the experimental data (54). For an experimental 2 < 1 the quality of fit is considered appropriate. However, an experimental 2 significantly lower than 1 may point to an overestimation of

.Whenever 2 < 1, the sl is not displayed in this work.

The variance of p was estimated from the sum of the squared differences at each ([e]o, t) between the p of each replicate and its corresponding samplemean, i.e. the mean of 4 replicates. Since there were no replicates in the case of Sn, its variance was computed from 1000 bootstrapped data sets (Box 2) of the original (t, [e]o, p) data. Each bootstrapped data set was built by re-sampling whole progress curves from the four replicates of each enzyme concentration. ( ep , Sn) sets were computed for each bootstrapped data set. Then, the variance of Sn was estimated as described above for p.

26

Table II: Parameter estimates and 2 goodness of fit of the mean model (MM: Eq.17) and the sensitivity model (SM: Eq.19) to progress curve data obtained with HIV-PR, GR and GST. Confidence limits are shown in Paper III.

E k (s-1) (10-3) B K Kps2 (sI)

MM 6.66 - 0.17 0.45 1.4 (<10-10)HIV-PR

SM - 1.66 0.58 4.48 0.29 MM 1.07 - 9.86 0.25 2.1 (<10-10)

GRSM - 0.80 11.0 0.17 0.32 MM 1.16 - 1.10 1.68 1.6 (<10-10)

GSTSM - 1.46 2.17 3.00 0.083

The estimated parameters with both models were similar (Table II). Fur-thermore, they were in the order of those reported previously even though the estimated Kps for HIV-PR was higher (53, 56, 57). However, the quality of fit according to the 2 test was very poor for the mean model. On the other hand, the quality of fit of the sensitivity model with the empirical parameter B was high. When this parameter was not included in the sensitivity model, the latter still described the shape of the experimental values. Nonetheless, there was a strong trend in the residuals because the fitted equation is shifted completely to one side (Fig. 2). The same results were obtained for the three experimental systems under study.

The origins of the persistent non-monotonic behavior of the sensitivity can be grasped with an intuitive analysis. Variations in the enzyme concen-tration generate variations in the reaction rate among progress curve data sets. This sort of asynchronicity is a dynamic event and, as such, it is non-existent at the beginning (p = 0) and at the end (p = 1) of the reaction. Thus, if progress curves are monotonous functions without inflections, it is ex-pected that the sensitivity reaches a maximum at some point between the beginning and the end of the reaction.

The lack of fit of the mean model and the need of the empirical parameter B in the sensitivity model are related to different aspects of the error struc-ture of the data.

The error structure of progress curve data The error structure of the data is defined by the type of distribution function (e.g. normal, Poisson, binomial), the scale parameter (e.g. standard devia-tion) of this distribution and the degree of statistical independence of the data. These properties can be analyzed in the residuals from the fitting of a model ( pp ). This is possible, because the error structure of p is invariant to the subtraction of a number, p , which is the estimated substrate conver-sion level obtained by fitting the mean model (Eq.17). The residuals from

27

the fitting of the mean model (Eq.17) to progress curve data from GR are given as example in Figure 3.

Some remarkable characteristics are evident in the residual plot (Fig. 3). Firstly, there are no systematic trends in the mean value of the residuals (red line), either as a function of the reaction progress or as a function of the en-zyme concentration (not shown). Furthermore, a similar residual plot was obtained for the residuals from the sample mean, implying that this profile is model-independent. In fact, progress curve replicates with a single [e]oshowed the same behavior in the residuals (not shown). This supports the adequacy of the mean model (Eq.17). The distribution of the residuals was normal at different points in the reaction, as assessed with the Lilliefors test (sl = 0.01). The residuals from each individual progress curve are distin-guishable as a sequence of points, as individual tracks on the plane. Finally, there is a clear heteroscedastic pattern, i.e. the dispersion of the residuals is not the same as the reaction progresses, that is characterized by a transient increase. This heteroscedastic pattern has been reported by Ronald Duggleby (58), an author with an extensive work in progress curve analysis. The resid-ual plots for the two other experimental systems (HIV-PR, and GST) showed similar features.

The residuals from each individual progress curve appear as individual tracks on the plane because a whole progress curve was obtained from moni-toring continuously a single reaction mixture. Thus, there is no statistical independency in a progress curve. The existence of data dependency in pro-gress curves obtained by continuous monitoring has been described before (59). Continuous monitoring of reactions is very popular in biochemistry and chemistry; it is labor saving and still very informative which is why it is performed whenever possible (59, 60). This practice implies that the errors in the preparation of a single mixture (e.g. errors in [S]o and [E]o) are the same for all the data in the progress curve obtained from it, but errors in the preparation of replicate mixtures are not the same. Thus, the data is depend-ent within a progress curve but independent among progress curves. Thus, when bootstrapping (Box 2) such a set of progress curves; the re-sampled

Figure 3: Residuals from the fitting of Eq.17 to progress curve data of GR. The red line represents the mean value of the residuals. The insets show histo-grams of the residuals at different sub-strate conversion levels. Note that the scale of the x-axis in these insets is the same. The residuals have a normal distribution at any point in the reaction, according to a Lilliefors test (sl = 0.01).

28

unit is a whole curve and not a single point in it. Data dependency was the reason why the empirical parameter B was required in the sensitivity model (Eq.19) (see next section).

The heteroscedastic pattern is characterized by a transient increase in the dispersion (Fig. 3). Besides, the dispersion is higher at the extreme of the reaction (p = 0 or p = 1) where the monitored signal is higher. This means that the dispersion is higher at the beginning than at the end of the reaction for GR, which has R = 0.03 (Table I and Fig. 3). The failure to take into account the heteroscedasticity may be one reason for the poor quality of fit of the mean model (Eq.17) to progress curve data according to the 2 test (see next section).

We assumed homoscedastic behavior (constant ) when it was clearly not the real situation. Estimates of the varying were required for a weighted non-linear least square fit of the mean model to progress curve data and to compute properly the experimental 2 (55). The challenge was: how to model the heteroscedastic dispersion of the data to obtain estimates of , i.e.the standard deviation (sd)?

A model for the standard deviation of progress curve data The transient increase in the dispersion of progress curve data is the result of random variations in the enzyme concentration operating through the sensi-tivity (Eq.19). It was already described how systematic variations in the en-zyme concentration (i.e. [e]o = 0.035) produced a transient increase in the differences among progress curves ( p). This effect is the same for randomvariations, i.e. noise, in [e]o. It translates into the transient increase in the dispersion of the residuals. Noise in the enzyme concentration is unavoid-able, for example, due to pipetting impreciseness. Furthermore, the random variations in the substrate concentration produce a similar effect. There is a transient component in the sensitivity of the substrate conversion level to variations in the substrate concentration,

pR

SnsSnsSSd

dpoo 1

122 [20]

where,

pKKKpKpKpK

Snsp

psp

111ln11 [21]

If the random variations in the enzyme and substrate relative concentrations were statistically independent but similar in magnitude, t, their contribution

29

to the transient increase in the dispersion of progress curve data, sdt, would be,

222

2t

ooot SSd

dped

dpsd [22]

When Eq.19, Eq.20 and Eq.21 are substituted in Eq.22, the result is a long expression. Unnecessarily long if the purpose is to estimate the standard deviation of the residuals without making any inferences from the estimated parameters. Certainly, the standard deviation of the residuals is not the proper data to estimate mechanistic parameters such as K and Kps. The error structure contains information from the underlying data generation mecha-nism but it may also be influenced by unrelated sources of noise. The tradi-tion has been to model the error structure empirically (55, 61-66) and it was followed here by using the simpler empirical expression,

22

5.12

6.021101ln

emt

W

t Wppsd [23]

Eq.23 keeps the essential from Eq.22, i.e. it is 0 at the beginning and at the end of the reaction, and has a maximum beyond 50% of substrate conver-sion. W defines the position of the maximum and together with emt, it de-termines the amplitude of the maximum. Additionally, emt is related to the relative dispersions (coefficient of variation) in the enzyme and the initial substrate concentration ( t). Both K and Kps are related to the empirical W and emt, but the functional relationship between them is complicated and irrelevant if the intention is to fit the experimental data. However, Eq.22 will prove useful to simulate the transient increase in the dispersion when its relationship to the mechanistic parameters K and Kps is relevant. This is the case in the development of a model for the detection limit of competitive inhibition (see next section).

Neither Eq.22 nor its empirical substitute (Eq.23) described the dispersion of residuals completely. At the beginning (p = 0) and the end (p = 1) of the reaction, these expressions are 0 whereas the dispersion in the residuals is not 0 at these points. In addition to the transient effect, there were others which we named calibration-like components since they reflect the similarity between a progress curve and a calibration curve. Most enzymatic assays are basically analytical chemistry assays, where the substrate and/or product concentrations are estimated from a monitored signal and a calibration rela-tionship (i.e. signal vs. concentration). A distinct feature of enzyme activity assays is the change of the analyte concentration over time, as a result of the ongoing enzymatic reaction. In this sense, a progress curve resembles a typi-

30

cal calibration curve with the analyte concentration axis (x-axis) transformed into time.

The dispersion of measurements in calibration curves is heteroscedastic and the standard deviation has been modeled with a variety of empirical expressions. All of them are positive functions that increase monotonically with the signal and have a non-zero value when the mean signal is 0 (55, 61-65, 67). This is the same effect we have already described, i.e. the dispersion of the residuals is higher in the extreme of the reaction where the signal is higher (Fig.3). Among these functions, the Rocke & Lorenzato two-component model (65) represents a useful formalization that neatly describes two common features of the dispersion in calibration curves, i.e. constant and proportional error components (see below). This model has been used in a series of reports dealing with weighted least squares and the detection limit of calibration curves in analytical chemistry (65, 66, 68). For a hypothetical analyte “A”, the Rocke & Lorenzato two-component model for the standard deviation of the signal in calibration curves is,

Asd AA2

02

02 [24]

The model has two components. One is the constant component, 2[A]=0, that

accounts for the signal noise in the absence of analyte. The other is the pro-portional component, 2

[A]>0, a coefficient of variation that describes an in-crease in the dispersion proportional to the increase in the analyte concentra-tion. To adapt the Rocke & Lorenzato two-component model to the realm of progress curves, it was assumed that the signal noise derived from the con-tribution of two analytes, substrate and product, to the signal. The contribu-tions of the analytes to the overall calibration noise were correlated to each other. Their concentrations were linked at any time by a conservation rela-tionship, i.e. [S]o= [S] + [P], and pipetting was probably an important source of dispersion. The proportional component was assumed to be the same for both analytes and the constant component contributed once, so that,

22

22

11

pcpc pR

sd [25]

The constant component ( c) is a term that contributes evenly all along the progress curve. It is constituted by the standard deviation of the signal in the absence of analyte (substrate and/or product) divided by the total signal win-dow of the assay, i.e. µ - µB. The proportional variance component ((1/(R-1) + p)2

p2) describes the increase in the dispersion as the signal increases.

31

Assuming that the calibration-like and the transient components are statis-tically independent, the resulting model for the standard deviation of pro-gress curve data is,

22

5.12

22

6.021101ln

11

emt

W

pc Wppp

Rsd [26]

The quality of fit of Eq.26 to the standard deviation of the residuals from the non-weighted mean model fitting was high (Fig. 4, and Table III).

Table III: Parameter estimates for the fitting of the standard deviation function (Eq.26) to the standard deviation of heteroscedastic residuals. Experimental 2 for the standard deviation function (SD) and for the weighted fitting of the mean model (WMM) using the estimated sd from Eq.26 as weights. Confidence limits are de-scribed in Paper III.

E c (10-2) p (10-2) t (10-2) W SD: 2 WMM: 2

HIV-PR 0.81 1.3 9.08 0.10 0.20 0.88GR 0.35 1.01 11.17 1.22 0.28 0.92

GST 0.61 1.20 9.86 0.69 0.21 0.93

The estimated standard deviations from the fitting of Eq.26 were used as weights to fit the mean model (Eq.17) to progress curve data. The quality of fit of the mean model after a weighted fitting was high (Table III). The re-sulting weighted residuals were homoscedastic (Fig.3 vs. Fig.4). The data dependency was also obvious in the weighted residuals and its effect on the robustness of the 2 test for goodness of fit was not assessed in this work.

Figure 4: I, Standard deviation of the re-siduals from the non-weighted fit of the mean model (Eq.17) to progress curve data for GR. Dots represent experimental data and the line, the fitted standard deviation function (Eq.26). II, Residuals from the weighted fitting of the mean model using the estimated standard deviation in I as weights.

The estimated parameters from the weighted fitting of the mean model were similar to those from the non-weighted fitting. However, these results are insufficient to state that the weighted fitting does not improve the quality of the estimated parameters. Improvement is expected but it may be marginal

32

(55). The quality of the parameters obtained from non-weighted and weighted fittings of the mean model can be analyzed in terms of efficiency. The relative efficiency of two estimators of a “true” parameter, htrue, from two different methods, 1 and 2, is the ratio of the dispersion of each estima-tor around the “true” parameter. The estimator with a lower dispersion is more efficient.

Box 2: Bootstrap and Monte Carlo generation of artificial data sets The Bootstrap is a technique where artificial q1, q2 …qH data sets with N data points are generated by random re-sampling with replacement from an original data set q0 with N statistically independent points obtained from experimenta-tion. These bootstrapped data sets can be seen as a kind of replicates. If a model is fitted to the original data set, the confidence limits of the estimated parame-ters can be obtained from the empirical distribution of the parameters estimated in the fitting of the model to the H bootstrapped data sets. The Monte Carlo simulation of artificial data sets is like playing Mother Nature (i.e. FSM). With a model for the mean values, a model for the error structure of the data, and “true” values of the parameter for both models, artificial q1, q2…qH data sets with N data points each are generated. These artificial data sets are used to obtain confidence limits for the “true” parameters used in the gen-eration of the data in the same way it is done with bootstrapped data sets. Since in the Monte Carlo procedure the experimenter knows the “true” values of the parameters, the efficiency of different estimation schemes of these parameters can be compared. Bootstrap and Monte Carlo procedures are two ways to bypass “the formidable wall of mathematics” placed in the middle of the road by traditional techniques of mathematical statistics (54, 69).

A Monte Carlo procedure (Box 2) was used to compute the relative effi-ciency of weighted and non-weighted fittings of the mean model. Artificial data sets simulating the experimental data for GR, were generated. The mean values of the substrate conversion level were computed with Eq.17, using the estimated parameters (Table II) and the time regressors of this ex-perimental system. For the sake of simplicity, four progress curve replicates for only one enzyme concentration ([e]o = 1) were included in each data set. The noise added to these mean values was generated according to the stan-dard deviation function (Eq.26). The dependency within a progress curve was reproduced as follows. From three normal distributions with mean 0 and standard deviations c, p and emt (Table III) a triplet ( c, p, emt) was sam-pled. This triplet was substituted in Eq.26 and the result was added to a sin-gle progress curve as a function of the mean value p. There was one triplet ( c, p, emt) per progress curve. An independent noise term was added to every point in a progress curve to simulate the contribution of the measuring devise. The amplitude of this dispersion was determined at a constant signal

33

value by repeated measurements and it was independent of the signal value. In this way 10000 artificial data sets were generated. The mean model (Eq.17) was fitted to each of the data sets by non-weighted and weighted fitting procedures. Weights were estimated with the standard deviation func-tion (Eq.26) as it was done above for the real data. The dispersion of the estimated parameters (k, K and Kps) around the “true” value was approxi-mately the same for both methods. Thus, the relative efficiency is near 1. This means that, in our experimental scenario, no improvement in the esti-mated parameters is achievable with the weighted fitting procedure. A similar Monte Carlo procedure but with the eight [e]o was used to generate independent data sets. The sensitivity model was fitted with and without the empirical parameter B. When the parameter B is included, the improvement in the quality of fit was much lower for independent than for dependent data sets. This explains why the inclusion of the empirical parameter B in the sensitivity model (Eq.19) produced a relevant improvement in the fit to our dependent experimental data sets. Now, we are in a better position to try to answer our original problem. The existence of a transient increase in the sensitivity of the signal to variations in the enzyme concentration has been clarified. This transient increase in the sensitivity may favor the detection at high substrate conversion levels (p > 0.5), of small reductions in the enzyme activity by the action of an inhibitor. However, due to the behavior of the sensitivity itself, the noise of the signal has also a transient increase that may obscure the detection of inhibition. How to describe the interplay of these factors?

Detection of inhibition in progress curves (Paper IV)

IDL-factor: Inhibition Detection Limit A model for the limit of detection of inhibition as a function of the substrate conversion level is needed to solve our original problem. With such a model, the substrate conversion level that optimizes the limit of detection can be identified. In order to develop this model we first need to define a variable for the extent of inhibition and a concept of detection.

A suitable definition of the extent of inhibition is the ratio between the in-trinsic strength of the inhibitor (Ki) and the inhibitor concentration ([I]), i.e.Ki /[I]. The higher this ratio, the weaker is the inhibition.

34

A total competitive inhibition is repre-sented in Scheme 3. Other assumptions included in this representation were de-scribed for the Scheme 2 and Eq.17 where no inhibitor was present, i.e the control of enzyme activity (pc). The mean model for the substrate conversion level in the pres-ence of inhibitor (pi) is,

Scheme 3

tekpKK

KI

pK oi

psii

ps 1ln1

1 [27]

This equation has been derived by assuming that the Ki of the inhibitor is higher than the enzyme concentration used in the assay. This may be untrue for tight binding inhibitors. However, the active inhibitor concentration commonly used in screening (high nM - low µM) is usually higher than the enzyme concentration (low nM), even for poorly soluble compounds, and this implies a very high inhibition that is easy to detect. Combining Eq.(27) and Eq.(17), a model for the Ki /[I] ratio results,

i

c

psic

ps

ii

ppKKppKK

pI

K

11ln11

1ln [28]

Restating our problem: in an end-point enzymatic assay, at which substrate conversion level of the enzyme activity control, pc, can the highest Ki/[I] be detected?

But, what is to detect? In a general sense, a given entity is detected if the signal it produces is unlikely to belong to the population of the reference of nullity. In the case of enzymatic assays used for primary screening of inhibi-tors, the reference of nullity is the control of enzyme activity. The popula-tion of enzyme activity controls is described by some probability distribu-tion. We have already gained some insight about this distribution, mainly regarding the way its standard deviation behaves as the reaction progresses (Eq.26). The standard deviation of the controls (sdp

c) is used to establish the signal cut-off. A compound associated to an assay signal beyond this thresh-

35

old is considered a “hit”. This signal cut-off (pidl) is the limit of detection and

it is generally set at three standard deviations from the controls (40, 70, 71),

cpci

dl sdpp 3 [29]

The probability that a signal beyond this limit belongs to the population of controls, pc, is dependent on the distribution function of the substrate conver-sion level. For a normal distribution, three standard deviations imply a prob-ability equal or lower than 1%.

Then, the limit of detection of the Ki/[I] ratio of an assay is obtained by substituting Eq.29 into Eq.28. This limit of detection is called the IDL-factor, i.e. Inhibition Detection Limit,

c

c

c

pc

c

pspps

pc

sdppKKsdKK

sdpIDL

311ln113

31ln [30]

The standard deviation of the enzyme activity controls (sdpc) was modeled

with Eq.26. Hereafter, we use the mechanistic expression (Eq.22) to model the transient component of this standard deviation function (Eq.26). Thus, the IDL-factor (Eq.30) is defined by 6 parameters ( c, p, t, R, K, Kps) and the substrate conversion level of the control, pc. The IDL function is a -shaped curve with a single maximum (IDLmax), the position of which repre-sents the substrate conversion level (pc

max) where the ability to detect com-petitive inhibition is maximized. Two typical examples of the behaviour of the IDL-factor as a function of the substrate conversion level are shown in Figure 5.

Figure 5: IDL as a function of pc for two hypo-thetical assays. In both cases K = 1 and Kps = 0.5. The quadruplet ( c, p, t, R) was (3.2 10-5;0.078; 0.019; 179) for the solid curve and (0.035; 0.013; 0.014; 0.016) for the dashed curve. The substrate conversion level, pc

max, at the maximal IDL-factor, IDLmax, is indicated by a vertical line for each curve.

Definition of assay populationsThe hypothetical assays, whose IDL functions are represented in Figure 5, are just two plausible instances. There are many others cases that may have a different behaviour. Thus, to reach any conclusion about the behaviour of the

36

IDL function for a generic end-point enzymatic assay, more cases should be analysed. One approach could be fully analytical by using differential calcu-lus tools to inquire the behaviour of the IDL-factor as a function of the 7 variables, i.e. c, p, t, R, K, Kps and pc (Eq.30). We instead preferred to sample several sextuplets of parameters ( c, p, t, R, K, Kps), and for each of them, to compute the IDL-factor as a function of pc in order to identify the value of the maximum (pc

max; IDLmax). Note, one sextuplet identifies one assay.

The model we are proposing here (Eq.30) has no precedent. Thus, we can just guess how a real end-point enzymatic assay looks like in terms of the sextuplet ( c, p, t, R, K, Kps), especially for the first four items which are conventions arising in this study. Any assay, and by extension any sextuplet, to be considered meaningful should be realistic and present a relevant qual-ity.

First, what is a realistic assay? A necessary condition for a realistic assay is physical possibility. There seems to be nothing physically impossible in the ranges of parameter values chosen for study (Table IV). Coefficient of variations (CV) like p and t were sampled from 1% to 15%. The CV of many automatic liquid dispensers is in the range from 1% to 5% (72, 73), but pipetting is just one of the sources of noise, albeit probably very important. Much lower values for c were allowed since the total signal windows of many assays can be much higher that the noise in the absence of analyte. Values of R were sampled in a range comprising from assays where the sig-nal changes relatively little from the beginning to the end of the reaction (e.g. fluorescence polarisation (33)) to assays with high relative signal change (e.g. scintillation proximity assay (29)).

Table IV: The parameter space sampled for the simulation of assays and the inclu-sion rule to ensure a relevant assay quality.

Parameter space Quality requirement -5 < log( c) < -0.8

-2 < log( p) < -0.8 -2 < log( t) < -0.8 R > 1: 0.3 < log(R) < 3.3 R < 1: -3.3 < log(R) < -0.3

1maxIDL

In order to give the same chance to all the values in the selected ranges, 20000 quadruplets ( c, p, t, R) were sampled in a logarithmic scale using a uniform pseudo-random distribution. From these randomly sampled 20000 quadruplets, one half had R > 1 and the other R < 1. Each of the quadruplets in these two sets of 10000 members was combined with a pair of values (K, Kps). K and Kps were not sampled randomly but regularly. That is, 21 values of K, from 0.1 to 10 regularly spaced (0.1) in a logarithmic scale were com-bined with just three values of Kps: 0, 1 and 10. These 63 different (K, Kps)

37

pairs were combined with each of the 10 000 quadruplets in each set, so that 126 assay populations, 10 000 members each, were finally obtained. These populations differed among themselves in the values of K, Kps and whether the signal increase (R > 1) or decrease (R > 1) with the reaction.

Second, what is a relevant assay for primary screening? The relevance of the assays was here defined by a quality requirement (Table IV). This inclu-sion rule means that only those assays with an optimal performance (IDLmax)able to detect compounds with Ki [I] were included for further considera-tion. Note that there is no guarantee that by measuring at pc

max such assays would detect an inhibitor with Ki [I]. Instead, a more adequate statistical interpretation is that: for an assay with a IDLmax 1, the probability of de-tecting a compound with Ki = [I] in a single measurement is at least 50%, provided that the distribution of this measurement is symmetric (e.g. nor-mally distributed) and that accurate estimates of the mean and the standard deviations of the controls are used (40). This probability increases with the number of replicates (74).

Detection limit and robustness in assay populations When the inclusion criteria (Table IV) was applied to the assay populations, the number of accepted assays was generally higher for R < 1 than for R > 1 assay types (Fig.6). This result is somewhat surprising since R > 1 assays are much more common than R < 1 assays.

Figure 6: The number of quality assays of the R >1 and R < 1 types, as a func-tion of log(K) at three differ-ent values of Kps (0, solid ; 1, dashed; 10 dotted).

For both assay types, the number of accepted assays decreased with the in-crease of K and Kps (Fig.6). This is a consequence of the corresponding in-crease in the competition that inhibitors face from substrate and/or products making inhibition weaker and detection harder. This competition impacts not only the quality of the maximum but also the distribution of pc

max (Fig.7).

38

Figure 7: Distribution of the substrate conversion level, pc

max, at the maximum of the IDL function, IDLmax, for 6 differ-ent assay populations, i.e. three different (K, Kps) pairs combined with two assay types, R >1 (solid line) and R < 1 (dashed line). In general, if the competition faced by the inhibitor fades as the reaction progresses (Kps < 1) the pc

max distribution is shifted to higher values of substrate conversion levels and the opposite if the competition increased (Kps > 1).

In all cases showed in Figure 7, the influence of the transient increase in the sensitivity of the substrate conversion level to variations in the active en-zyme concentration was present. The transient increase in the sensitivity “attracts” the pc

max distribution to the region of 50%-90% of substrate con-version in consonance with the position of the sensitivity maximum. This influence was clearer when the competition faced by the inhibitor was con-stant during the entire course of the reaction (Kps = 1) as shown in the middle panel of Figure 7.

One general feature of the pcmax distribution was its high dispersion (Fig.

7). This dispersion is due to the differences in the error structure parameters of the assays, ( c, p, t, R), which determines the way the signal noise varies during the reaction according to Eq.26. Then, to identify the substrate con-version level that maximizes the ability of a certain assay to detect inhibi-tion, it is essential to know the error structure parameters ( c, p, t, R) of the assay. If such knowledge were not available or easy to obtain, this depend-ence would limit the possibility of identifying the optimal substrate conver-sion level of the assay.

This dependence was relieved by analyzing the overall detection limit of an assay population since a population is defined just by K, Kps and assay type, i.e. R > 1 or R < 1. The first step in this analysis, was to normalize the IDL-factor of each assay respective its maximum performance in order to give the same weight to all members in the population,

maxIDLIDLnIDL [31]

nIDL is the normalized IDL-factor and it is also a function of pc and the set of parameters that characterize an assay. It is clear that the nIDL is equal to

39

or lower than 1. Each assay is described by a nIDL, so that a population of assays is also a population of nIDL. How does a population of nIDL behave as a function of pc? A minimal characterization of any population should include a measure of central position and a measure of dispersion. For cen-tral position we used the median (50th percentile) and for dispersion we used the 5th and the 95th percentiles; which encompass 90% of the population. In Figure 8, it is shown how these three percentiles behaved as a function of pc

for two assay populations that differ in the assay type (R > 1 or R < 1).

Figure 8: Median (solid) and 5th and 95th percentiles (dashed) of nIDL versus the substrate conversion level (pc) for two assay populations (both K = 1, Kps=1). The vertical lines indicate the substrate conver-sion level where the median reach a maximum (po

c)which is not exactly the same, but close to, where the dispersion reach a minimum. In this region of high median-low dispersion, the probability of near opti-mal performance for any assay is maximal.

From this representation (Fig.8) it was clear that there is a narrow region, identified by po

c, where the median of the population is very close to one, and the population is very low dispersed as judged from the closeness be-tween the 5th and the 95th percentiles. This means that by setting the controls of enzyme activity, pc, at this consensus substrate conversion level, po

c, it is maximized the probability of near optimal IDL for any assay in the popula-tion irrespective of c, p, t and the exact value of R. Furthermore, as the three percentiles change slowly around po

c it means that the robustness of the assays, i.e. the resilience of the performance to variations in pc, is also maximized. po

c is a function of K, Kps and the assay type (R > 1 or R < 1). This relation is shown in Figure 9. The position of po

c was beyond the initial rate region in all cases. Moreover, many assays, mainly of the R < 1 type, with a very poor performance in the initial velocity region becomes quality assays at po

c (Fig. 8).

Figure 9: The consensus substrate conversion level, po

c, for R >1 and R < 1 assay types, as a function of log(K) at three different values of Kps(0, solid ; 1, dashed; 10 dotted).

40

IDL-factor vs. Z’-factor According to the nomenclature used in this study, the Z’-factor (Eq.1) can be rewritten as,

cppc sdsdp

Z 031' [32]

where sdp=0 is the standard deviation of the substrate conversion level pc = 0. This function (Eq.32) increases monotonically as the pc increase, reaching its maximum value at the end of the reaction. If we follow the typical “the-higher-Z’-the-better” criterion for setting the control, it would be set at the end of the reaction (pc = 1), where no detection is possible. This means that in the Z’-guided assay optimisation some complementary criteria are re-quired. This may be relaxed by the realization that for many assays, mainly of the R > 1 type, the Z’-factor stabilizes at a near maximum value for sub-strate conversion levels much lower than 1 (not shown).

More important for the users of the Z’-factor is to be aware that, accord-ing to its relationship with the IDL-factor (Fig.10), the Z’-factor is not an unequivocal qualifier of the ability of an assay to detect inhibition. The rea-son behind this is the purely phenomenological conception of the Z’-factor, i.e. it does not make any assumption and does not give any clue about the underlying states of the system. As discerning such states is the main pur-pose of the screening (i.e. is there inhibition?) the IDL-factor would be more suitable than the Z’-factor as an indicator of the assay quality. Despite the advantages of the IDL-factor, it must be noted that its appropriate expression is dependent on the target/inhibitor system kinetics, while the Z’-factor is not. The Z’-factor is therefore a more general indicator.

Figure 10: Relationship between the Z’-factor and the IDL-factor at po

c for two assay populations (both K = 0.1, Kps= 1). Although there is a clear trend in the rela-tionship between the two variables, the data was a highly dispersed. The features were present in all assay populations. Assayswith Z’ = 0.3, which are usually classified as low performance, may have the same ability to detect inhibition (IDL) as assays with Z’ = 0.6, which are usually considered excellent (37). This situation is more sig-nificant for the assay type R < 1 than for R >1.

41

An answer. Its limitations The probability of detecting any competitive inhibition effect is maximized when the substrate conversion level of the controls of enzyme activity is beyond the initial velocity region, i.e. beyond 10% of substrate conversion into products. Furthermore, if the performance of an end-point assay is a disaster in the initial velocity range, don’t be saddened; there could be a radical improvement at higher substrate conversion levels. This is very likely for assays where the signal decrease as the reaction goes on. I have seen already some authors doing this (75-77). In fact, don’t be surprised if the much less popular R < 1 assays have, on average, a better performance that the common R > 1 type. The key is the substrate conversion level of the controls.

This consensus substrate conversion level, poc, does not depend on how

the dispersion of your measurements look like, as long as they are truly ran-dom, but just on the initial saturation level of the enzyme by the substrate ([S]o/KM) on the extent of product inhibition (KMs/KMp) and on whether the signal increase (R > 1) or decrease (R < 1) in the reaction. Besides, at po

c, the robustness of the detection ability to variations in the substrate conversion level of the controls is also maximized. You cannot locate po

c while optimiz-ing the assay based on the Z’-factor. Moreover, it may happen that an assay with a low Z’-factor (e.g. 0.3) is as good as an assay with a high Z’-factor (e.g. 0.6). For locating po

c you require the model for the IDL-factor. If you have a detailed knowledge on the error structure of the end-point assay ( c,

p, t), fine tuning of the performance can be achieved, again by using the model for the IDL-factor and computing the particular substrate conversion level, pc

max, for the maximal detecting ability, IDLmax.Now, if you set the enzyme activity controls beyond the initial velocity

region, the reaction will take longer and this may affect the productivity of your screening. For example, if the controls are set at 70% of substrate con-version it will take from 7 to 11 times longer than if you set them at 10%. This may be overcome by increasing the enzyme concentration and/or using a better substrate (higher kcat/KM).

Harder to answer is at which substrate conversion to set the controls if the free enzyme species (E) inactivates faster that the bound ones (ES, EP & EI) or if there are edge effects in the microplates and there is a poor randomiza-tion of controls and screened samples (70)? These two are real situations, and so far I have not an answer. Such an answer may be the goal of a future study.

42

Summary in Swedish

I levande organismer sker det ständigt en mängd av kemiska reaktioner. I detta avseende styrs vi av kemiska reaktioner. Många substanser, som vi kan kalla S, omvandlas till andra vi kan kalla P. Dessa omvandlingar sker medan jag skriver och du läser. För att röra mina fingrar, för att tänka, för att röra dina ögon, för att andas. Enzymer är proteiner som katalyserar (skyndar på) de här kemiska reaktionerna. Annars går de för långsamt för att vi ska kunna överleva. Det finns massor av enzymer i alla levande organismer, eftersom det behövs ett enzym för varje typ av kemisk reaktion. Många sjukdomar (t ex malaria) kan behandlas genom att man stänger av enzymer som katalyse-rar reaktioner som är kopplade till sjukdomen. Det här kan man göra genom att använda en enzymhämmare, en förening som interagerar med enzymet och förhindrar det att omvandla S till P. Således är också många bra läkeme-del enzymhämmare. Aspirin, penicillin och Viagra (just det!) är alla enzym-hämmare. Många forskare letar efter hämmare till vissa kritiska enzymer. De letar efter hämmare i stora bibliotek av naturliga och syntetiska föreningar. För att veta om ett visst ämne X, från ett sånt här bibliotek, fungerar som hämmare av ett enzym E, använder man sig av en enzymatisk analys. I en sån analys tittar man på hur substratet S omvandlas till produkten P med hjälp av enzymet E. Genom att mäta någon signal, t ex färg, som förändras när S omvandlas till P kan man se hur reaktionen framskrider. Om reaktio-nen går långsammare när X finns i blandningen kommer signalen vid en given tidpunkt att vara lägre än utan X. I så fall är X en hämmare till enzy-met. Även om effekten är liten kan den ändå vara väldigt betydelsefull. En-zymatiska analyser måste därför optimeras för att öka chansen att detektera hämning, även om den är svag. I avhandlingen har jag arbetat med frågan om optimering av såna analyser. Hur mycket substrat ska ha omvandlats till produkt när man jämför signalen från reaktionen med eller utan X?

Många olika faktorer spelar roll för skillnaden mellan reaktionskurvorna med och utan hämmare. En av dem är hur känslig signalen är för variationer i enzymkoncentrationen medan reaktionen fortskrider. Känsligheten visade sig börja på 0 och sen uppnå ett maximum när 60-70% av substratet om-vandlats och går sen ner till 0 igen mot slutet av reaktionen. En annan faktor är den konkurrens som blir mellan X och S respektive P. Om X binder till samma ställe på enzymet som S och P, kommer det att vara svårare för X att binda om S och/eller P binder väldigt hårt. Men eftersom S omvandlas till P med tiden, har det betydelse om det är S eller P som binder hårdare, eftersom

43

konkurrensen med X minskar beroende på om det är ett överskott av den starkare eller den svagare bindaren. Ytterligare en faktor i analysen är hur spridningen av signalen förändras under reaktionen. Ju högre spridningen är desto svårare blir det att detektera små skillnader mellan reaktionerna med, respektive utan, X.

För att ta hänsyn till alla dessa faktorer, har vi arbetat med olika matema-tiska ekvationer i det här arbetet. Är du rädd för dem? Jag frågar eftersom väldigt många är rädda för ekvationer. Tro mig när jag säger det: du behöver inte vara det. Ekvationerna i arbetet används som modeller för att beskriva och förutsäga resultaten från experiment, experiment som också är modeller av en mer komplex verklighet. Den här förenklade verkligheten kan beskri-vas med många ord, minst lika många som i föregående stycke. Men med en ekvation kan vi beskriva verkligheten med bara några symboler och den kan fortfarande ge en massa information och insikt! Vi har gjort det och utveck-lat en modell som vi kallar för IDL-faktorn. IDL står för Inhibition Detection Limit, alltså detektionsgräns för hämning. Den här modellen relaterar för-mågan att upptäcka hämning till graden av omvandling av S till P. Det här var ju precis det vi ville, eller hur?

Med hjälp av den här modellen kunde vi förutsäga att det alltid är opti-malt att göra jämförelsen med och utan X vid en punkt när en stor andel av substratet omvandlats till produkt. Det innebär alltså att mycket mer än 10% av substratet ska ha omvandlats. I nästan 100 år har alla mätningar i enzyma-tiska analyser utförts vid initial hastighet, dvs när 0-10% av S omvandlats. Om man vill kunna detektera svaga hämmande effekter av förening X är den gamla traditionen dock inte så bra. Den optimala substratomvandlingen kan vara så mycket som 70% när konkurrensen med substrat och produkt är kon-stant med tiden, vilket visar att känsligheten som beskrivits ovan är en viktig faktor. Modellen har också gett andra intressanta resultat. Till exempel är enzymatiska analyser där man tittar på hur en signal ökar när substratet om-vandlas väldigt vanliga och man brukar föredra dem framför analyser där signalen minskar. Våra resultat visar dock att de senare är att föredra om man ska jämföra reaktioner med och utan X vid den optimala nivån av sub-stratomvandling. Till sist kan vi också konstatera att den vanligaste kvalitets-indikatorn för analyser, Z’-faktorn, bland forskare som letar efter hämmare, bör användas med försiktighet. Även om den i allmänhet är mycket använd-bar visar vår modell att den ibland kan vara vilseledande.

Våra resultat kan bidra till en förändring av hur man utformar och optime-rar enzymatiska analyser för att identifiera hämmare. Forskare kanske arbe-tar utifrån en tradition som behöver förändras.

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Acknowledgements

There are people and plenty of unforgettable experiences I need to acknowl-edge. The reader must be aware that this is just a pale surrogate.

Al país y a la gente de donde yo vengo. Mis profesores y mis colegas en Cuba que viajando desde un sueño loco y bello hasta una realidad agotadora, han construido y mantienen con vida la enseñanza y la práctica de ciencia para todos. Que una riqueza como esta no sea demolida por la desesperanza o el fuego ciego de una venganza. Una cultura científica es tan disfrutable y productiva como una cultura musical. Por mi parte, sé de un compromiso que por ser mío y conmigo ha de ser cumplido a mi aire, no al del soldado que no soy.

A Bill Kong, Wichy, Lissi y otros por informales y enriquecedores. Professor Helena Danielson thanks for everything! For opening the doors

when I wrote to you three years ago. I was lucky and I came to Sweden where, strangely enough, I didn’t need air conditioning. Once here, you granted me independence, support, understanding and a strong push towards a commitment with intelligibility.

I’m grateful to “The Swedish Institute” and to the “Sven & Lilly Lawski Foundation” for the scholarships they have granted me.

Manon, Mauricio “El Jefe”, Alfre y Ale gracias por recogerme, cobijarme y llenar algunas de mis horas con paz y rabia (against the machine, of course!). Al Lo (siempre rico tenerte cerca, compa!), Migue, y Marianao, con el extra de los cómplices! Comrade Kristorsht Torshkrist, thanks for hitting hard the ball in the end of the 9th.

An early invitation to dance, and the commitment to a permanent low in-tensity embarrassment. From Russia, again, two times the passion for sci-ence. The painful beauty of a forest in winter, the snow-scooter (Yes!) and of the silent approach of the elder. The ultimately notion of measure expressed exquisitely in “si, si, pero solo un poco” warmly baked by four hands. The incarnation of sincere politeness. The pending intoxication in the woods just relieved with a dose of compassion. The sweet couple of believers, in their silence and their faith. A gentile invitation to the cinema. A practice of dif-ference: halving the motherhood of a child. The Xp7-003 agents from the reptilian headquarters. La cantadera de Silviadas a viva voz. Las gauchas salvadoras, que las quiero por gauchas y por salvadoras. The long time de-sired but just recently arrived happy noise from the ficka room. Real and dreamed lovers, thanks. Professors from Uppsala University and particularly

45

from the Department of Biochemistry and Organic Chemistry who let me know, in their lagom and talented Swedish way, a meaning of tradition.

All these and the daily unexplainable have happened to me here since I arrived almost three years ago on my 28th birthday. Sweden, where there is much more interest in making solidarity an institution than in making mar-ginality a crime.

A los de siempre hasta que la música deje de sonar: mi Pura del Alma y mi Puro, ejemplares y amorosos. A mis familias, la de Cuba, la redescubierta allá en el Norte, mi Alina con sus 10 999 virgenes que también son las mias y aquellos que se han colado por las puertas del amor. Por cierto, desde ya, a Mariana, que seguramente querrá lo que quiere Mariana, y tambien querrá tener por buen rato a la Madrecupulistrepondrandilis, asi que pa’lante coño! A mis pocos buenos amigos que merodean mi ser en la distancia.

A lo vital que se entierra, nace, y destierra con Leticia, la hembra más dulce del Universo.

And finally, to my beloved Lord, The Flying Spaghetti Monster (FSM), for the inspiration in the pursuit of intricate excuses to live for.

46

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Acta Universitatis UpsaliensisDigital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 184

Editor: The Dean of the Faculty of Science and Technology

A doctoral dissertation from the Faculty of Science andTechnology, Uppsala University, is usually a summary of anumber of papers. A few copies of the complete dissertationare kept at major Swedish research libraries, while thesummary alone is distributed internationally through theseries Digital Comprehensive Summaries of UppsalaDissertations from the Faculty of Science and Technology.(Prior to January, 2005, the series was published under thetitle “Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology”.)

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