Poisson Indicator and Fano Factor for Probing Dynamic Disorder in

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Poisson Indicator and Fano Factor for Probing Dynamic Disorder in Single Molecule Enzyme Inhibition Kinetics Srabanti Chaudhury J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/jp506141v • Publication Date (Web): 14 Aug 2014 Downloaded from http://pubs.acs.org on August 16, 2014

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The Journal of Physical Chemistry

Poisson Indicator and Fano Factor for Probing Dynamic Disorder in Single Molecule Enzyme Inhibition Kinetics

Srabanti Chaudhury* Department of Chemistry, Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pune 411008, Maharashtra, India

Abstract: We consider a generic stochastic model to describe the kinetics of single molecule enzyme inhibition reactions in which the turnover events correspond to conversion of substrate into a product by a single enzyme molecule in the presence of an inhibitor. We observe that slow fluctuations between the active and inhibited state of the enzyme or the enzyme substrate complex can induce dynamic disorder which is manifested in the measurement of the Poisson indicator and the Fano factor as functions of substrate concentrations for different inhibition reactions. For a single enzyme molecule inhibited by the product we derive a single-molecule Michaelis-Menten equation for the reaction rate, which shows a dependence on the substrate concentration similar to the ensemble enzymatic catalysis rate as obtained from bulk experimental results. The measurement of Fano factor is shown to be able to discriminate reactions following different inhibition mechanisms and also extract kinetic rates.

Keywords: enzyme inhibition, first passage time distribution, dynamic disorder

*email address: [email protected], Phone number: 912025908140 1

ACS Paragon Plus Environment


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I. Introduction The study of enzyme kinetics utilizing single-molecule fluorescence microscopy has enabled detection of many interesting aspects of enzymatic activity that usually remain hidden when probed at the ensemble level.1-3

Recent experiments of single molecule detection of a

flourogenic product have revealed that the distribution of turnover time is exponential at low substrate concentrations and non- exponential at high substrate concentrations.4 Such multi exponentiality leads to temporal fluctuations in the catalytic rates, a phenomenon known as dynamic disorder2. Single molecule kinetic theories have suggested that the dynamic disorder is a manifestation of multiple competing time scales in the reaction. In order to rationalize the phenomena of dynamic disorder, Xie and coworkers have proposed a n- state discrete model of single molecule Michaelis Menten(MM) kinetics. In this model, a free enzyme and the enzyme substrate complex can exist in any one of the n conformers given by E1, E2, . . . , En and ES1, ES2, . . . ESn)5. If the conformational fluctuations between the different ES conformers is slow or comparable to the rate of product formation, then the

turnover time distribution follows

Poissonian statistics at low substrate concentration and non –Poissonian statistics at high substrate concentrations. Also, it has been shown that for a simple single molecule enzymatic the rate equation is similar to ensemble MM equation even in the presence of dynamic disorder.4,5 The catalytic activity of a simple enzymatic reaction can be inhibited by a substance which inhibits the rate of the enzyme catalyzed reaction.6 Many drugs act as enzyme inhibitors and play an important role in the regulation of metabolism. For example, an inhibitor can mask the activity of an enzyme and can kill a pathogen thereby acting as a drug. Inhibition reactions can be classified into many types namely competitive inhibition, uncompetitive inhibition and non competitive inhibition.6 For example, the inhibitor can bind reversibly to the substrate to 2


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form an enzyme inhibitor complex which can prevent substrate binding. This type of inhibition is known as competitive inhibition as shown in Figure 1a where the substrate and the inhibitor competes for the same binding site because of their structural resemblance or any conformational change in the enzyme that may prevent substrate binding or vise versa. The steady state MMkinetics for this reaction mechanism is6 v=

v max [S ]  [I ]   + [S ] K s 1 + K i  

where vmax is the maximum velocity in the absence of the inhibitor [I], K s =

3


(1)

[S ][E ] , K = [I ][E ] . [ES ] i [EI ]


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Figure 1: Reaction schemes for (a) competitive inhibition (b) uncompetitive inhibition (c) uncompetitive inhibition by product

An inhibitor can also bind to the enzyme substrate complex to form ESI complex that is catalytically inactive. This takes place when a conformational change in the enzyme occurs after substrate binding which favors the binding of the inhibitor to the enzyme substrate complex ES as shown in Figure 1b. Such type of inhibition is known as uncompetitive inhibition and the kinetics is described as6 v=

v max [S ]  [I ]   K s + [S ]1 +  Ki 

(2)

In non competitive inhibition the inhibitor can bind simultaneously to the substrate and the ES complex. In this work we consider such different types of inhibition mechanisms and ask the following question: Do the kinetics of inhibited reaction show dynamic disorder at the single molecule level? To address this question theoretically, one need to analyze the temporal behaviors of the catalytic activity of a single enzyme. The most accessible characteristics of reactivity fluctuations in individual enzyme molecules relate to the second moment of turnover time statistics and is defined by the randomness parameter R which is given as R =

t2 − t t

2

2

.

In the context of single enzyme kinetics, t is the time between successive catalytic turnover events and averaging is over a large number of such observed single molecule events. When the waiting time distribution is a single exponential decay function, then randomness parameter R is equal to unity. Any deviation of R from this predicted value is an indication of dynamic disorder, 4


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i.e., temporal fluctuations of reaction rates. It has been shown both experimentally and theoretically that for single molecule enzymatic reactions in the absence of any inhibitors, the randomness parameter is greater than unity at high substrate concentration.4,7 The randomness parameter of single-turnover waiting times serves as a measure of temporal fluctuations for single molecule Au- nanoparticle catalysis and connections have been made to experimental data.8 The randomness parameter has also been calculated for periodic sequential kinetic schemes with arbitrary forward and backward periodic rate constants that can serve as a model of molecular motor.9-10 In the context of inhibition reaction, Dua et al. proposed a chemical master equation approach11 for the time evolution of the joint probability distributions of the free enzyme, the enzyme substrate, the enzyme inhibitor and the enzyme substrate inhibitor complex. Using suitable initial conditions, these coupled equations for the transition probabilities can be solved to obtain expressions for the waiting time distribution for the enzymatic turnovers. From the first and the second moments of the waiting time distribution, the mean waiting time and the randomness parameter were calculated for different types of inhibitions. If f(t) is the probability density function of the waiting time, then

∞

t = ∫ tf (t )dt and 0

∞

t 2 = ∫ t 2 f (t )dt . Since the 0

expressions for the waiting time distributions for inhibition reactions are quite cumbersome, these equations are solved numerically to obtain the mean time and the randomness parameter for both competitive as well as uncompetitive inhibition reactions.11 Such numerical solution requires estimates of all the transition rates between different possible states. Also the master equations become more and more complicated to solve analytically if we have reactions that can involve more than one intermediate state. In order to avoid such complications, we have

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considered here a simple analytical model of single enzyme kinetics based on the first passage time distribution between successive catalytic turnover events. This model was introduced earlier by us12 to study solute transport through an ion channel and derive expressions for the first passage time distribution and the Poisson indicator(P) as defined as

P=

t2 − 2 t t

2

(3)

2

The Poisson indicator is closely related to the randomness parameter and serves as a measure of dynamic disorder in single molecule MM reactions5. If the statistics of turnover times are

(

exponential then variance t 2 − t

2

) is equal to the square of the mean and P = 0 . In the context

of ion channel transport kinetics the mean flux has the same dependence on the solute concentration irrespective of the number of internal states with the channel. But the Poisson indicator has a different qualitative dependence on the solute concentration for a channel with one internal state (analogous to simple MM mechanism) and that for a channel with many internal states. Also for a channel with more than one internal state, P has simple generic functional dependence on solute concentration irrespective of the number of internal states.12 In addition to Poisson indicator, we also focus on another important characteristic of molecular fluctuations namely the Fano factor defined by F=

n2 − n

2

n

(4)

where n is the total number of enzymatic turnover events during a specified measurement time interval. If the statistics of turnover events is Poissonian, then F= 1. In the context of photon statistics, the Fano factor and the Poisson indicator are related to the Mandel’s parameter which

6


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describes the bunching and antibunching of emitted photons.13,14 Functional dependencies of the Poisson indicator and the Fano factor on the substrate concentration, [S], can provide valuable information about the nature of the inhibition reaction and the temporal fluctuations in reaction rates(dynamic disorder). This paper is organized as follows. In Section II, we discuss about our kinetic model and calculate the first passage time distribution for single enzyme kinetics. In Section III, we derive expressions for the first passage time distribution and the Poisson indicator for both competitive and uncompetitive inhibition reaction mechanisms. We discuss about a single molecule experiment where individual enzyme molecules are trapped in lipid bilayer vesicles and the fluorogenic product itself results in allosteric inhibition. Here we employ our theoretical model to compare with bulk experimental data and test the Michaelis-Menten equation at the singlemolecule level in the presence of an inhibitor. In section IV we discuss about the event averaged probability distribution and its relationship with Fano factor. We calculate the Fano factor for both types of inhibition reactions. Finally, in Section V, we summarize our results.

II. First passage time distribution and Poisson indicator A key measurable quantity in single molecule experiments is the time distribution function φ (t ) between successive single molecule events (first passage time distribution) in renewal processes.15 For example, in enzymatic reactions, this key quantity is the probability distribution function (PDF) of catalytic turnover events which are referred to as monitored

()

transitions. More precisely, given the moment of one monitored transition, φ t dt is the probability of observing the next monitored transition between time t and t+dt after this time

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moment. In this article, we will assume that monitored transitions correspond to events when product molecules are formed with regeneration of the free enzyme. Such events are actually detectable in single molecule fluorescence experiments. For example in the single molecule experiment on the catalytic activity of the enzyme beta galactosidase, each catalytic turnover event corresponding to a monitored transition is characterized by a burst of fluorescence from the fluorogenic product.4 Let us consider a simple enzymatic reaction,

in which the substrate S is converted into product P via an intermediate complex ES that the substrate creates with enzyme molecule E. It is generally assumed that the kinetic rate for binding of the substrate to the free enzyme is proportional to the substrate concentration [S]. Let kE1[S] be the rate for binding of the substrate S to the free enzyme E, with constant parameter kE1 independent of [S]. In correspondence to this process, we introduce the probability per unit time,

QE1 (t ) of the event that at time t, after the product is formed and the enzyme is regenerated, a new solute molecule binds to the enzyme for the first time to form the ES complex. Explicitly,

QE1 (t ) is exponentially distributed: QE1 (t ) = k E1 [S ]exp(− tk E1 [S ])

(5)

This type of probability is known as the waiting time distribution function. Cao and Silbey have proposed the self consistent pathway formalism to derive expressions for the waiting time distribution in terms of the elementary kinetic rates for simple kinetic models.16 A generalized

8


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model of chemical kinetics for the non-Poisson enzyme reaction was proposed by Sung et al. and was applied to a more complex reaction networks.17 Let QR (t ) be the probability per unit time of the event that the enzyme substrate complex will form the product at time t to the right, i.e. by making a monitored transition. We assume that all elementary reactions, except the monitored ones are, in principle, reversible, so there is also a finite probability per unit time, QL (t ) , that the enzyme substrate complex will dissociate to the substrate and the enzyme at time t without making the monitored transition. In both cases, after the free enzyme is regenerated the process renews. Now the probability of having a monitored transition at time t is equal to the probability to create the ES complex at time t', during time interval dt, times the probability to leave the complex after some time t- t'. As shown in Appendix A the probability φ (t ) is given by

φ (s ) =

QE1 (s )QR (s ) 1 − QE1 (s )QL (s )

(6) ∞

where φ (s ) is the Laplace transform of φ ( t ) , i.e. by φ ( s ) = ∫ e − stφ ( t ) dt . 0

This is a formal solution because only the functional form of QE1 (s ) is known and is given by

Q E 1 (s ) =

k E1 [S ] s + k E1 [S ]

(7)

while the explicit forms of QL (s ) and QR (s ) are actually not needed as neither of them depend on the concentration of the substrate [S]. Substituting QE1 (s ) from eq 7 into eq 6 and taking the derivative of eq 6, t = ( −1) lim ∂φ ( s ) ∂s , we obtain the average first passage time s →0

9



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t =

A + B, S   

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(8)

where A = 1 k E1

(9)

B = − (b + c ) a

where a = QR (0),1 − a = QL (0) , b and c are constants that are the first derivatives of QR (0) and

QR (0) respectively, at s= 0. Eq. 8 is nothing but the Michaelis- Menten equation where t

−1

is equal to reaction velocity for

ensemble measurements. One can also obtain the higher moments of the first passage time ∂ n φ (s ) which cannot be inferred from deterministic methods. Finally s → 0 ∂s n

distribution t n = (− 1) lim n

the Poisson indicator for a simple MM enzyme reaction can be obtained using eq 3 as shown in Appendix 2 and is given by the following functional form P=

− ρ [S ]

(ς + [S ])2

(10)

where

ρ=

k2 k E1

k + k2 ς = 1E k E1

(11)

III. Single molecule inhibition kinetics Using the above methodology we can calculate the first passage time distribution for single molecule enzyme inhibition reactions. We first consider the case of competitive inhibition where the enzyme can bind reversibly either to the substrate S or to the inhibitor I to form ES or EI 10


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complexes respectively as shown in Figure 1a. The substrate binding is followed by the release of the product and the free enzyme is regenerated. Following eq 6 the first passage time distribution function for this reaction motif is given by

φ (s ) =

Q R (s )Q E1 1 1 − Q 1L (s )Q E1 1 − Q L2 (s )Q E21

(12)

where QE1 1 (s ) and QE21 (s ) describe the transitions from the E to ES and E to EI respectively, and depend on the concentration of the substrate and the inhibitor

QE1 1 (s ) =

k3 [I ] k1 [ S ] , QE21 (s ) = s + k1 [S ] + k 3 [I ] s + k1 [S ] + k 3 [I ]

(13)

QL1 (s ) and QL2 (s ) are the backward transition rates from the complex state ES and EI to the

empty state E respectively, and QR (s ) describes the monitored transition from ES to P respectively ∞

Using the relation t

n

= ∫ dtt nφ ( t ) = ( −1) lim ∂ nφ ( s ) ∂s n , we calculate the first and second n

s →0

0

moments and also the Poisson indicator given by eq 3

tcompetitive =

[ S ] + A + B[ I ] C[ S ]

(14)

and

Pcompetitive =

[S ](− α + β [I ]) (B + [S ] + A[I ])2

11


(15)


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The inhibitor can also bind to the ES complex instead of the free enzyme to form a complex ESI which is catalytically inactive (uncompetitive inhibition). For this reaction motif, the first passage time distribution function is given by

φ (s ) =

QR (s )QE1 (s ) 1 − QL (s )QE1 − Q23 (s )Q32 (s )

(16)

where QR (s ) describes the monitored transition from ES to P, QE1 (s ) and Q23 (s ) describe the transitions from the state E to ES and the state ES to ESI respectively, were only QE1 (s ) depend on the [S], QE1 (s ) =

k1 [ S ] , QL (s ) and Q32 (s ) are the backward transition rates from the s + k1 [S ]

intermediate state ES to E and ESI to ES respectively. Using eq 16 we calculate the mean reaction velocity and the Poisson indicator in the case of uncompetitive inhibition as shown in Appendix A2 and these are given by

tuncompetitive =

[S ] + [I ][S ]η + µ C [S ]

(17)

Puncompetitive =

[S ](− κ − [I ](q − λ [S ])) (µ + [S ](1 + η [I ]))2

(18)

From eq 14 and eq 17 we find that the mean first passage time shows the same dependence on the substrate concentration for both competitive and uncompetitive inhibition schemes. Hence the mean dynamics is not sufficient to distinguish between the two reaction schemes. But the Poisson indicator has a different functional dependence on [S] at a given inhibitor concentration as shown in eq 16 and 18. Figure 2 shows P as a function of [S] for both competitive and uncompetitive inhibition for a given inhibitor concentration calculated using eq 15 and eq 18 respectively. For both the inhibition schemes, the Poisson indicator is a non12


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monotonous function of the substrate concentration and is not equal to zero at intermediate substrate concentrations. At low substrate concentrations, as obtained from eq 15 and eq 18, the Poisson indicator vanishes for both the inhibition schemes. For competitive inhibition mechanism, the Poisson indicator goes to zero even at large substrate concentrations, indicating that the turnover statistics of a single-enzyme at high [S] is governed by a mono-exponential distribution with a characteristic decay constant. In a recent experiment, where the inhibition kinetics of single-β-galactosidase in the presence of the inhibitor D-galactal and the substrate resorufin-β-D-galactopyranoside was studied using fluorescence spectroscopy18 it was found that for competitive inhibition the autocorrelation functions for single-enzyme substrate turnovers showed single exponential decay. This is in contrast to previously reported fluorescence spectroscopy measurements where the distribution of the time between successive catalytic turnover events is multi exponential at high substrate concentrations due to protein conformational fluctuations occurring at timescales comparable to reaction times.4 These findings suggest that dynamic disorder in single molecule inhibition kinetics is due to the presence of slow fluctuations between the active and inhibited state of the enzyme and not due to protein conformational fluctuations. If [I] = 0, for both competitive and uncompetitive mechanisms the expression for Poisson indicator is equivalent to eq 10 for a simple MichaelisMenten reaction. Similar to inhibition kinetics reported here, in the context of motor proteins, the variation of the randomness parameter as a function of the applied force is different for the basic kinetic scheme, the kinetic model with side branches at each site, and the model with arbitrary death rates present at each site.9-10

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Figure 2: (a) Poisson indicator P as a function of substrate concentration [S] for competitive inhibition at β= 8, [I] = 20 (dashed line), β= 5, [I] = 20 (dotted line) and β= 8, [I] =0 (solid line) (b) Poisson indicator P as a function of substrate concentration [S] for uncompetitive inhibition at (a) λ= 10.8, [I] = 20 (dashed line) (b) λ= 2.8, [I] = 20 (dotted line) (c) λ= 2.8, [I] =0 (solid line)

Inhibition of enzymatic reactions by product molecules has also been observed experimentally at the single molecule level. Individual HRP enzyme molecules encapsulated within lipid membrane catalyzes the oxidation of the nonfluorescent substrate Amplex Red in the presence of hydrogen peroxide to form the fluorescent product resorufin that remains trapped in the vesicle interior.19 The generation of resorufin product by individual enzyme molecules is obtained by monitoring the increase in fluorescence signal as the reaction progressed. Several control experiments were performed and it was confirmed that the product molecules allosterically inhibit the individual enzyme molecules. A series of bulk enzymology experiments were performed, each with the same concentration of substrate and at varying H2O2 14


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concentrations. The concentration of the product was fixed from 0 to 15 µM for each series. The Eadie- Hofstee plot of the initial velocity v as a function of v/[S],6 shows non- competitive inhibition by the product itself where the product can bind simultaneously to the free enzyme and the ES complex.6 But within the vesicles, the substrate is always present in excess compared to the enzyme, so the free enzyme species can be effectively ignored and it can be assumed that the product will only bind to the ES to form a ESP complex as shown in Figure 1c. Using our theoretical formalism, for this reaction motif the first passage time distribution is given by eq 16 where Q23 (s ) and Q32 (s ) describe the forward and backward transitions from the state ES to ESP and ESP to ES respectively and the single molecule reaction rate is given as

t

−1

=

A′[S ] B ′ + [S ](1 + C ′[P ])

(19)

where A’, B’, and C’ are constants. This single molecule rate equation resembles the ensemble averaged MM kinetics for uncompetitive inhibition by product. Figure 3 shows a comparison of v vs v/[S] at different product concentration for the bulk experiments and our model. With proper choice of kinetic parameters, we get good agreement between the bulk and our single molecule results. Thus the ensemble reaction kinetics for inhibition reaction holds good even at the single molecule level.

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Figure 3: Bulk kinetic measurements of HRP enzymatic activity in the presence of resorufin. The initial enzymatic velocity is measured in the presence of varying H2O2 concentrations [S] and a fixed initial concentration of the product resorufin at [P] =0 µM (circles), [P]= 3µM (squares), [P] = 9µM (stars) and [P]= 15µM (triangles). Solid lines are fit to experimental data points using eq 19 for different A’ values at B’= 1.3 and C’= 0.13

IV. Event average distribution function and Fano factor In this section we would be considering another type of single molecule measurement which is the probability distribution for the number of catalytic turnover events within a time bin.20,21 This is also known as the event average probability distribution22 and is related to the fluorescence intensity distribution. As discussed in reference 12, a convenient way to study this distribution is the generating function method23,24 ∞

Z (χ , t ) = ∑ Pn (t )e χn = e ω ( χ )

(20)

n =0

where Pn (t ) is the probability distribution of obtaining n number of turnover events occurring in time t after many repetitions of the measurement, and in the Laplace domain it is given by 16


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Pn (s ) =

1 − φ (s ) n φ (s ) s

(21)

χ is called the counting parameter, ω ( χ ) is the cumulant generating function. Its derivatives with respect to χ give the cumulants of the distribution Pn, such as the mean n and the variance

σ2: n =

∂ω (χ ) ∂χ

2 2 − n χ =0 , σ = n

2

=

∂ 2ω (χ ) ∂χ 2

(22)

χ =0

The Fano factor is defined to be the ratio of the variance to the mean, i.e. F=

σ2 n

(23)

Recently, the Fano factor was studied by Mugler et al.25 for general enzymatic reactions with two internal states. It was shown that measuring the Fano factor as a function of substrate concentration could distinguish among all possible two-state enzymatic kinetics reactions. In this work we explore whether measurements of the Fano factor can differentiate among different inhibition reaction mechanisms. In order to achieve this, we find a relation between the turnover time probability distribution ϕ(t) and the cumulant generating function and then express the Fano factor in terms of derivatives of ϕ (t) such that 12

 ∂φ s −  ∂s ∂s 2  s=0 F= 2  ∂φ s     ∂s   s=0 

()

()

∂2φ s

()

   s=0 

2

. (24)

Using eq 24 in eq 12 and eq 16, the Fano factor for competitive and uncompetitive inhibition schemes as shown in Figure 1a and 1b are obtained as 17



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Fcompetitive = 1 +

[S ](β [I ] − α ) (B + [S ] + A[I ])2

Funcompetitive = 1 +

where δ =

[S ](δ [S ][I ] − κ − γ [I ]) (µ + [S ](1 + η [I ]))2

Page 18 of 30

(25)

(26)

2k 3 (k 2 + k −3 ) 2k (1 + k 2 ) ,γ = 3 and A, B, α, β , κ, µ, η are defined in Appendix B. 2 k1 k − 3 k −3

At saturating substrate concentration, that Fcompetitiv e (S → ∞ ) = 1 and Funcompetit ive (S → ∞ ) = 1 + ∆ For a simple MM reaction, ([I]= 0) FMM = 1 −

2 ρ [S ]

(ζ + [S ])2

(27)

where ρ and ς are defined in eq 11. Since different reaction mechanisms have different statistical properties our measurement of Fano factor can be used to differentiate inhibition reaction mechanisms based on experimental observations. Figure 4 shows the differences in the Fano factor curves among the two inhibition schemes and the simple MM reaction. We find that the saturation value of F at high substrate concentration is equal to one for competitive inhibition and greater than one for the uncompetitive reaction scheme. This clearly demonstrates that estimation of Fano factor can be effectively utilized to qualitatively differentiate between the two types of inhibition reaction schemes.

18


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Figure 4: Fano factor F as a function of substrate concentration [S] for competitive inhibition (dashed line), uncompetitive inhibition (dotted line) and simple MM kinetics (solid line). Parameter values are A= 1, B= 1, α= 10, β=10, µ= 1, η= 1, δ = 10, κ = 10, γ= 1

Apart from differentiating reaction schemes, the plot for the Fano factor versus substrate concentration can be used to extract the kinetic rates. For example, for a simple MM reaction, all the three kinetic rates can be extracted from the qualitative plots of the reaction rate as a function of [S] and the minimum value of the Fano factor. For more complex reaction mechanisms, as shown in Figure 1a and 1b one can use the analytical expressions as given in eq 27 and eq 28 and fit to the experimental data. However, in such complex reaction schemes, all rates may not be determined individually, but rather as a combination from the measurement of the reaction rate and Fano factor. Nonetheless, calculating Fano factor can provide useful direct correlation with experimental data which can be beneficially exploited for rationalizing mechanistic aspects of enzymatic reactions.

19



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V. Conclusion Noise can lead to important consequences in biological systems. Noise plays an important role in understanding a reaction mechanism.26-28 In this work, we explore the role of noise in enzymatic inhibition reactions. We have used our generic model based on the first passage time distribution formalism to calculate the characteristics of molecular fluctuations in inhibition reactions. In the presence of inhibitors the Poisson indicator is greater than zero at intermediate to high substrate concentrations. Our results indicate that dynamic disorder may arise due to the fluctuations between the active and inhibited state of the enzyme or enzyme substrate complex on timescales which are longer or comparable to the time scales of the reaction. Also, we find that the Poisson indicator is equal to zero at high substrate concentrations for competitive inhibition. This suggests that protein conformational fluctuations which are a potential source of dynamic disorder in single molecule MM reaction may not affect the catalytic rate in the presence of inhibitors. A comparison of our theoretical results with bulk experimental data also verifies that the single molecule rate equation for inhibition kinetics is similar to the ensemble enzymatic turnover rate for the case when the enzyme is inhibited by the product. We also calculate the Fano factor and show that the measurement of the Fano factor can provide a qualitative distinction between competitive and uncompetitive inhibition reaction mechanisms. Thus, one can compare the calculated Fano factor with the experimental data and identify the nature of the inhibition reaction. The analytical expressions for the Poisson indicator and the Fano factor as a function of substrate concentration can be used to make connections between experimental data and theoretical models. This theoretical formalism can also be applied to the study of motor proteins. Similar to enzymatic reactions, stochastic models of molecular motors involve a number of sequential kinetic steps that depend on the concentration of the substrate and the 20


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applied mechanical force. Currently, we are extending our theoretical formalism to include the effects of protein conformational fluctuations which can lead to dynamic disorder at high substrate concentration along with the interconversions between different active and inhibited state of enzyme conformers.

Acknowledgement The author acknowledges Prof. Gilad Haran at the Weizmann Institute of Science for sharing their experimental data. S. C acknowledges support from IISER Pune.

Appendix A Consider a generic model of enzymatic reaction.

It consists of a sequence of reversible transitions between states and a final irreversible transition to form the product P. The first reversible step corresponds to the binding of the substrate S to the free enzyme E and this is the only step that depends on the substrate concentration. The first passage time distribution in the Laplace domain can be written as

φ (s ) =

Q1P (s ) 1 − Q11 (s )

(1)

where Qij (s ) is the notation for the waiting time distribution corresponding to the transition from state i to state j in the Laplace space. Here Q1P (s ) is the waiting time distribution for the transition from state 1 to the product, and Q11 (s ) represents the waiting time distribution for the transition from and back to state 1. Here state 1 represents the free enzyme. Eq. A1 can be written as an infinite sum 21



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(

φ (s ) = Q1P (s ) 1 + Q11 (s ) + Q11 (s )2 + ....

)

Page 22 of 30

(2)

The first term in the summation corresponds to the transition from state 1 to product P without returning to state 1, the second term corresponds to the transition from state 1 to product P and returning to state 1 exactly once, and so on. For a simple Michaelis- Menten enzymatic reaction, where we will consider only one intermediate state,

Q1P (s ) = QE1 (s )QR (s ), Q11 (s ) = QE1 (s )QL (s )

(3)

Substituting this in eq A1 we have

φ (s ) =

QE1 (s )QR (s ) 1 − QE1 (s )QL (s )

(4)

Appendix B Example: Michaelis-Menten kinetics Let us consider a MM enzyme catalytic reaction

For this simple reaction, the waiting time distributions can be written as

Q E1 (s ) =

k E1 [S ] s + k E1 [S ]

Q L (s ) =

k −1 s + k −1 + k 2

Q R (s ) =

k2 s + k −1 + k 2

(1)

Substituting eq B1 in eq 6 we obtain 22


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φ (s) =

k 2 k E1 [ S ]

sk1E + ( k2 + s ) ( k E1 [ S ] + s )

,

(2)

and the Poisson indicator is given by

P=−

2k2  S  k +k  k E1  2 1E +  S   k E1 

2

.

(3)

The waiting time distributions in eq 12 for Figure 1a is given by

Q 1L (s ) =

k −1 s + k −1 + k 2

Q L2 (s ) =

k −3 s + k −3

Q R (s ) =

k2 s + k −1 + k 2

(4)

Substituting eq B4 in eq 11 and eq 12 we have

[S ] + [I ] k 3 (k −1 + k 2 ) + (k −1 + k 2 ) k1 k − 3 k 2 [S ]

t competitive =

k1

(5)

[S ] [I ] 2k 2 k 3 (k −1 + k2 2 − k −3 ) − 2k 2  

Pcompetitive =

With

A=





k1 k − 3

k1 

 k 3 (k −1 + k 2 ) (k −1 + k 2 )   [I ] + + [S ] k1 k − 3 k1  

2

(6)

k3 (k −1 + k 2 ) (k + k ) 2k 2k k (k + k − k ) , B = −1 2 , C = k 2 , α = 2 , β = 2 3 −1 2 2 −3 we arrive at the k1k −3 k1 k1 k1k −3

functional forms for tcompetitive and Pcompetitiv e given in eq 14 and eq 15 respectively. The waiting time distributions in eq 16 for Figure 1b is given by 23



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Q 1E (s ) =

Page 24 of 30

k1 [S ] s + k1 [S ]

Q23 (s ) =

k 3 [I ] s + k −1 + k 2 + k 3 [I ]

Q L (s ) =

k −1 s + k −1 + k 2 + k 3 [I ]

Q3 2 (s ) =

k −3 s + k −3

Q R (s ) =

k2 s + k −1 + k 2 + k 3 [I ]

(7)

Substituting eq B7 in eq 16 we have

[S ] + [I ][S ] k 3



Substituting κ =

(k −1 + k 2 )

k −3 k 2 [S ]

t uncompetitive =

Puncompetitive =

+

[S ] [I ][S ] 2k 2 k2 3 − 2k 2

k1

−

(8)

2k 2 k 3  [I ] k1 k − 3 

k1 k −3    k (k + k 2 )  [I ][S ] 3 + −1 + [S ] k −3 k1  

(9)

2k k 2k k k 2k 2 (k + k 2 ) in eq B9 we arrive , C = k 2 , q = 2 3 , λ = 2 2 3 ,η = 3 , µ = −1 k1 k1 k −3 k −3 k1 k −3

at the functional forms of eq 17 and 18.

24


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References 1. Lu, H. P.; Xun, L.; Xie, X. S., Single-molecule enzymatic dynamics. Science 1998, 282, 1877–1882 2. van Oijen, A. M.; Blainey, P. C.; Crampton, D. J.; Richardson, C. C.; Ellenberger, T.; Xie, X. S., Single-molecule kinetics of γ exonuclease reveal base dependence and dynamic disorder. Science 2003, 301, 1235–1239. 3. Zhang, Y.; Song, P.; Fu, Q.; Ruan, M.; Xu, W., Single-molecule chemical reaction reveals molecular reaction kinetics and dynamics. Nat. Comm. 2014, 5, 4238. 4. English, B. P.; Min, W.; van Oijen, A. M.; Lee, K. T.; Luo, G.; Sun, H.; Cherayil, B. J.; Kou, S. C.; Xie, X. S., Ever-fluctuating single enzyme molecules: Michaelis-Menten equation revisited. Nat Chem Biol 2006, 2, 87-92. 5. Kou, S. C.; Cherayil, B. J.; Min, W.; English, B. P.; Xie, X. S., Single-Molecule Michaelis-Menten Equations. J. Phys. Chem. B 2005, 109, 19068-19081. 6. Segel, I. H., Enzyme Kinetics: Behavior and Analysis of Rapid Equilibrium and Steady State Enzyme Systems. Wiley: New York, 1993. 7. Yang, Y.; Cao, J.; Silbey, R. J.; Sung, J., Quantitative Interpretation of the Randomness in Single Enzyme Turnover Times. Biophys J. 2011, 101, 519–524 8. Xu, W.; J.S., K.; Chen, P., Probing the catalytic activity and heterogeneity of Aunanoparticles at the single-molecule level. Phys. Chem. Chem. Phys. 2009, 11, 2767–2778. 9. Derrida, B., Velocity and diffusion constant of a periodic one-dimensional hopping model J. Stat. Phys. 1983, 31, 433-450. 10. Kolomeisky, A. B.; Fisher, M. E., Periodic sequential kinetic models with jumping, branching and deaths. Physica A 2000, 279, 1-20. 11. Saha, S.; Sinha, S.; Dua, A., Single-molecule enzyme kinetics in the presence of inhibitors. J. Chem. Phys. 2012, 137, 045102. 12. Chaudhury, S.; Cao, J.; Sinitsyn, N. A., Universality of Poisson Indicator and Fano Factor of Transport Event Statistics in Ion Channels and Enzyme Kinetics. J. Phys. Chem. B 2013, 117, 503−509. 13. Mandel, L., Sub-Poissonian photon statistics in resonance fluorescence. Opt. Lett. 1979, 4, 205. 14. Short, R.; Mandel, L., Observation of sub-Poissonian photon statistics. Phys. Rev. Lett. 1983, 51, 384. 15. Cox, D. R., Renewal Theory. Methuen & Co Ltd: London, 1962. 16. Cao, J.; Silbey, R. J., Generic Schemes for Single-Molecule Kinetics. 1: Self-Consistent Pathway Solutions for Renewal Processes. J. Phys. Chem. B 2008, 112, 12867–12880. 17. Jung, W.; Yang, S.; Sung, J., Novel Chemical Kinetics for a Single Enzyme Reaction: Relationship between Substrate Concentration and the Second Moment of Enzyme Reaction Time. J. Phys. Chem. B 2010, 114, 9840-9847. 18. Gorris, H. H.; Rissin, D. M.; Walt, D. R., Stochastic inhibitor release and binding from single-enzyme molecules. Proc. Natl. Acad. Sci. U.S.A. 2007, 104, 17680. 19. Piwonskia, H. M.; Goomanovskya, M.; Bensimonc, D.; Horovitzb, A.; Harana, G., Allosteric inhibition of individual enzyme molecules trapped in lipid vesicles Proc. Natl. Acad. Sci. USA 2012, 109 E1437-E1443. 25



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20. Gopich, I. V.; Szabo, A., Statistics of transitions in single molecule kinetics 2003. J. Chem. Phys. 2003, 118, 454. 21. Gopich, I. V.; Szabo, A., Theory of the statistics of kinetic transitions with application to single-molecule enzyme catalysis. J. Chem. Phys. 2006, 124, 154712. 22. Cao, J., Event-averaged measurements of single-molecule kinetics. Chem. Phys. Lett. 2000, 327, 38–44. 23. Gardiner, C. W., Handbook of stochastic methods: For Physics, Chemistry, and the natural sciences. Springer: New York, 1996. 24. van Kampen, N. G., Stochastic processes in physics and chemistry. Elsevier Science: New York, 1992. 25. de Ronde, W. H.; Daniels, B. C.; Mugler, A.; Sinitsyn, N. A.; Nemenman, I., Mesoscopic statistical properties of multistep enzyme-mediated reactions. IET Systems Biology 2009, 3 429– 437. 26. Hasty, J.; Pradines, J.; Dolnik, M.; Collins, J. J., Noise-based switches and amplifiers for gene expression. Proc Natl Acad. Sci U S A. 2000, 97, 2075-2080. 27. Lu, T.; Ferry, M.; Weiss, R.; Hasty, J., A molecular noise generator Phys. Biol. 2008 5 036006. 28. Lu, T.; Shen, T.; Zong, C.; Hasty, J.; Wolynes, P. G., Statistics of cellular signal transduction as a race to the nucleus by multiple random walkers in compartment/phosphorylation space. Proc Natl Acad Sci USA 2006 103, 16752–16757.

Table of contents

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179x362mm (300 x 300 DPI)



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78x34mm (300 x 300 DPI)


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69x54mm (300 x 300 DPI)



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74x61mm (300 x 300 DPI)


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Poisson Indicator and Fano Factor for Probing Dynamic Disorder in

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