9840
J. Phys. Chem. B 2010, 114, 9840–9847
Novel Chemical Kinetics for a Single Enzyme Reaction: Relationship between Substrate Concentration and the Second Moment of Enzyme Reaction Time Won Jung, Seongeun Yang, and Jaeyoung Sung* Department of Chemistry, Chung-Ang UniVersity, Seoul 156-756, Korea ReceiVed: January 8, 2010; ReVised Manuscript ReceiVed: May 20, 2010
We report a robust quadratic relation between the inverse substrate concentration and the second moment, , of the catalytic turnover time distribution for enzyme reactions. The results hold irrespective of the mechanism and dynamics of the enzyme reaction and suggest a novel single molecule experimental analysis that provides information about reaction processes of the enzyme-substrate complex and ergodicity of the enzyme reaction system, which is beyond the reach of the conventional analysis for the mean reaction time, . It turns out that - 22 is linear in inverse substrate concentration for an ergodic homogeneous enzyme system given that the enzyme substrate encounter is a simple rate process, and its value at the high substrate concentration limit provides direct information about if any non-Poisson reaction process of the enzyme-substrate complex. For a nonergodic heterogeneous reaction system, the corresponding quantity becomes a quadratic function of the inverse substrate concentration. This leads us to suggest an ergodicity measure for single enzyme reaction systems. We obtain a simple analytic expression of the randomness parameter for the single catalytic turnover time, which could provide a quantitative explanation about the previously reported randomness data of the β-galactosidase enzyme. In obtaining the results, we introduce novel chemical kinetics applicable to a non-Poisson reaction network with arbitrary connectivity, as a generalization of the conventional chemical kinetics. I. Introduction A modern cutting-edge single molecule spectroscopic technique enables one to observe individual reaction events of a single enzyme molecule or a molecular motor complex.1–4 One of the advantages of such single molecule experiments over the conventional experimental technique is that the former provides statistical fluctuations of single molecule observables such as the number of product molecules and the time elapsed during a single reaction event, while the latter provides only the mean values of the observables. Therefore, in this field, it is an important task to establish systematic theoretical methods for information mining from the statistical fluctuation of single molecule experimental observables.5 Until now, with few exceptions,5 most of the single-molecule experimental data are analyzed by theoretical methods within the paradigm of the rate constant concept. Representative examples include the master equation approach6,7 or the stochastic simulation method proposed by Gillespie for the number distribution of reactant or product molecules8,9 and the methods based on the conventional chemical kinetic equation for the reaction time distribution of a single molecule reaction event.10–16 The essential assumption of these theoretical methods is that each of the elementary reaction processes composing a single molecule reaction is a Poisson process that can be characterized by a single rate constant. However, the recently reported reaction time distributions of a single enzyme molecule indicate that the assumption may not be true for an enzyme molecule of which reactivity fluctuates over wide range of time scales.1,2,17 In most theories for description of nonclassical kinetics of a chemical reaction, the reaction is assumed to be a Poisson process in which the rate coefficient is dependent on a slow * To whom all correspondence should be addressed. E-mail:
[email protected].
dynamical variable.17–25 As a matter of fact, the latter model can provide a correct asymptotic behavior of the mean reactant number density even for the case with an arbitrary non-Poisson reaction process at a time scale much longer than the individual reaction time scale.26 However, statistical fluctuation of the product number for the case with a non-Poisson reaction process can be quite different from that for the case with the Poisson reaction process, whether or not the reaction process is coupled to another dynamical process.27 Recently, Cao and Silbey presented the self-consistent pathway approach specifically for the reaction time distribution of a single molecule reaction composed of arbitrary renewal reaction processes,28 and the authors clearly showed that the mean reaction time of the single molecule reaction event is equivalent to that of the corresponding single molecule reaction scheme composed of Poisson reaction processes. In the present work, we will show that the fluctuation of the single molecule reaction time is sensitive to non-Poisson reaction processes constituting the single molecule reaction, and it contains valuable information about the reaction dynamics and the ergodicity of the single molecule reaction system, which is beyond the reach of the conventional experimental analysis. In this work, we first report a general relationship between the substrate concentration and the second moment of the reaction time distribution (RTD) for the classic enzyme (E)-substrate (S) reaction scheme
E + S h ES f E + P
(1)
where the forward (ES f E + P) or the backward (E + S r ES) reaction of the enzyme-substrate complex, ES, can be an arbitrary stochastic process. We find that the second moment of RTD is inversely proportional to the square of the substrate concentration. The latter quadratic relation is as robust as the
10.1021/jp1001868 2010 American Chemical Society Published on Web 07/15/2010
Chemical Kinetics for a Single Enzyme Reaction well-known Michaelis-Menten equation, a linear relation between the inverse substrate concentration and the first moment of RTD.29 By analyzing the second moment of the enzyme reaction time distribution with our result, one can obtain such quantitative information of the reactions of the ES complex as the mean survival time of the ES complex, the probability of the forward (backward) reaction of the ES complex, and the mean number of the backward dissociation events of the ES complex during a single enzyme reaction and the randomness parameter that characterizes the stochastic properties of the reactions of the ES complex. Another important issue we address here is the effect of nonergodicity of a single molecule reaction system on the statistical fluctuation of the single molecule reaction time, which has been largely neglected in contemporary single molecule experimental analysis. We find that the randomness parameter Q defined by ( -2 2)/2 vanishes for the ergodic and homogeneous enzyme system in the low substrate concentration limit, but it does not for the nonergodic or the heterogeneous enzyme reaction system. This finding leads us to suggest the low substrate concentration limit value of the randomness parameter as a homogeneity measure for single enzyme reaction systems. We find that the randomness parameter data of the β-galactosidase enzyme system reported in ref 17 can be explained by the simple analytic expression of the randomness parameter we obtain for the ergodic homogeneous system, whereas the behavior of the randomness parameter of the β-galactosidase enzyme in the low substrate concentration regime is qualitatively different from that of the nonergodic heterogeneous enzyme system. This paper is organized as follows. In the next section, we introduce a generalized chemical kinetics for the non-Poisson enzyme reaction, which can be applied to a more complex nonPoisson reaction network in a straightforward manner. In Section IIIA, we discuss the second moment of the reaction time distribution and other related single molecule experimental observables of the single enzyme reaction for an ergodic homogeneous enzyme reaction system. Then, in Section IIIB, we discuss the behavior of the single molecule observables for a nonergodic heterogeneous enzyme reaction system. Here, we also compare our theoretical results to the single molecule experimental data of the β-galactosidase enzyme reported in ref 17. Finally, in Section IV, we summarize the present work. II. Generalized Chemical Kinetics for the Non-Poisson Enzyme Reaction Let our system be composed of a number of enzyme molecules of the same kind. Enzyme molecules may have different catalytic efficiency from each other depending on their microscopic state. However, if our observation time is long enough and the enzyme system is ergodic, the observed reaction time distribution (RTD) for every enzyme in the system would be the same. At first, we will derive our result for such a homogeneous enzyme system and then generalize the result for a heterogeneous enzyme system. We begin by introducing a generalization of the conventional chemical kinetics, for description of enzyme reaction 1 composed of non-Poisson elementary processes. The present approach can be straightforwardly extended to a more complex non-Poisson reaction network with an arbitrary topology. Let fI(t)dt denote the probability that the single enzyme system arrives at chemical state I [∈(E + S, ES, E + P)] in time interval (t, t + dt). The RTD, ψ(t), for enzyme reaction 1 is nothing but fE+P(t) given that the E + S state is prepared at time zero with
J. Phys. Chem. B, Vol. 114, No. 30, 2010 9841 probability 1. With this initial condition, fI satisfies the following equations
fE+S(t) )
∫0t dτφ-1(t - τ)fES(τ) + δ(t)
fES(t) )
∫0t dτφ01(t - τ)fE+S(τ)
fE+P(t) )
∫0t dτφ2(t - τ)fES(τ)
(2)
Here, φ10(t) denotes the RTD for the substrate-enzyme encounter process, E + S f ES, which satisfies the normalization condition ∫0∞dtφ10(t) ) 1; φ10(t)dt is the probability that a substrate molecule encounters the enzyme molecule in time interval (t, t + dt) given that the encounter process begins at time zero. φ-1(t) and φ2(t), respectively, denote the RTDs for two competing reactions, the backward reaction ES f E + S and the forward reaction ES f E + P, of the ES complex. They satisfy the following normalization condition: ∫0∞dt[φ-1(t) + φ2(t)] ) 1. φ-1(t) and φ2(t) are given by 0 0 (t) ∫t∞dτφ20(τ) and φ2(t) ) φ20(t) ∫t∞dτφ-1 (τ), where φ-1(t) ) φ-1 0 φ-1(2)(t) denotes the normalized RTD for the ES f E + S (ES f E + P) reaction in the absence of the other competing process ES f E + P (ES f E + S).30 Solution of eq 2 can be easily obtained in the Laplace domain as (n) ˆfE+S (s) ) Ω(s) ˆfES(s) ) Ω(s)φˆ 01(s) ˆfE+P(s) ) Ω(s)φˆ 01(s)φˆ 2(s) ) ψ ˆ (s)
(3)
where Ω(s) is given by Ω(s) ≡ [1 - φˆ 10(s)φˆ -1(s)]-1. Noting that [φ10(s)φ-1(s)]n is the RTD for n time consecutive repetition of the association-dissociation reaction between enzyme and substrate molecules, we obtain the Laplace domain expression for the generation time distribution f J(n) of chemical state J after just n time repetition of the association-dissociation reaction between enzyme and substrate molecules as follows (n) ˆf E+S (s) ) [φˆ 01(s)φˆ -1(s)]n (n) ˆf ES (s) ) [φˆ 01(s)φˆ -1(s)]nφˆ 01(s) (n) ˆf E+P (s) ) [φˆ 01(s)φˆ -1(s)]nφˆ 01(s)φˆ 2(s)
(4)
f (n) J can also be a single molecule observable if each chemical state can be resolved in the single molecule experiment. ˆfJ(s) in eq 3 is ∞ ˆ (n) f J (s). the same as ∑n)0 Another single molecule observable is the probability PJ(t) that the enzyme reaction system is in chemical state J at time t during a single enzyme reaction event, given that the reaction event begins at time 0. PJ(t) is related to fJ(t) by
PJ(t) )
∫0t dτΦJ(t - τ)fJ(τ)
(5)
Here, ΦJ(t) is the survival probability of state J or the probability that the system in state J generated at time 0 does not suffer any more change of state for time t: ΦE+S(t) ) 1 - ∫0t dτφ10(τ), ΦES(t) ) 1 - ∫0t dτ[φ-1(τ) + φ2(τ)], and ΦE+P(t) ) 1. The last equality follows because the system in the E + P state does
9842
J. Phys. Chem. B, Vol. 114, No. 30, 2010
Jung et al.
not suffer any more change for the single enzyme reaction event. From eqs 3 and 5, we obtain
PˆE+S(s) ) PˆES(s) )
1 - φˆ 01(s) Ω(s) s
1 - φˆ -1(s) - φˆ 2(s) Ω(s)φˆ 01(s) s
1 PˆE+P(s) ) Ω(s)φˆ 01(s)φˆ 2(s) s
(6)
The probabilities given in eq 6 satisfy the normalization condition
A. Ergodic Homogeneous System. We assume that the encounter process of substrate molecules to the enzyme molecule, E + S f ES, is a simple Poisson process with a constant bimolecular reaction rate. On the other hand, the backward (forward) reaction process, ES f E + S(P), of the ES complex is not assumed to be a simple Poisson process because the reaction is possibly a multistep reaction involving intermediate states, and the reaction rate of the ES complex would fluctuate in line with conformational dynamics of the ES complex. The functional form for RTD, ψ(t), of reaction 1 involving non-Poisson reaction processes can be quite different from that of reaction 1 composed of simple Poisson reaction processes only. ˆ (s)/∂(-s)n, Substituting eq 3 into ) ∫0∞dttnψ(t) ) lim ∂nψ sf0 we obtain the first two moments of ψ(t) as follows:
)
PˆE+S(s) + PˆES(s) + PˆE+P(s) ) 1/s To show the connection of the present approach to the classical chemical kinetics, we present the time evolution equation for PI, which can be easily derived from eqs 2 and 5,31
nj + 1 + nj + k1[S]
(
nj + 1 ) 2 k [S] 1
)
2
+2
(8)
nj + 1 ( + 2nj ) k1[S] 2
2 > + + 2nj ( + nj < t-1> ) + nj < t-1
(9) Pˆ˙ E+S ) -κˆ 1(s)PˆE+S(s) + κˆ -1(s)PˆES(s) P˙ˆES ) κˆ 1(s)PˆE+S(s) - [κˆ -1(s) + κˆ 2(s)]PˆES(s) P˙ˆE+P ) κˆ 2(s)PˆES(s)
(7)
where Pˆ˙ I denotes the Laplace transform of dPI(t)/dt. In eq 7, the ˆ E+S(s), κˆ -1(s) ) φˆ -1(s)/ rate kernels are given by κˆ 1(s) ) φˆ 10(s)/Φ ˆ ES(s). When reaction processes conˆ ES(s), and κˆ 2(s) ) φˆ 2(s)/Φ Φ stituting enzyme substrate reaction 1 are Poisson processes, i.e., when φ01(t) ) k1[S]exp(-k1[S]t), φ0-1(t) ) k-1 exp(-k-1t), and φ02(t) ) k2 exp(- k2t), the rate kernels become constant: κˆ 1(s) ) k1[S], κˆ -1(s) ) k-1, and κˆ 2(s) ) k2, so that eq 7 reduces to the conventional chemical kinetic equations. Therefore, eq 7 is the generalization of the classical chemical kinetic equations for enzyme-substrate reaction 1 composed of non-Poisson reaction processes. Note that eq 7 conforms to the classical chemical kinetic equations at long ˆ J(s) exists. As a matter of fact, the times as long as lim Φ sf0 reduction of eq 7 to the classical chemical kinetic equations at long times is not universal. For example, the long-time power-law relaxation kinetics observed for diffusion-influenced reversible reactions cannot be described by the classical chemical kinetics; the correct kinetic equations describing the relaxation behavior of reversible reactions in the liquid phase involve rate kernels as eq 7 does, but they do not reduce to the classical chemical kinetics even at long times.22,32,33 III. Second Moment of Reaction Time Distribution of Enzyme Reaction In this section, we will focus on the relationship between the second moment of the RTD of the enzyme reaction 1 and the stochastic properties of the reaction processes of ES complexes. If our enzyme reaction system is ergodic and our observation time is long enough, the observed RTD for every enzyme in the system would be the same. In the subsequent section, we will discuss the second moment of RTD and other related SM experimental observables for such a homogeneous system.
Here, nj denotes the average number of the backward reactions of the ES complex occurring during a single enzyme reaction 1, which is given by nj ) p-1/p2 with p-1(2) being the reaction probability, ∫0∞dtφ-1(2)(t), of the ES complex for the backward (forward) reaction, ES f E + S(P).34 p-1 and p2 are subject to the normalization condition, p-1 + p2 ) 1, so that 1 + nj ) p2-1. In eqs 8 and 9, denotes the mean backward (forward) reaction time in the presence of the competing forward (backward) reaction, defined by
∫0∞ dttφ-1(2)(t)/pj The linear relation, given in eq 8, between the mean reaction time and the inverse of substrate concentration [S]-1 is equivalent to the Michaelis-Menten relation between the reaction rate kobs () -1) and substrate concentration, [S]. From eq 8, we obtain
kobs )
kmax[S] [S] + KM
(10)
where kmax ) (nj + )-1 and KM ) 1/(p-1 + p2)(1/k1). When the backward and the forward reactions of enzyme-substrate complexes are both simple Poisson processes, we have p-1(2) ) k-1(2)/k-1 + k2, and ) ) 1/(k-1 + k2). Substituting the latter equations into eq 10, we recover the conventional MM equation for a single enzyme, derived from the classical chemical kinetics
kobs )
k2[S] 0 [S] + KM
(for Poisson reaction processes)
(11) 0 0 denotes the conventional MM constant: KM ) (k-1 + Here KM k2)/k1. Note that, since eq 10 for the ES complex suffering arbitrary non-Poisson reaction processes is isomorphic to eq 11
Chemical Kinetics for a Single Enzyme Reaction
J. Phys. Chem. B, Vol. 114, No. 30, 2010 9843
for the ES complex undergoing Poisson reaction processes, the relation between the mean reaction time or mean reaction rate kobs and [S] cannot provide any information about nonPoisson reaction processes of the ES complex. In comparison, the quadratic relation, eq 9, between the second moment of the RTD of reaction 1 and the inverse substrate concentration [S]-1 is more sensitive to if any non-Poisson reaction process of the ES complex. We can rewrite eqs 8 and 9 as
)
(
)
(12)
( )
(13)
KM +1 p2 [S]
-2 2 ) -2
KM qES + p2 [S] p2
Here denotes the mean survival time of the ES complex defined by ≡ p-1 + p2. In terms of , KM and kmax are read as (k1)-1 and p2/, respectively. In eq 13, qES denotes the parameter that characterizes the stochastic property of the reaction processes of the ES complex, which is defined by qES ) p2( - 22) + p-1( - 2). qES vanishes if both the forward and the backward reaction processes of the ES complex are simple Poisson processes. The 0 (t), for the backward (forward) Poisson reaction RTD, φ-1(2) process of the ES complex in the absence of the competing 0 (t) ) k-1(2) forward (backward) reaction is given by φ-1(2) exp(-k-1(2)t), so that the corresponding RTD, φ-1(2)(t) 0 (t)∫∞t dτφ2(-1)(τ)], of the ES complex in the presence [≡ φ-1(2) of the competing reaction is given by φ-1(2)(t) ) k-1(2) exp[-(k-1 + k2)t] and the backward (forward) reaction probability becomes p-1(2) [≡ ∫0∞dtφ-1(2)(t)] ) k-1(2)/(k-1 + k2). In this case, both φ-1(t)/p-1 and φ2(t)/p2 are given by φ-1/p-1 ) φ2/p2 ) (k-1 + k2)exp[-(k-1 + k2)t]. That is to say, for the ES complex undergoing simple Poisson reaction processes, the mean reaction time [) ∫0∞dttφ-1(t)/p-1] of the backward reaction, ES f E + S, in the presence of the competing forward reaction is the same as the mean reaction time [) ∫0∞dttφ2(t)/p2] of the forward reaction process, ES f E + P, in the presence of the competing backward reaction; both and are given by (k-1 + k2)-1, the same as (≡ p-1 + p2).35 For such an ES complex with simple Poisson reaction processes only, the second moment of RTD is given by twice the square of the first moment, i.e., [≡ ∫∞0 dtt2φ-1(2)(t)/p-1(2)] ) 2(k-1 + k2)-2 ) 22, and eq 13 assumes a very simple form
P - 2 2
P2
2 )k2k1[S]
(14)
Here, P denotes the nth moment of reaction 1 when both the forward and the backward reactions of the ES complex are the simple Poisson processes. Note that P - 2P2 vanishes in the high substrate concentration limit. However, in general, lim [S]f∞ - 22 does not vanish unless the ES complex undergoes simple Poisson reaction processes. Therefore, a finite value of lim [S]f∞ - 22 signifies the presence of a non-Poisson reaction process of the ES complex. By analyzing the first two moments of RTD of enzyme reaction 1 with use of eqs 12 and 13, we can extract such quantitative information about the reaction dynamics of the ES complex as and qES/p2 in addition to KM and p2/ (≡ kmax). If we assume that φ-1(t) and φ2(t) can be approximated
0 0 as φ-1(2) = p-1(2)φES (t), with φES (t) being an arbitrary normalized n > and dwell time distribution of the ES complex, both - 22. By analyzing the where qES is given by qES ) j
u 1 + ju
)] n
(16)
where j and j are defined by ∫ drτ(r)Fj(r) and j ) ∫ dr(τ(r) - j)nFj(r) with τ(r) and Fj(r) being the mean dwell time, defined by τ(r) ≡ [k-1(r) + k2(r)]-1, of the ES complex at microscopic state r and the reaction probability weighted state distribution of the ES complex, defined by Fj(r) ≡ pj(r)PC(r)/pj, respectively. Here, p2(-1)(r) denotes the forward (backward) reaction probability of the ES complex at conformation r, defined by p2(-1)(r) ) k2(-1)(r)/[k-1(r) + k2(r)], which is related to p-1(2) (≡ ∫∞0 dtφ-1(2)(t)) by p2(-1) ) ∫ drp2(-1)(r)PC(r). It is remarkable that eq 16 provides the universal expression for the RTD of the generalized Poisson process with a slowly fluctuating rate coefficient, which holds regardless of the functional form for state-dependent reaction rate k-1(2)(r) and state distribution PC(r) of the ES complex. Noting that ) ∂n[φˆ j(u)/pj]/∂(-u)n|u)0 (j ) -1,2), one can immediately obtain ) j and ) 2(2j + j) from eq 16. Note that qj (≡ - 22) of Model A always has a positive value given by twice the variance j of the mean dwell time, τ(r)
9844
J. Phys. Chem. B, Vol. 114, No. 30, 2010
Jung et al.
Figure 1. (a) Dependence of on KM/[S] and (b) dependence of - 22 on KM/[S] for the homogeneous enzyme system. KM is given by (k1)-1. Equations 12 and 15 are used to plot the lines. qES characterizes the stochastic property of the reaction process of the ES complex; qES ) 0 for the ES complex with Poisson reaction; qES > ( - 22). Since the values of KM and kmax of qES (≡ M[S]-2 + 2(2 M < β < t2.M)[S]-1 + M
Figure 2. Dependence of randomness parameter Q defined in eq 18 on substrate concentration for the homogeneous enzyme reaction system. Equation 19 is used to plot the lines. qES has the same meaning as in Figure 1. Q0 (≡ lim Q) vanishes, but Q∞(≡ lim Q) does not for [S]f∞ [S]f∞ the ES complex with a non-Poisson reaction process.
In the latter case, randomness parameter Q has a negative value for any value of [S]. In Figure 2, we show the dependence of Q on [S]/KM given in eq 18. For simplicity, we consider the 0 (t), for which is the same case with φ-1(2)(t) ) p-1(2)φES as . The values of the unspecified parameters are the same as those used in Figure 1. For the ergodic homogeneous enzyme system considered in the present section, randomness parameter Q0 [≡ lim Q([S])] [S]f0 in the low substrate concentration limit vanishes, i.e., Q0 ) 0, irrespective of the reaction dynamics of the ES complex. However, as shown in the next section, it is not the case for the nonergodic heterogeneous enzyme system. B. Nonergodic Heterogeneous System. Until now, we have considered the case where the enzyme system is ergodic and our observation time is long enough so that the observed RTD for every enzyme in the system is the same. When our reaction system is nonergodic or when the observation time is not long enough, each enzyme may have different RTDs from each other during the observation time. From now on, let us discuss the second moment of RTD of enzyme reaction 1 for such nonergodic heterogeneous systems. Let our system contain M enzyme molecules, each of which is under our observation at the single molecule level. For the nonergodic heterogeneous (j) (t) and the forward RTD φ2(j)(t) system, the backward RTD φ-1 are dependent on enzyme index j (1 e j e M). Each individual enzyme reaction system has its own RTD, ψi(t), for enzyme reaction 1, which is given by eq 3 with φ-1 and φ2 being (j) and φ2(j), and the first two moments j and replaced by φ-1 2 n >, j of ψj(t) should be given by eqs 8 and 9 with p2, M[S]-2
- 2[ M and M denote M - 2M and M - M M, respectively. Note that eq 22 is a quadratic function of [S]-1 with the coefficient of [S]-2 given by twice the variance of enzyme-to-enzyme fluctuation of βj ) 1/(k1p2)j.37 In Figure 3, we show the dependence of ,t2.M 2 on [S]-1 for the nonergodic heterogeneous enzyme 2 ,t.M reaction system in which the reaction probability p2 is 2-1(1 + ∆) for a half of the enzymes but is 2-1(1 - ∆) for the other half. All the other parameters are assumed to be the same as those used in Figure 1 for every enzyme in the system. Note that the curvature increases with the value of heterogeneity parameter ∆. As in Figure 1, the intercept of the curve is dependent on qES, which measures the non-Poissonian character of the reactions of ES complex. If we define by ≡ [,t2.M - 2,t.2M]/,t.2M, the expression for is given by
)
0 + ∞x(x - η) (x + 1)2
(23)
where x denotes [S]/KM with KM being equal to M/M. 0 and ∞ denote lim and lim , respectively, [S]f0
[S]f∞
given by 2(M/2M) and M/2M - 2. In eq 23, η designates 2[(
