Nonclassical Kinetics of Clonal yet Heterogeneous Enzymes - The

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Non-Classical Kinetics of Clonal Yet Heterogeneous Enzymes Seong Jun Park, Sanggeun Song, In-Chun Jeong, Hye Ran Koh, Ji-Hyun Kim, and Jaeyoung Sung J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.7b01218 • Publication Date (Web): 13 Jun 2017 Downloaded from http://pubs.acs.org on June 16, 2017

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The Journal of Physical Chemistry Letters

Non-Classical Kinetics of Clonal yet Heterogeneous Enzymes

Seong Jun Park1,3, Sanggeun Song1-3, In-Chun Jeong1-3, Hye Ran Koh1,2, Ji-Hyun Kim1*, and Jaeyoung Sung1-3* 1

National Creative Research Initiative Center for Chemical Dynamics in Living Cells,

Chung-Ang University, Seoul 06974, Korea. 2

Department of Chemistry, Chung-Ang University, Seoul 06974, Korea.

3

National Institute of Innovative Functional Imaging, Chung-Ang University, Seoul 06974,

Korea.

Author Information S.J.P. and S.S. contributed equally to this work. The authors declare no competing financial interests. Correspondence and requests for materials should be addressed to J.S. ([email protected]) and J.-H.K. ([email protected]).

1

ACS Paragon Plus Environment


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Abstract Enzyme-to-enzyme variation in the catalytic rate is ubiquitous among single enzymes created from the same genetic information, which persists over the lifetimes of living cells. Despite advances in single-enzyme technologies, the lack of an enzyme reaction model accounting for the heterogeneous activity of single enzymes has hindered a quantitative understanding of the non-classical stochastic outcome of single enzyme systems. Here we present a new statistical kinetics and exactly solvable models for clonal yet heterogeneous enzymes with possibly nonergodic state dynamics and state-dependent reactivity, which enable a quantitative understanding of modern single enzyme experimental results for the mean and fluctuation in the number of product molecules created by single enzymes. We also propose a new experimental measure of the heterogeneity and nonergodicity for a system of enzymes. Table of Contents (TOC) Graphic

KEYWORDS: stochastic kinetics, single enzyme, enzyme kinetics, conformational dynamics, Nonergodicity, dynamic heterogeneity, heterogeneous single enzyme reaction

2


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As one twin may run faster than their sibling, a single enzyme may perform catalytic reactions faster than another, even though both were materialized from the same genetic information. Modern single molecule experiments clearly showed a wide enzyme-to-enzyme variation among a clonal population of single enzymes in, for example, the catalytic turnover rates of β -galactosidase and phospholipase A21-3, the RNA or DNA unwinding rate by RNA helicase A or HCV NS3 Helicase4-5, and the velocity of kinesin motor movement on microtubules6. Surprisingly, enzyme-to-enzyme variations in catalytic activities have been found to persist throughout experimental measurements, exhibiting unexpectedly slow relaxation dynamics. The quasi-static variation in enzyme activities originates from a coupling between the enzyme catalytic rate and the microscopic state of the enzyme and/or the surrounding environment, which is often hidden or beyond direct measurement. An enzyme’s structural conformation is a representative example of microscopic state variables coupled to enzyme activity. The primary sequence, which varies from enzyme to enzyme among clonal enzymes due to errors in transcription and translation processes7, is another example of the microscopic state variable on which enzyme activity may depend8. In the case of enzyme reactions occurring in living cells, enzyme activity also depends on cell state variables such as inhibitor or activator concentration, the phase of the cell cycle, and the nutrition state or growth condition9,10. Variation in the catalytic activity among enzymes is inevitable even in highly homogenous environments. When it comes to enzymatic reactions occurring in living cells, heterogeneity in enzyme catalytic activity is even greater due to additional couplings between the enzyme activity and heterogeneous cell states11. This is actually an important source of 3



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cell-to-cell variation in gene expression levels and consequent phenotypic variations among a clonal population of cells12-15. In catalytic nanoparticle systems as well, a large dispersion in the catalytic activity has been observed across nanoparticles, due to heterogeneity in the microscopic state of nanoparticle surface and its coupling to catalytic activity, and the rate parameters of the catalytic reaction were found to have a broad distribution over the nanoparticles16. The enzyme-to-enzyme variation in catalytic activity among enzymes with the same primary sequence is in conflict with the fundamental hypothesis of chemical kinetics founded by van’t Hoff17. According to this hypothesis, molecules of the same kind have the same reactivity. Because the Michaelis-Menten (MM) kinetic scheme18, the conventional enzyme reaction kinetic scheme, is based on this hypothesis and does not take into account enzymeto-enzyme variation in catalytic activity coupled to the microscopic state of enzymes, the MM kinetics scheme cannot provide a quantitative explanation of the stochastic chemical dynamics of single enzyme systems. For example, while the mean enzymatic turnover time of a single β -galactosidase was found to obey MM kinetics19-22, the variance or the randomness in the enzymatic turnover time cannot be explained by the MM kinetics scheme19. A successful quantitative explanation of the randomness in the enzymatic turnover time of β -galactosidase was achieved by Yang, Cao, Silbey, and Sung23, with the use of a recently developed chemical kinetics for renewal enzyme processes, where the reactions of the enzyme-substrate complex are modelled as renewal processes23-25. A renewal process is the simplest stochastic process that can represent a non-Poisson process. While a Poisson reaction process is characterized by the rate constant, a renewal reaction process is characterized by the distribution of waiting time between successive reaction events, which 4


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reduces to a Poisson process when the reaction waiting time is exponentially distributed26. Utilizing the renewal enzyme reaction model, but with an exponential waiting time distribution of the enzyme-substrate dissociation reaction, Reuveni, Urbakh, and Klafter investigated the dependence of the enzymatic turnover rate on the enzyme-substrate dissociation rate for enzyme reactions in the low dissociation rate limit27. Considering several kinetic schemes of enzyme reactions, Moffitt and Bustamante examined what information can be extracted from the substrate concentration dependence of the mean and variance in the enzymatic turnover time28. More recently, Barato and Seifert have analysed the third and fourth moments of the enzymatic turnover time distribution for a renewal enzyme reaction scheme29. For non-renewal enzyme reaction schemes, Jung, Yang, and Sung and, independently, Moffitt and Bustamante investigated the substrate concentration dependence of the mean and randomness in the enzymatic turnover time distribution28,30. Despite these recent advances in the quantitative analysis of the enzymatic turnover time distribution, there has been little progress in a quantitative understanding of the product number distribution of an enzyme reaction, or counting statistics of enzyme reaction events. The product number distribution of enzyme reactions carries valuable information about the heterogeneity and the dynamics of enzyme states. When a reaction process is a renewal process, Cox’s renewal theory provides the exact relationship between the product number distribution of the reaction and the reaction waiting time distribution26. However, renewal theory cannot explain the anomalous features in the counting statistics of enzyme reaction events, which include a slow decrease in the mean enzyme reaction rate1-2, in a time scale much longer than the mean enzymatic turnover time, and a quadratic time dependence of the variance in the product number6. These features were clearly observed in recent 5



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femtoliter microfluidic droplet-based single enzyme experiments on β -galactosidases1-2 and single molecule tracking experiments on the movement of kinesin motors along microtubules6. To the best of our knowledge, currently available theories and enzyme reaction models are unable to provide a satisfactory explanation of the anomalous dynamics in the counting statistics of single enzyme reactions, and it remains unknown how the product number distribution is related to enzyme state dynamics on which enzyme activity depends9,23. In this Letter, we present a new statistical kinetics for a generic model of a single enzyme with a state dependent activity, enabling us to understand the anomalous dynamics observed in the product number distribution of single enzymes. We further propose a new measure of the heterogeneity, the ergodicity, and the dynamics of enzyme activity, in terms of the mean and variance in the product number. This measure provides otherwise inaccessible information about heterogeneity in enzyme activity and ergodicity in enzyme state dynamics on which enzyme activity depends. The dynamics of enzymes with primary sequence variation is nonergodic, as long as the primary sequence of each enzyme is conserved during the course of enzyme dynamics. Given that enzyme activity is dependent on the primary sequence of the enzyme, the primary sequence variation must be reflected in this ergodicity measure, and as the sensitivity of enzyme activity to the primary sequence increases, the more clearly primary sequence variation is reflected in this measure. Our model of the single enzyme reaction is represented by ϕ1 ( t ) ϕ2,Γ ( t ) → E + S ← ES  →E + P

(1)

ϕ−1,Γ ( t )

where E, S, ES and P denote enzyme, substrate, enzyme-substrate complex, and product 6


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molecule, respectively. In eq 1, ϕ1 (t ) denotes the reaction time distribution of E + S → ES or the distribution of time, t , elapsed to complete reaction E + S → ES , given that enzyme E is prepared at time 0. Likewise, ϕ −1,Γ (t ) or ϕ 2,Γ (t ) denote, respectively, the distribution

of time elapsed to complete the dissociation reaction E + S ← ES or the catalytic reaction

ES → E + P of the enzyme-substrate complex ES at state Γ , given that ES is prepared at time 0. Here Γ collectively represents every state variable coupled to enzyme activity. The normalization condition for ϕ1 (t ) , ϕ−1,Γ (t ) and ϕ 2,Γ (t ) are given by

∫

∞

0

dtϕ 2,Γ (= t ) p2 (Γ)

,

and

∫

∞

0

(t ) p−1 (Γ) dtϕ −1,Γ =

,

respectively.

∫

∞

0

dtϕ1 (t ) = 1 ,

p2 (Γ)

and

p−1 (Γ) [ =1 − p2 (Γ) ] denote the probability of the catalytic reaction, ES → E + P , and the dissociation reaction, E + S ← ES , of the enzyme-substrate complex, which are, in general, dependent on state variable Γ . For the sake of simplicity, we assume that the dynamics of the enzyme-substrate encounter reaction is independent of enzyme state, and ϕ−1,Γ (t ) p−1 (Γ) and ϕ2,Γ (t ) p2 (Γ) are given by the same function, φES (t ) , the lifetime distribution of the enzyme-substrate complex. The enzyme reaction model in eq 1 is more general than the previously proposed models, in that it takes into account the general dependence of enzyme activity on the state, Γ , of enzymes and the surrounding environment. In the special case where p2 (Γ) is the

same for any state Γ , the enzyme reaction process, given in eq 1, reduces to a renewal process so that our enzyme model obeys the kinetics for the renewal enzyme processes in refs 25

and

30

. The kinetics for a renewal enzyme process further reduces to MM kinetics

7



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k1 k2  → ES  → E + P , when describing the conventional enzyme kinetics scheme, E + S ←  k−1

the elementary reaction processes composing the enzyme reaction are Poisson processes with exponential reaction time distributions, i.e., when ϕ1 (t ) = k1[ S ]e − k1 [ S ]t , ϕ ES (t ) = e − ( k−1 + k2 )t , and

p2 = 1 − p−1 = k2 ( k−1 + k2 ) . For the enzyme reaction model given in eq 1, we obtain analytic results for the mean and variance in the number of product molecules created in time interval (0, t ) (See the Supporting Information, Method 1). When the reactivity fluctuation of single enzymes is fast enough, the results obtained for our model reduce to the results for the renewal enzyme process, reported in refs 23, 25 and 30. It is known that, for a renewal enzyme process, the mean product creation rate obeys the classic MM equation, d n(t ) dt

≅

kmax [ S ] [S ] + K M

(2)

at times longer than the mean enzymatic turnover time. In eq 2, kmax , [ S ] , and K M denote, respectively, the maximum rate of enzyme reaction, the substrate concentration, and the Michaelis-Menten constant. According to recent experiments, however, the reactivity fluctuation of a single enzyme occurs in a time scale much longer than, not only the mean enzymatic turnover time, but also the measurement time scale1-3. In this case, our model no longer reduces to a renewal enzyme reaction model; depending on the initial condition, the mean enzymatic turnover rate predicted by our model can persistently deviate from the MM equation at times much longer than the mean enzymatic turnover time. For example, let us consider an ensemble of enzymes 8


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that successfully initiated enzyme reaction at time 0. The state distribution of the enzymes in this ensemble is different from the equilibrium distribution, because enzymes with a greater reactivity are more likely to be included in this ensemble. The average reaction rate of the enzyme ensemble obeys the following equation (See the Supporting Information, Method 2): *

d n(t )

≅

dt

where

kmax [ S ] k2 (t )k2 (0) 2 [S ] + K M k2

( t >>

eq

t1

)

(3)

eq

*

n(t )

denotes the number of product molecules created by the initially activated

enzymes in time interval (0, t ) . In eq 3, the catalytic rate, k2 (t ) , is defined by p2 (Γ(t )) t ES

k2

eq

= p2

,

eq

so t ES

that with

the

equilibrium

p2

being the average of p2 (Γ) over the equilibrium or

eq

catalytic

steady-state distribution Peq (Γ) of enzyme state Γ .

rate,

k2 (t )k2 (0)

k2

eq

eq

,

is

given

by

denotes the equilibrium

time-correlation function of the catalytic rate whose value decreases from the initial value k22 (0)

eq

(= k ) 2 2 eq

to the final value

k2 (t )

eq

k2 (0)

(= k ) . 2

eq

2

eq

k2 (t )k2 (0)

eq

is related

to the conditional probability density, G (GG , t | 0 ) , that the enzyme is found at state Γ at time

t

,

k2 (t )k= 2 (0) eq

given t ES

−2

∫

that

it

was

initially

at

Γ0

as

follows:

d GGGGGGG d 0 p2 ( )G ( , t | 0 ) p2 ( 0 ) Peq ( )

Equation 3 explains the decrease in the reaction rate of the initially activated enzymes over time, which has been observed in recent single enzyme experiments1-2. This is demonstrated in Figure 1 for a two-state enzyme model undergoing a slow stochastic transition between the active and inactive states. The slow decrease in the mean product 9



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creation rate reflects the slow relaxation dynamics of the enzyme state dependent catalytic rate. We found that the two-state enzyme model shown in Figure 1 provides a successful quantitative explanation of the experimental result reported in ref 2, which clearly shows the slow decrease in the mean turnover rate of initially activated β -galactosidases (Figure S1). The time-dependent relaxation of the catalytic rate emerges whenever the initial state distribution of enzymes deviates from the equilibrium distribution or the steady-state distribution (See eq. M2-7 in the Supporting Information, Method 2). The mean enzyme reaction

rate,

given

in

eq

M2-7,

{

d n(t ) dt ≅ N E [ S ] ([ S ] + K M )  × k2

eq

approaches

the

MM

equation

following

}

+ ∫ d Γk2 (Γ)d P(Γ, t ) . Here, δ P(Γ, t ) denotes

the deviation of the enzyme-state distribution from the equilibrium or steady-state distribution, Peq (Γ) , which vanishes at long times for any initial state distribution. In obtaining eq 3, we assume that the enzyme-substrate association reaction is a simple Poisson process, represented by ϕ1 (t ) = k1[ S ]e − k1 [ S ]t , with substrate concentration in excess of enzyme. However, we make no assumption regarding the catalytic reaction, ES → E + P , or the dissociation reaction, E + S ← ES , of the enzyme-substrate complex so that the lifetime distribution, φES (t ) , and the catalytic reaction probability, p2 (Γ) , are arbitrary. For the general enzyme reaction model, the analytic expressions of K M and kmax are obtained as K M = ( k1 t ES

)

−1

and kmax = k2

eq

N E , where

t ES ,

k2

eq

, and N E denote the mean

lifetime of the enzyme-substrate complex, the mean catalytic rate of the enzyme-substrate complex, and the number of single enzymes, respectively (See the Supporting Information, Method 2). Equation 3 holds at times longer than the mean enzymatic turnover time, 10


t1 ,

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defined as = t1

−1

k2

eq

(1 + K M

[S]) (See the Supporting Information, Method 2).

When the initial state of an enzyme is distributed according to the equilibrium or the steady-state distribution, the conventional MM equation holds at any time longer than the mean enzymatic turnover time,

t1 . Even when the initial distribution of enzyme state

deviates from the equilibrium distribution, the mean enzyme reaction rate reduces to MM kinetics after the enzyme state distribution relaxes to the equilibrium distribution. For example, eq 3 reduces to the conventional MM equation at long times when k2 (t )k2 (0)

eq

≅ k2

2 eq

. In Figure 1b, this is demonstrated for the two-state enzyme model.

This result is in direct contradiction with the assertion made in ref

31

, stating that the MM

equation does not hold for a mesoscopic enzyme system. In the presence of a strong heterogeneity in K M , the mean enzyme reaction rate can deviate from the MM equation even in the steady-state20, which can be analysed with use of the method presented in refs

32

or 33. Fluctuation in the product number of enzyme reactions carries valuable information about the enzyme reaction dynamics, and the heterogeneity and ergodicity of the enzyme system. We find that Θ n (t ) ≡ σ n2 (t ) − n(t )  n(t )

2

is a particularly useful measure of the

heterogeneity and ergodicity for a system of single enzymes, where σ n2 (t ) denotes the variance in the number of product molecules created by single enzymes in time interval

(0, t ) . For enzymes initially at the equilibrium state or at the steady-state, are given by

n(t ) ≅ kmax [ S ] ([ S ] + K M ) t and

11


n(t )

and Θ n (t )


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Θ n (t ) ≡ ≅

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σ n2 (t ) − n(t ) n(t )

2

( t >>

A([S]) 2η  A([ S ])  t  + 1 −  ∫0 dt f (t ) + A([ S ]) f (t )   t t  t   2 k2 2

t1

)

(4)

where A([ S ]) is defined by A([ S ]) ≡ t ES ([ S ]RtES − 2 K M ) ([ S ] + K M ) . η k22 and RtES designate, respectively, the relative variance, δ k22 parameter

for

(

2 RtES = t ES − t ES

the 2

)

lifetime 2

t ES

of

t ES

the

k2

2

, of k2 (t ) and the randomness

enzyme-substrate

complex,

defined

by

− 1 (See the Supporting Information, Method 3). In eq 4, f (t )

t

is given by f (t ) = ∫ dtfk2 (t ) with φk2 (t ) being the normalized time-correlation function of 0

k2 (t ) , i.e., φk2 (t ) = δδδ k2 (t ) k2 (0) k22 . Since k2 (Γ(t )) = p2 (Γ(t )) t ES , η k22 and φk2 (t ) are the same as the relative variance and the normalized time correlation function of p2 (Γ(t )) , respectively. At short times when the relaxation of the enzyme state coupled to enzyme activity does not occur significantly, i.e., when φk2 (t ) ≅ 1 , f (t ) is approximately the same as t so that eq 4 reduces to

(

Θ n (t ) ≅ η k22 + 1 + ηk22

) A([tS ])

( φk2 (t ) ≅ 1, t >> t1 )

(5)

By analyzing the short-time dependence of Θ n (t ) on the substrate concentration with eq 5, we can extract the value of ηk22 (= η p22 ) , K M = ( k1 t ES

)

−1

and

k2

eq

(=

p2

eq

t ES

t ES , and RtES in addition to the values of

),

which can be obtained by analyzing the

12


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dependence of the mean enzyme reaction rate on the substrate concentration or from the conventional Lineweaver-Burk plot analysis. Combining these results, we are then able to separately estimate the values of k1 , t ES ,

2 p2 , δ t ES , and δ p22 .

Equation 5 can also be obtained for enzymes with statically heterogeneous catalytic rates (See the Supporting Information, Method 4, 5, and 6). In this case, Θ n (t ) approaches

ηk2 at long times, which causes the variance, σ n2 (t ) , in the number of reaction events to 2

increase with time in a quadratic manner. For any statically heterogeneous enzyme system, it

(

)

is easy to show that the variance σ n2 (t ) ≡ n 2 − n

2

components

Information,

σ n2= (t )

(See

σ n2 (t | Γ) +

the

( n(t )

Supporting

− n(t ) Γ

)

2

can be represented by the sum of two

. Here σ n2 (t | Γ)

Method

4,

5):

denotes the mean of the variance

in the number of reaction events occurring in each single enzyme trajectory in time interval (t0 , t + t0 ) , which increases linearly with time t (Figure S2b). This term represents the randomness in the reaction event number in a single enzyme, which persists in the absence of enzyme-to-enzyme variation in catalytic rate. On the other hand,

( n(t )

Γ

− n(t )

)

2

denotes the variance in the mean number of reaction events catalyzed by single enzymes in time interval (t0 , t + t0 ) , which is proportional to t 2 (Figure S2c). The latter term is proportional to enzyme-to-enzyme variation in the catalytic rate. Each contribution of

σ n2 (t | Γ)

and

( n(t )

Γ

− n(t )

)

2

to σ n2 (t ) is shown in Figure S1 (d) for a model of

enzymes with statically heterogeneous catalytic rates. The quadratic time dependence of

σ n2 (t ) was previously observed for several varieties of kinesin motor proteins6. This finding 13



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indicates that the mean catalytic rate of kinesin motors differs from kinesin to kinesin in a quasi-static manner at least in the experimental measurement’s time scale (See the Supporting Information, Note 3). The long-time limit value of Θ n (t ) is quite sensitive to the ergodicity of enzyme state dynamics. It vanishes for ergodic enzyme systems, but not for a nonergodic enzyme system. In the case of ergodic enzyme systems, the enzyme state dynamics spans all possible enzyme states from any initial state at sufficiently long times, i.e., lim G (GG , t | 0 ) = Peq (G) , so that we t →∞

∞ 2 have lim k2 (t )k2 (0) − k2 = lim φk2 (= t ) 0 . As long as t k2  ≡ ∫ dtfk2 (t ) = f (∞)  is finite,  0  t →∞ t →∞

Θ n (t ) in eq 4 vanishes, following t −1 relaxation behaviour,

Θ n (t ) ≅

A([ S ]) + 2ηk22t k2 t

,

(ergodic system)

(t >> t k2 , t1 )

(6)

at long times where f (t ) ≅ f (∞) =t k2 . Equation 6 is the universal relaxation behaviour emerging for any ergodic system with finite τ k2 . On the other hand, for a nonergodic enzyme system, the enzyme state dynamics does not span all the enzyme states. In this case, we have lim G (GG , t | 0 ) ≠ Peq (G) for any Γ 0 so that lim φk2 (t )  ≡ φk2 (∞)  may not vanish, and the t →∞

t →∞

long time limit value of Θ n (t ) is finite, i.e.,

Θ n (t ) ≅ ηk22 φk2 ( ∞ ) + 1 + ηk22 φk2 (∞) 

A([ S ]) , (nonergodic system) t

(7)

This result is easily obtained from eq 4 for the nonergodic enzyme system, for which

14


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t f (t )  ≡ ∫ dtfk2 (t )  is approximately given by f (t )  fk2 (∞)t at long times. This result  0 

suggests Θ n (t ) as a good measure of nonergodicity for a single enzyme system (Figure 2b). In the special case where the enzyme system is ergodic but τ k2 does not exist, Θ n (t ) vanishes in the long time limit, but the asymptotic relaxation dynamics of Θ n (t ) deviates from the t −1 relaxation behaviour, depending on the details of the state dynamics or the functional form of φk2 (t ) . For example, for a two-state enzyme with the state dynamics being a renewal process whose waiting time distribution has a power-law tail proportional to t − (α +1) ,

(

∞

we obtain φk2 (t ) ∝ t −α so that t k2 = ∫ dtφk2 (t ) 0

) does not exist when 0 < α < 1 . In this case,

it has previously been established that a weak ergodicity breaking takes place34. For the model with a weak ergodicity breaking, we obtain Θ n (t ) ∝ t −α

(0 < α < 1)

(system with a weak ergodicity breaking)

from eq 4 at long times, as shown in Figure 2a (See the Supporting Information, Method 3). The anomalous power-law relaxation of Θ n (t ) also emerges when the enzyme state dynamics follows fractional Brownian motion or the generalized Langevin equation with fractional Gaussian noise35. The stochastic enzyme kinetics presented in this Letter can be utilized for the development of a new theory on the stochastic transport of molecules and colloids in disordered environments, which is awaiting publication at time of writing. This theory encompasses various theories of transport in disordered systems, such as polymer transport theory36, the fractional Fokker-Planck equations or the continuous time random walk 15



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model35,37, and recent theories on diffusing diffusivity models38-39. In this transport theory, the equivalent of Θ n (t ) can be easily identified, which is comparable to the ergodicity breaking (EB) parameter, or the relative variance of the time-averaged mean square displacement over an ensemble of transport trajectories, introduced in refs 35 and 40. Quantitative analysis of the EB parameter and Θ n (t ) ’s equivalent in transport theory will provide us with valuable, new information about the ergodicity and the dynamics of disordered transport systems. Enzyme-to-enzyme variation in the catalytic rate of the gene-expression machinery proteins such as RNA polymerase and ribosome has an important consequence on the variation in the mRNA or protein levels that causes phenotypic variations among a clonal population of cells. An application of the present enzyme kinetics to a quantitative analysis of gene expression variability in various cell systems has been achieved and will be reported separately. In Summary, we have presented a new stochastic kinetics theory and an exactly solvable model for single enzymes, taking into account enzyme-to-enzyme variation in the catalytic rate, which is ubiquitous even among clonal enzymes. Our model explains recent single enzyme experimental results in which the mean product creation rate of initially activated single enzymes gradually decreases over time in time scales much longer than the single enzymatic turnover. In addition, we provided a new experimental measure, Θ n (t ) defined in eq 4, of ergodicity for single enzyme systems. Quantitative analysis of Θ n (t ) will be useful in answering the open question of whether or not state dynamics of single enzymes is ergodic and, if it is, how long does it take for single enzymes to span all enzyme states. By analyzing the time dependence of Θ n (t ) and

n(t )

on the substrate concentration, we are

16


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also able to extract quantitative information about the enzyme-substrate association reaction rate, the mean and variance in the lifetime of the enzyme-substrate complex, and the probability of the enzyme-substrate complex undergoing the catalytic reaction or the dissociation reaction, which cannot be obtained by the conventional Lineweaver-Burk plot analysis. Supporting Information. The supporting Information is available free of charge on ACS Publication website at DOI: 6 items in Supplementary Methods for derivation of key results, 4 items in Supplementary Notes for additional discussions, 3 items in Supplementary Figures: Quantitative analysis of β-galactosidase activity in femtoliter droplets, Counting statistics of product molecules by a single enzyme with a statically heterogeneous activity, The non-ergodicity measure and the time correlations function of catalytic rate fluctuation.

Acknowledgment Authors are grateful to Mr. Luke Bate for his careful reading of our manuscript. This work was supported by the Creative Research Initiative Project program (2015R1A3A2066497) funded by National Research Foundation (NRF) of the Korean government; the NRF grant (MSIP) (2015R1A2A1A15055664); and the Priority Research Center Program through the NRF (2009-0093817), funded by the Korean government.

17



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Figures

Figure 1. (a) The number n(t )* of product molecules created by initially activated single enzymes with slow transition dynamics between “on” state and “off” states. The value of the success probability, p2 , of the catalytic turnover is 1 (0) for the enzyme at the “on (off)” state. (Colored lines) simulation time traces for n(t )∗ ; (circles) the mean product number, *

n(t ) , calculated from the simulation results; (black line) the prediction of eq 1. The value of n(t )∗ shows a wide enzyme-to-enzyme variation. In the model calculation, ϕ1 (t ) and

ϕ 2( −1) (t ) in eq 1 are chosen as k1[ S ]e − k [ S ]t and p2( −1) tES 1

−1

e

− t / tES

, respectively. The

concentration, [ S ] , of substrate molecules is set equal to the value of the MM constant, K M = ( k1 〈t ES 〉 )

(k

off

e

− ( kon + koff ) t

lifetime,

−1

.

+ kon

For

) (k

on

this

model,

k2 (t )k2 (0)

k2

in

eq

1

is

given

by

+ koff ) . (See the Supporting Information, Note 1). The mean

tES , of an enzyme-substrate complex is the unit time. The rate, kon , of the 21



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transition from the “off” to the “on” state is 7.5 × 10−4

tES

−1

Page 22 of 23

. The transition rate, koff , from

the “on” to the “off” state is 3kon . (b) Lineweaver-Burk (LB) plot showing the substrate concentration dependence of

( n(t )

*

dt

)

−1

at different times. The slope of the line in the

LB plot increases with time, which is given by the mean product number

n(t )

*

k2

k2 (t )k2 (0) . (c) Time-dependence of

for various values of fluctuation rates, kon and koff . The

value of kon is listed in the legend; the value of koff is 3kon . (d) The time dependence of the mean product creation rate, d n(t )

*

dt , which decreases over time. These results of the

present enzyme model are consistent with the experimental results reported in refs 1 and 2. In (b)-(d), the MM kinetics is recovered only in the long time limit or in the fast fluctuation limit.

22


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(

Figure 2. Time dependence of Θ n (t ) ≡ (σ n2 (t ) − 〈 n(t )〉 ) 〈 n(t )〉 2

)

for a system of single

enzymes. (a) Θ n (t ) , for an ergodic enzyme system with lim φk2 (t ) = 0 , vanishes in the long t →∞

∞ time limit. If t k2  ≡ ∫ dtφk2 (t )  exists, Θ n (t ) follows the t −1 relaxation behaviour at long  0 

times. Otherwise, the relaxation dynamics of Θ n (t ) depends on the details of the functional form of φk2 (t ) . Dotted lines are the long-time asymptotes in each case. (b) Θ n (t ) for a system of enzymes with three different states, Γ 0 , Γ1 and Γ 2 . The probability p2 of the catalytic turnover is dependent on the state, i.e., p2 (Γ 0 ) =, 0.7 , and p2 (Γ 2 ) = 0 p2 (Γ1 ) = 1.

c(d ) designates the transition rates between Γ 0 (Γ1 ) and Γ1 (Γ 2 ) . The line and circles represent the prediction made by eq 4 and the stochastic simulation results, respectively.

lim Θ n (t ) does not vanish for a non-ergodic enzyme system whose state dynamics does not t →∞

span all the possible states.

23


Nonclassical Kinetics of Clonal yet Heterogeneous Enzymes - The

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