The inside of a cell is a highly complex environment. In the past two decades, detailed measurements of the chemical and biophysical properties of the cytoplasm have established that the conditions in which intracellular reactions occur are, by and large, very different than those typically maintained in laboratory conditions. One of the outstanding differences between in vivo and in vitro conditions, is that in the former, biochemical reactions typically occur in minuscule reaction volumes [1]. For example, in eukaryotic cells, many biochemical pathways are sequestered within membrane-bound compartments, ranging from ~50 nm diameter vesicles to the nucleus, which can be several microns in size [2]. It is also found that the total concentration of macromolecules inside both prokaryotic and eukaryotic cells is very large [3, 4], of the order of 50 - 400 mg/ml which implies that between 5% and 40% of the total intracellular volume is physically occupied by these molecules [5]. The concentration of these crowding molecules is highly heterogeneous (see for example [6]), meaning that typically one will find small pockets of intracellular space, characterized by low macromolecular crowding, surrounded by a "sea" of high crowding; such pockets of space may serve as effective compartments where reactions may occur more easily than in the rest of the cytosol. Analysis of experimental data for the dependence of diffusion coefficients with molecular size suggests the length scale of such effective compartments is in the range 35-50 nm [7], a size comparable to that of the smallest vesicles. The significant crowding also suggests that frequently an active means of transport such as vesicle-mediated transport, may be more desirable than simple diffusion as a means of intracellular transport.

The volume of a spherical cavity of space of diameter 50 nm is merely ~6.5 × 10^-20 liters, an extremely small number compared to the typical macroscopic reaction volumes of in vitro experiments (experimental attolitre biochemistry is still in its infancy - see for example [8]). These very small reaction volumes imply that at physiologically relevant concentrations (nano to millimolar), the copy number of a significant number of intracellular molecules is very small [1] and consequently that intrinsic noise cannot be ignored; for example 255 μM corresponds to an average of just 10 molecules in a 50 nm vesicle and fluctuations about this mean of the order of 3 molecules [9].

The traditional mathematical framework of physical chemistry ignores the basic physical properties of the intracellular environment. Kinetics are described by a set of coupled ordinary differential equations which implicitly assume (i) that the reaction compartment is so large that molecular discreteness can be ignored and that hence integer numbers of molecules per unit volume can be replaced by a continuous variable, the molar concentration. Since the number of molecules is assumed to be very large, stochastic fluctuations are deemed negligible and the equations are hence deterministic; (ii) the reaction compartment is well-stirred so that homogeneous conditions prevail throughout [9]. Both assumptions can be justified for reactions occurring in a constantly stirred reactor of macroscopic dimensions. However if diffusion is the dominant transport process inside the compartment then the homogeneity assumption holds only if the volume is small enough so that in the time between successive reactions, a molecule will diffuse a distance much larger than the size of the compartment. This comes at the expense of the first assumption. It hence appears natural that for intracellular applications, the first assumption, namely that of deterministic kinetics cannot be justified a priori. The second assumption can be justified if reactions are localized in sufficiently small parts of the cell and in particular for reaction-limited processes i.e. those for which the typical time for two molecules to meet each other via diffusion is much less than the typical time for them to react if they are in close proximity. For such conditions, a molecule will come within reaction range several times before participating in a successful reaction, in the process sampling the compartment many times which naturally leads to well-mixed conditions [9–11].

In this article we seek to understand the magnitude of deviations from the classical kinetic equations in small intracellular compartment volumes. We specifically focus on the case of reaction-limited enzyme reactions which allows us to relax the first assumption of physical chemistry while keeping the second one; this makes the mathematics tractable. We quantify deviations from classical kinetics in the context of the Michaelis-Menten (MM) equation; this is the cornerstone of present day enzyme kinetics and is a derivative of the traditional deterministic mathematical framework based on ordinary differential equations. In steady-state metabolic conditions, it is predicted to be exact. Thus this equation is ideal as a means to accurately test the effects of small-scale compartmentation on chemical kinetics. We consider three successive biological models of intracellular enzyme kinetics, each one building on the biological detail and realism from the previous one (Figure 1). The models incorporate the intrinsic noisiness of kinetics in small compartments, the details of the substrate transport process to the compartment (diffusion or active transport) and the presence of intra-compartmental molecules other than substrate molecules which may modulate the enzyme-catalyzed process e.g. inhibitors. On the macroscopic level, i.e. for large volumes, the steady-state kinetics of all models conform to the MM equation. We test whether this equation holds on the on the length scale of small intracellular compartments by deriving the dependence of the ensemble averaged rate of product formation on the ensemble-averaged substrate concentration from the corresponding stochastic models in the steady-state. It is shown via both calculation and stochastic simulation that at these small length scales the MM equation breaks down, being replaced by a new more general equation. Practical consequences of this breakdown are illustrated in the context of the calculation of the effective dosage of enzyme inhibitor drug needed to suppress intra-compartmental enzyme activity by a given amount. To make our approach accessible to readers not familiar with stochastic equations and their analysis, in the Results/Discussion sections we mainly focus on the biological/biophysical context and implications of the models together with the main mathematical results which are verified by simulation. Detailed mathematical derivations and the methods of simulation are relegated to the Methods section.

In this last section we discuss some fine points regarding: (i)the assumptions behind the use of master equations which throws light on the range of use of the derived mesoscopic equations, (ii) the use of the system-size expansion to perturbatively solve the master equation and (iii) the assumption of steady-state metabolic conditions. We conclude by placing our work in the context of previous recent studies of stochastic enzyme kinetics and discuss possible experiments to verify some of the conclusions we have reached.

We have implicitly assumed throughout the article that a single (global) master equation model suffices to capture the deviations from classical kinetics due to fluctuations in chemical concentrations inside a single subcellular compartment. As noted by Baras and Mansour [23], "the global master equation selects the very limited class of exceptionally large fluctuations that appear at the level of the entire system, disregarding important nonequilibrium features originated by local fluctuations." Hence the results presented here necessarily underestimate the possible deviations from classical kinetics, in particular the local fluctuations due to diffusion of molecules inside the compartment. These local fluctuations are typically small for reaction-limited processes (as in this article) but significant for diffusion-limited ones. To capture them effectively, one would be required to spatially discretize the compartment into many small elements and describe the reaction-diffusion processes between these elements by means of a multivariate master equation [12, 23]. The latter is known as a reaction-diffusion master equation; typically it does not allow detailed analytical investigation as for a global master equation and one is limited to stochastic simulation. Use of the global master equation is also restricted for compartments which are not too small: in particular the linear dimensions of the compartment should be larger than the average distance traveled by a molecule before undergoing a successful reaction with another molecule i.e. the length scale is much larger than that inherent in molecular dynamics simulation [23].

We have applied the systematic expansion due to van Kampen to perturbatively solve the master equation. It is sometimes a priori assumed that because this expansion is about the macroscopic concentrations, it cannot give information regarding the stochastic kinetics of few particle/small volume systems. This is true if one restricts oneself to the expansion to order Ω⁰ i.e. the linear-noise approximation; this is commonly the case found in the literature since the algebra becomes tedious if one considers more terms. However we note that as argued and shown by van Kampen himself [12], terms beyond the linear-noise approximation in the system-size expansion add terms to the fluctuations that are of order of a single particle relative to the macroscopic quantities and are essential to understanding how fluctuations are affected by the presence of non-linear terms in the macroscopic equation (substrate-enzyme binding in our case). In our theory we went beyond the linear-noise approximation. We find that the predicted theoretical results are in reasonable agreement, in many cases (comparison of bold and italic values in Tables 1, 2 and 3), with stochastic simulations of just a few tens of enzyme molecules in sub-micron compartments, which justifies our methodology.

We have also imposed metabolic steady-state conditions inside the subcellular compartment. Technically this is convenient since in such a case one does not deal with complex transients. Also since under such conditions the MM equation is exact from a deterministic point of view, it provides a very useful reference point versus which to accurately compute deviations due to intrinsic noise. In reality one may not always have steady-state conditions inside cells, this depending strongly on the rate of substrate input relative to the maximum rate at which the enzyme can process substrate. Another possibility is that one is dealing with a batch reaction i.e. one in which a number of substrate molecules are transported at one go and just once to the subcellular compartment (e.g. via vesicle-mediated transport) and the reaction proceeds thereafter without any further substrate replenishment. This latter scenario is compatible with the presentation of the MM equation typical in standard physical chemistry textbooks. The MM equation is then an approximation (not exact as in steady-state case) to the deterministic kinetics, when substrate is present in much larger concentration than enzyme. This case is currently under investigation using the same perturbative framework used in this article.

We note that this is not the first attempt to study stochastic enzyme kinetics. The bulk of recent studies [24–27] have focused on understanding the kinetics of a Michaelis-Menten type reaction catalyzed by a single enzyme molecule. Deviations from classical kinetics were found to be most pronounced when one takes into account substrate fluctuations [26]. These pioneering studies were restricted to a single-enzyme assisted reaction which reduces complexity thereby making it ideal from a theoretical perspective; since the reaction is dependent on just a single enzyme molecule one also finds maximum deviations from deterministic kinetics. In reality, it is unlikely to find just one enzyme molecule inside a subcellular compartment - as mentioned in the introduction a physiological concentration of just a few hundred micromolar would correspond to few tens inside the typically smallest subcellular compartment. It is also the case that diffusion may not always be the main means of substrate transport to the compartment and that the reaction maybe more complex than the simple Michaelis-Menten type reaction of these previous studies. The present study fills in these gaps by using a systematic method to derive approximate and relatively simple analytic expressions for mesoscopic rate equations describing the kinetics of the general case of N enzyme molecules in a subcellular compartment with or without active transport of substrate and in the presence of enzyme inhibitors. Most importantly our approach shows the effects of intrinsic noise on the kinetics can be captured via effective ordinary differential equations. This enables quick estimation of the magnitude of stochastic effects on reaction kinetics and thus gives insight into whether a model or parts of a model should be designed to be stochastic or deterministic without the need for extensive stochastic simulation. In the present study, this approach enabled us to readily compute, for the first time, the deviations from deterministic kinetics for a broad range of realistic in vivo parameter constants (Tables 1, 2 and 3), a task which would be considerably lengthy if one had to rely solely on data obtained from ensemble-averaged stochastic simulations.

We conclude by briefly discussing possible experiments which can verify the predictions made in this article. It is arguably not an easy task to perform the required experiments in real-time in a living cell. A viable alternative would consist of monitoring reaction kinetics inside single artificially-made vesicles. Pick et al [8] have shown that the addition of cytochalasin to mammalian cells induces them to extrude from their plasma membrane minuscule vesicles of attolitre volume with fully functional cell surface receptors and also retaining cytosolic proteins in their interior. The change in the intra-vesicular calcium ion concentration in response to surface ligand binding was measured using fluorescence confocal microscopy (FCM). Since the vesicle sizes are of typical small sub-cellular compartment dimensions (1 attolitre corresponds to a spherical vesicle of approximate diameter 120 nm) and FCM allows the measurement of the concentration of a fluorescent probe (via a calibration procedure), this experimental technique appears ideal to verify the predictions of Model I and of Model III for the case of diffusive substrate transport. Model II and Model III with vesicle-transport of substrate are probably much more challenging to verify since one then needs to construct the in vitro equivalent of microtubules. This is within the scope of synthetic biology and may be a possibility in the next few years.

We here provide full details of the calculations reported in the Results section. The system size-expansion which is at the heart of the analysis has to-date not been applied extensively to biological problems and thus we go into some detail in its elucidation in Sub section I, which is dedicated exclusively to Model I. For other recent applications of the general method in the context of reaction kinetics, see for example [28] and [29]. Subsections II and III (treating Model II and Model III, respectively) naturally build on the results of the first subsection and thus we only give the main steps of the calculations in these last two cases. Sub section IV has a brief discussion of the simulation methods used to verify the theoretical results.

Model I: Michaelis-Menten reaction occurring in a compartment volume of sub-micron dimensions. Substrate input into compartment is modeled as a Poisson process

The reaction scheme is . The stochastic description of this system is encapsulated by the master equation which is a differential equation in the joint probability function π describing the system:

(13)

where π = π(n _C , n _P , n _S ), n _X is the integer number of molecules of type X (where X = C, P, S), Ω is the compartment volume, and are the step operators defined by their action on a general function g(n _X ) as: g(n _X ) = g(n _X ± 1). Note that the relevant variables are three, not four: the integer number of molecules of free enzyme (n _E ) is not an independent variable due to the fact that the total amount of enzyme is conserved. The master equation cannot be solved exactly but it is possible to systematically approximate it by using an expansion in powers of the inverse square root of the volume of the compartments. This is generally called the system-size expansion [12].

The method proceeds as follows. The stochastic quantity, n _X /Ω, fluctuates about the macroscopic concentrations [X]; these fluctuations are of the order of the square root of the number of particles:

(14)

Note that since n _E + n _C = constant, it follows that n _E = Ω[E] - Ω^1/2ϵ _C . The joint distribution function and the operators can now be written as functions of the new variables, ϵ _X , giving: π = Π(ϵ _C , ϵ _P , ϵ _S , t) and ; using these new variables the master equation Eq. (13) takes the form:

(15)

where

(16)

(17)

(18)

Note that in Eq. (18) terms which involve products of first and second-order derivatives, third-order derivatives or higher have been omitted - these do not affect the low-order moment equations which we will be calculating.

Analysis of Ω^1/2 terms

The terms of order Ω^1/2 are the dominant ones in the limit of large volumes. By equating both terms of this order on the right and left hand sides of Eq. (15) and using Eq. (16), one gets the deterministic rate equations:

(19)

(20)

(21)

These are exactly those which one would write down based on the classical approach whereby one ignores molecular discreteness and fluctuations. This is an important benchmark of the method since it shows that it gives the correct result in the limit of large volumes. On a more technical note, the cancelation of these two terms of order Ω^1/2 makes Eq. (15) a proper expansion in powers of Ω^-1/2. By imposing steady-state conditions we have the Michaelis-Menten (MM) equation:

(22)

where v _max = k₂ [E _T ] is the maximum reaction velocity, [E _T ] = [E] + [C] is the total enzyme concentration which is a constant at all times and K _M = (k₁ + k₂)/k₀ is the Michaelis-Menten constant.

Analysis of Ω⁰ terms

To this order, the master equation is a multivariate Fokker-Planck equation whose solution is Gaussian and thus fully determined by its first and second moments. The equations of motion for these moments can be straightforwardly obtained from the master equation to this order, leading to a set of coupled but solvable ordinary differential equations:

(23)

(24)

where,

(25)

Note that the matrices and vectors in the above equations have been reduced to a simpler form by the application of a few row operations. Note also that these equations are independent of ϵ _p since the product-forming step is irreversible and hence the fluctuations in substrate and complex are necessarily decoupled from its fluctuations. At the steady-state, it is found that ⟨ϵ _S , _C ⟩ → 0. From Eq. (14), it is clear that this implies that to this order the average number of substrate molecules per unit volume, ⟨n _S /Ω⟩, is simply equal to the macroscopic concentration, [S]. The same applies for complex molecules. Hence to this order in the system-size expansion there cannot be any corrections to the macroscopic equations or to the MM equation. By writing the macroscopic concentrations in Eqs. (24) and (25) in terms of k _in and solving, one obtains the variance and covariance of the fluctuations about the steady-state macroscopic concentrations. We here only give the result for the covariance since this will be central to our discussion later on:

(26)

where α = k _in /v _max is the normalized reaction velocity of the enzyme.

Analysis of Ω^-1/2 terms

The system-size expansion is almost never carried out to this order because of the algebraic complexity typically involved, however it is crucial to find finite volume corrections to the deterministic rate equations and in particular to the MM equation. Using the master equation to this order, the first moment of the complex concentration is governed by the equation of motion:

(27)

Now the production of product P from complex occurs through a decay process which necessarily has to be described by a linear term of the form: k _in = k₂⟨n _C /Ω⟩ (the steady-state condition). Since the steady-state macroscopic complex concentration is equal to [C] = k _in /k₂, then it follows that to any order in the expansion we have ⟨ϵ _C ⟩ = 0. This is always found to be the case in simulations as well. Hence it immediately follows from Eq. (27) that the average of fluctuations about the macroscopic substrate concentration are non-zero and given by:

(28)

From a physical point of view, this indicates that the steady-state concentration of substrate in the compartment is not equal to the value predicted by the MM equation (i.e. [S]) and hence the non-zero value of the average of the fluctuations about [S]. The real substrate concentration inside the compartment is obtained by substituting Eqs. (28) and (26) in Equation (14), leading to:

(29)

An alternative mesoscopic rate equation replacing the MM equation

The renormalization of the steady-state substrate concentration indicates the breakdown of the MM equation; this phenomenon occurs because of non-zero correlations between noise in the substrate and enzyme concentrations, ⟨ϵ _S ϵ _C ⟩, which the MM equation implicity neglects. To obtain the alternative to the latter, one needs to obtain a relationship between the normalized reaction velocity, α and the real substrate concentration ⟨n _S /Ω⟩; writing [S] in terms of α and substituting in Eq. (29), one obtains this new relation:

(30)

(31)

Note that in the limit of large volumes, the second term on the left hand side of Eq. (30) becomes vanishingly small and we are left with the MM equation. In the results section the quantity on the right hand side of Eq. (30) is referred to as α _M since this is the normalized reaction velocity which would be predicted by the MM equation given the measured substrate concentration ⟨n _S /Ω⟩ inside the compartment. A quick estimate of the magnitude of the error that one is bound to incur by using the conventional MM equation can be obtained by substituting α = 1/2 (i.e. enzyme is half saturated with substrate) in Eqs. (30) and (31), solving for α _M and then using this value to compute the fractional error e = 1 - α _M /α. This leads to the simple expression:

(32)

We finish this section by noting that Eq. (30) will be found to be valid generally and not only for the simple Michaelis-Menten scheme treated in this section; the details of the reaction network come in through the form of Eq. (31) which is reaction-specific.

Model II: Michaelis-Menten reaction occurring in a compartment volume of sub-micron dimensions. Substrate is input into compartment in groups or bursts of M molecules at a time

A natural generalization of Model I which has direct biological application is when substrate molecules are fed into the compartment M at a time with mean rate . The total mean substrate input rate is then equal to k _in = M . The master equation for this process is Eq. (13) with the first term on the right hand side replaced by . This leads to the following change in the expression for a₂ (Eq. 17):

(33)

Note that since the expression for a₁ (Eq. 16) is unchanged, the deterministic equations are precisely the same as those of Model I. However now the fluctuations about the macroscopic substrate concentration are enhanced by a factor M; consequently the entries in the vector B in Eq. (25) need the change k _in → k _in M. The analysis proceeds in the same manner as before. The mesoscopic rate equation replacing the MM equation is now given by Eq. (30) together with:

(34)

The fractional error rate evaluated at α = 1/2 gives:

(35)

This clearly shows that generally larger deviations from the predictions of the MM equation are expected in this case compared to those computed for Model I.

Model III: Michaelis-Menten reaction with competitive inhibitor occurring in a compartment volume of sub-micron dimensions. Substrate input as in two previous models

Competitive inhibition is modeled by the set of reactions: , where k₄ = [I] and [I] is the inhibitor concentration (similar models have been studied by Roussel and collaborators [30, 31] in the context of biochemical oscillators though these assume M = 1). In the rest of this section, we change the notation of enzyme-inhibitor complex from EI to V, just to make the math notation easier to read. The substrate input into the compartment is considered to occur as in Model II since this encapsulates that of Model I as well. The master equation for this system is:

(36)

The change of variables from n _X to ϵ _X is done as before, however note that now the conservation law for enzyme is different than in the two previous models. The total enzyme concentration is now equal to [E _T ] = [E] + [C] + [V] from which it follows that n _E = Ω[E] - Ω^1/2(ϵ _C + ϵ _V ). The description is chosen to be in terms of numbers of molecules of types C, S and V and thus E being a dependent variable does not show up explicitly in the step operators of the master equation above.

Due to the significant number of changes in the terms of the expansion from those of previous models, we will show the equivalent of Eqs. (15)-(18) in full. The master equation in the new variables ϵ _X is given by:

(37)

where

(38)

(39)

(40)

Analysis of Ω^1/2 terms

As for previous models, these terms give the macroscopic equations. Equating both terms of this order on the right and left hand sides of Eq. (37) and using Eq. (38), one obtains:

(41)

(42)

(43)

(44)

In the steady-state we have the Michaelis-Menten (MM) equation:

(45)

where β = [I]/K _i and K _i = k₃/ is the dissociation constant of the inhibitor.

Analysis of Ω⁰ and Ω^-1/2 terms

The equations for the first moments are easily obtained and we shall not reproduce them here; suffice to say that at steady-state, it is found that ⟨ϵ _S , _C , _V ⟩ → 0 which implies that to this order in the system-size expansion there cannot be any corrections to the macroscopic equations or to the MM equation. The addition of a new species, V, does however substantially increase the algebraic complexity in the equations of motion for the second moments computed using terms up to order Ω⁰. In particular the matrix A is now a 6 × 6 matrix, rather than the 3 × 3 matrix of the previous two models.

(46)

where,

(47)

and

(48)

In the above equations we have defined k' = k₃ + k₄. Note also that the system of equations has been simplified through the application of a few row operations.

Now to next order, i.e. Ω^-1/2, the first moments of the concentrations of molecules of type C and V are governed by the equation of motions:

(49)

(50)

As in previous models, since the production of product P from complex occurs through a decay process, it follows that at steady-state, ⟨ϵ _C ⟩ = 0 which also implies ⟨ϵ _V ⟩ = 0 from Eq. (50). Hence it follows from Eq. (49) that ⟨ϵ _S ⟩ = [⟨ϵ _S ϵ _C ⟩ + ⟨ϵ _S ϵ _V ⟩]/Ω^1/2 [E]. The two cross correlators can be estimated to order Ω⁰ by solving Eqs. (46)-(48). The non-zero value of ⟨ϵ _S ⟩ implies a renormalization of the substrate concentration inside the compartment and hence to a new rate equation replacing the MM equation. This is obtained exactly in the same manner as previously shown for Model I. The mesoscopic rate equation is found to be given by Eq. (30) together with:

(51)

where the numerator coefficients are given by:

(52)

(53)

(54)

(55)

(56)

and the denominator coefficients by:

(57)

(58)

(59)

(60)

(61)

(62)

Note that = 0 such that at α = 0, there is no correction to the MM equation i.e. α _M = 0 also. The case β = 0 reduces to Model II, i.e. f(α) is given by Eq. (34).

Stochastic simulation

In this section we briefly describe the simulation methods used to verify the theoretical results which are described in detail in the Results section. All simulations were carried out using Gillespie's exact stochastic simulation algorithm, conveniently implemented in the standard simulation platform, Dizzy [32].

The data points in Figure 2 were generated by iterating the following four-step procedure: (i) pick a value for α between 0 and 1. This gives the substrate input rate k _in = α v _max ; (ii) run the simulation and measure the ensemble-averaged substrate concentration, ⟨n _S /Ω⟩ = [S*] at long times; (iii) compute α _M using the MM equation, α _M = [S*]/([S*]+ K _M ); (iv) compute the absolute percentage error R _p = 100|(1 - α _M /α)|. The solid curves in Figure 2 were obtained by numerically solving the cubic polynomial in α given by Eqs. (7) and (8) in the Results section for given values of α _M and then using the above expression for R _p . Figure 3 is generated in the same manner as Figure 2, except that: in step (i) we fix M and pick a value for α between 0 and 1. Since k _in = M , the required simulation parameter is = α v _max /M; step (iv) is not needed. The solid curves were obtained by numerically solving the cubic polynomial in α given by Eqs. (7) and (9) in the Results section for given values of [S*]. The y-axis for this figure is v/v _max = α _M for the MM equation and v/v _max = α for the stochastic model. Figure 4 is obtained by numerically solving the quintic polynomial in α given by Eqs. (7) and (12) in the Results section together with the coefficients given by Eqs. (52)-(62) in the present section; the inhibitor concentration, [I], is varied while the substrate concentration, [S*], is kept fixed. The substrate concentration is chosen so that at [I] = 0, v/v _max = 0.909 in all cases. Note that for models I and II, α _M = [S*]/([S*] + K _M ) while for Model III, α _M = [S*]/([S*] + (1 + β)K _M ). Note that the error bars are very small on the scale of the figures and thus are not shown.