Investigating the robustness of the classical enzyme kinetic equations in small intracellular compartments
© Grima; licensee BioMed Central Ltd. 2009
Received: 2 June 2009
Accepted: 8 October 2009
Published: 8 October 2009
Classical descriptions of enzyme kinetics ignore the physical nature of the intracellular environment. Main implicit assumptions behind such approaches are that reactions occur in compartment volumes which are large enough so that molecular discreteness can be ignored and that molecular transport occurs via diffusion. Though these conditions are frequently met in laboratory conditions, they are not characteristic of the intracellular environment, which is compartmentalized at the micron and submicron scales and in which active means of transport play a significant role.
Starting from a master equation description of enzyme reaction kinetics and assuming metabolic steady-state conditions, we derive novel mesoscopic rate equations which take into account (i) the intrinsic molecular noise due to the low copy number of molecules in intracellular compartments (ii) the physical nature of the substrate transport process, i.e. diffusion or vesicle-mediated transport. These equations replace the conventional macroscopic and deterministic equations in the context of intracellular kinetics. The latter are recovered in the limit of infinite compartment volumes. We find that deviations from the predictions of classical kinetics are pronounced (hundreds of percent in the estimate for the reaction velocity) for enzyme reactions occurring in compartments which are smaller than approximately 200 nm, for the case of substrate transport to the compartment being mediated principally by vesicle or granule transport and in the presence of competitive enzyme inhibitors.
The derived mesoscopic rate equations describe subcellular enzyme reaction kinetics, taking into account, for the first time, the simultaneous influence of both intrinsic noise and the mode of transport. They clearly show the range of applicability of the conventional deterministic equation models, namely intracellular conditions compatible with diffusive transport and simple enzyme mechanisms in several hundred nanometre-sized compartments. An active transport mechanism coupled with large intrinsic noise in enzyme concentrations is shown to lead to huge deviations from the predictions of deterministic models. This has implications for the common approach of modeling large intracellular reaction networks using ordinary differential equations and also for the calculation of the effective dosage of competitive inhibitor drugs.
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 . 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 . 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 . The concentration of these crowding molecules is highly heterogeneous (see for example ), 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 , 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 ). 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  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 .
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 . 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].
Model I: Michaelis-Menten reaction occurring in a compartment volume of sub-micron dimensions. Substrate input into compartment occurs via a Poisson process
Hence the prediction of the MM equation is only correct, i.e. α = α M , in the limit of infinitely large compartment volumes, in which case the second term on the left hand side of Eq. (7) will become vanishingly small and can be neglected. For finite compartment volumes, the MM equation is not exact (except in the two limiting cases of α M → 0 and α M → 1) but is at best an approximation, even though steady-state conditions are imposed; this is at odds with the prediction of the conventional deterministic theory. An inspection of Eqs. (7) and (8) shows that the magnitude of the deviations from the MM equation depends on the two non-dimensional quantities: (i) K M Ω, a measure of the rate at which enzyme-substrate combination events occur relative to the rate of decay of complex molecules; (ii) [E T ]Ω, the total integer number of enzyme molecules in the compartment.
As shown in the Methods section, the MM equation is found to implicity assume that the noise about the macroscopic substrate and enzyme concentrations is uncorrelated (this assumption has generally been found to be at the heart of many macroscopic models - for example see ); properly taking into account these non-zero correlations leads to the corrections encapsulated by Eqs. (7) and (8). These correlations are expected to be small in two particular cases: (i) if K M is large; in this case when substrate molecules combine with an enzyme to form a complex, the latter dissociates very quickly back into free enzyme and thus successive enzyme-substrate events to the same enzyme molecule are bound to be almost independent of each other. The opposite situation of small K M would imply that the bottleneck in the catalytic process is the decay of complex rather than enzyme-substrate combination; if a successful combination occurs, the next substrate to arrive to the same enzyme molecule would have to wait until the complex decays, naturally leading to correlations between successive enzyme-substrate combination events. (ii) if the total number of enzyme molecules is large; in such a case, at any one time, the noise about the macroscopic concentrations will be the sum total from a large number of enzymes, each at a different stage in the catalytic process and each independent from all others, which naturally dilutes any temporal correlations.
Maximum Percentage error in reaction velocity from prediction of the MM equation for Model I.
K M = 10 μM
The theory is always found to underestimate the actual deviations predicted by simulations; hence the theoretical expressions provide a quick, convenient way by which one can generally estimate a lower bound on the deviations to be expected from the MM equation without the need to perform extensive stochastic simulation.
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
Model I captures the basics of a general enzyme-catalyzed process occurring in a small intracellular compartment. In this section we build upon this model to incorporate further biological realism. In particular, in the previous model we assumed that substrate input can be well described by a Poisson process, where one molecule at a time is fed into the compartment with some average rate k in . This is the simplest possible assumption and approximates well the situation in which molecules are brought to the compartment via normal diffusion. However there are many situations where this may not be the case; we now describe two such cases.
The intracellular condition of macromolecular crowding limits the Brownian motion of molecules in the cytoplasm, this being reflected in the relatively small diffusion coefficients measured in vivo compared to those known in vitro for moderately to relatively large molecules. Experiments with inert tracer particles in the cytoplasm of Swiss 3T3 cells show that the in vivo diffusion coefficient is an order of magnitude less than that in vitro for molecules with hydrodynamic radius 14 nm and diffusion becomes negligibly small for molecules larger than approximately 25 nm ; similar results have been obtained in Xenopus neurons  and skeletal muscle myotubes . If diffusion is considerably hindered, one expects active transport to become a more desirable mode of transport. Indeed there exists ample evidence for the active transport of macromolecules: they are typically packaged in a vesicle or a granule which is then transported along microtubules or by some other means. It is also found that each vesicle or granule typically contains several of these molecules (examples are: mRNA molecules - several estimated per granule [20, 21]; cholesterol molecules which are transported in low-density lipoproteins  - approximately 1500 per lipoprotein).
Generally an active means of transport is not exclusively linked with the transport of large substrate molecules. The cell being a highly compartmentalized and dynamic entity requires for its survival the precise transport of certain molecules from one compartment to another and a regulation of this transport depending on its current physiological state. Brownian motion leads to an isotropic movement of molecules down the concentration gradient and to a consequent damping of the substrate concentration with distance. In contrast active transport provides a directed (anisotropic) means of transport with little or no loss of substrate with distance, is independent of the concentration gradient and it is also easily amenable to modulation.
Maximum Percentage error in reaction velocity from prediction of the MM equation for Model II.
K M = 10 μM
Model III: Michaelis-Menten reaction with competitive inhibitor occurring in a compartment volume of sub-micron dimensions. Substrate input as in previous models
Maximum Percentage error in reaction velocity from prediction of the MM equation for Model III.
K M = 10 μM
M (burst size)
Discussion and Conclusion
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 , "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 .
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 , 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 . 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  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  and . 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
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 .
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 Ω1/2 terms
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
where α = k in /v max is the normalized reaction velocity of the enzyme.
Analysis of Ω-1/2 terms
An alternative mesoscopic rate equation replacing the MM equation
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
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
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] - Ω1/2(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.
Analysis of Ω1/2 terms
where β = [I]/K i and K i = k3/ is the dissociation constant of the inhibitor.
Analysis of Ω0 and Ω-1/2 terms
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.
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).
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 .
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.
It is a pleasure to thank Arthur Straube and Philipp Thomas for interesting discussions. The author gratefully acknowledges support from SULSA (Scottish Universities Life Sciences Alliance).
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