J. Phys. Chem. B 2001, 105, 7099-7103
7099
Microscopic Self-Organization in Biochemical Reactions: A Lattice Model Hongli Wang, Qi Ouyang,* and Yi-an Lei Department of Physics, Mesoscopic Physics Laboratory, Peking UniVersity, Beijing 100871, People’s Republic of China ReceiVed: January 22, 2001; In Final Form: April 30, 2001
The interesting spontaneous synchronization of enzymatic turnover cycles proposed by Mikhailov and Hess ( Hess, B.; Mikhailov, A. J. Phys. Chem. 1996, 100, 19059.) has been investigated with a lattice model. We demonstrate that good synchronization activities of enzymatic turnover cycles found in the original stochastic automaton can be developed in this more realistic model when the correlation length created by the fast diffusion of regulatory particles is large enough. Results presented here indicate that the microscopic selforganization is realistically possible in actual biochemical systems if enzymatic turnover times have a weak dispersion.
1. Introduction One of the most striking features of a living cell is its high degree of self-organization. Its origin has been always fascinating. For an ordinary cell, the aqueous cytosol bounded by a membrane is enclosed in a small volume typical of few microns. It contains macromolecules including proteins, enzymes, ribonucleic acids, as well as organic compounds and ions, and also has a number of sub-cellular compartments, each of which having a specific function. In the eyes of a chemist, the cell can be considered as a tiny reactor where thousands of chemical reactions take place simultaneously. The reaction and diffusion processes are carried on under strong nonequilibrium conditions since a living cell is an open system, constantly exchanging energy and materials with its environment. This opens up the possibility of self-organization.1 The best examples along this line are probably the patterns of calcium oscillations and spatial waves observed within various cells.2,3 Other examples include microtubule formations.4 The self-organization processes in living cells were usually understood in terms of classical chemical kinetics.5 Hess and Mikhailov6 have suggested, however, that due to the particular space and time scales that are typical for intracellular processes, self-organization behavior in living cells is not a reduced copy of patterns and oscillations in macroscopic reaction-diffusion systems. It would be occasionally necessary for the classical concepts of chemical kinetics to be revised when processes inside living cells are considered.7 In a recent series of publications,8,9,10,11 Mikhailov and his colleagues demonstrated such possibilities. They investigated an interesting coherent behavior in biochemical reactions taking place in small volumes, and pointed out that macromolecules in the process of biochemical reactions typically experience complex conformational changes. When the reactions take place in small volumes of sizes possibly comparable to cells or cellular compartments, diffusive transport and mixing time of substrates and small intermediate molecules commonly used by macromolecules can be much shorter than characteristic time of conformational changes of macromolecules. Small molecules could have thus played a role of messengers conveying reaction information on individual macromolecules, and establishing strong correlation in the system. The population of macromol-
ecules together with small molecules under this situation has been called molecular network, and coherent behaviors that arise were classified as microscopic self-organizations, which distinguishes itself from those in classical chemical kinetics. A general criterion for microscopic self-organization in chemical reactions involving small numbers of molecules and occurring in relatively small volumes has been formulated. To illustrate their general analysis, Mikhailov et al. considered simple models of enzymatic reaction with allosteric product activation10 and inhibition11 as examples. Under proper conditions, the model showed coherent behavior of mutual synchronization of individual molecular turnover cycles. The purpose of this paper is to examine this interesting microscopic self-organization behavior with a more realistic model. The situation considered in ref 9 is an extreme where the diffusion of molecules was treated as an instantaneous event. The diffusive transport of regulatory particles has been discarded, and the whole allosteric activation reaction has been simplified as a stochastic substrate binding of regulatory molecules to enzymes and subsequent stepping ahead of internal states of activated enzymes. Another extreme situation is the classical reaction kinetics where reactions are considered as instantaneous events. We investigate an intermediate situation where the rate of diffusive transport and mixing of regulatory particles is high but limited. Enzymatic activation reactions are considered on a lattice and the regulatory product molecules perform random walks on it. We show in this more realistic model, that the enzymatic reaction cycles can develop good synchronization activities under appropriate parameter values. 2. Model The biochemical reaction considered in this paper is a simple enzymatic reaction with allosteric product activation:9
S + E f ES f E + P, P f 0
(1)
where S is a substrate and E is an enzyme. The binding of S by E is allosterically activated byproduct P, i.e., when a P particle is captured by an enzyme and occupies the allosteric activation site on its surface, the enzyme changes to a new conformation, under which the binding of substrates to enzymes is promoted.
10.1021/jp010239h CCC: $20.00 © 2001 American Chemical Society Published on Web 06/22/2001
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Figure 1. Schematic map of lattice model. Pentagons represent enzymes at initial state and solid circles denote the activated enzymes. The small black dots are regulative P particles that do random walks on the lattice.
This allosteric product activation thus introduces the reaction a feedback mechanism. A complete reaction cycle can be summarized as several stages: An enzyme E is first activated by binding a substrate molecule S to form a substrate-enzyme complex ES, then the complex undergoes a sequence of conformational transformations to convert a substrate into a product molecule P. After the product has been released at a certain stage, the enzyme changes its conformation to its initial state to complete the cycle. During the recovery stage the enzyme is not ready for binding a substrate and denies starting a new cycle. Thus, the cycle can be divided into two periods: before and after the product is released. In the above model, the product P decays while activating enzymes allosterically. Consider that the reaction 1 is occurring in a very small volume of a few micrometers in size. Because of the small size of the reactor, P diffuses comparatively fast. It could have crossed the reaction volume several times during a single enzymatic turnover cycle. It has been estimated that, in a volume of a micrometer size, the time needed for a small molecule, after it is released, to be found anywhere in the reactor with equal probability is on the order of several milliseconds; while an enzymatic turnover cycle can be tens of milliseconds. Unlike the usual situation in classical reaction kinetics where reactions are considered as instantaneous events, in a small volume, the internal conformational transformation takes an important part of the whole reaction processes. At this point, the population of enzymes, products and substrates forms a molecular network, whose dynamics requires a different type of description from the usual macroscopic reaction-diffusion equations. The extreme situation where the diffusion of regulatory particles is infinitely fast and is regarded as an instantaneous process has been considered previously in the stochastic automata model.9,10 To investigate an intermediate situation between this extreme and that of the classical chemical kinetics, we here study the enzymatic reaction 1 with a more realistic lattice model. We consider the reactions of N enzymes distributed at random on a 100 × 100 square lattice. Because of their huge mass compared to the regulatory P particles and substrates, the enzymes are supposed to be immobile. P particles diffuse fast randomly on the lattice and decay at a certain rate. The substrate is assumed to be adequate, and its concentration is maintained constant. A schematic map for the lattice model is depicted in Figure 1. As in the original model,9 the internal conformational evolution of enzymes is simplified as a motion along a single internal conformational coordinate. For the
Figure 2. Simulation results with a relatively low diffusion rate. The histogram (top panel) depicts the distribution of the internal phases of enzymes over the cycle; the curve (bottom panel) is the time dependence of the product population. Parameters: N ) 800, w0 ) 0.001, w ) 0.15, g ) 0.3, K ) 120, K′ ) 60, D ) 50. τ is the enzymatic turnover time.
convenience of computer simulation, a single molecular turnover cycle τ has been discretized into K equal small steps (∆t ) τ/K), and the phase for each enzyme molecule is denoted by an integer φ(0 e φ e K). When a P molecule runs into an enzyme at initial state, it allosterically activates the enzyme with a certain probability w. The enzyme stays at initial state (φ ) 0) until it is activated. Once activated it’s internal phase φ shifts from 0 to 1, and then step ahead from 1 up to K in a deterministic manner. The enzyme returns to the initial state when it reaches the final phase K, and waits for a next activation. A P particle is released at a specific phase φ ) K′ (0 < K ′< K) within the cycle, then wanders on the lattice to trigger other reactions until it dies out. P decays with probability g per time step. An enzyme can also be activated spontaneously with a small probability w0. We carried out computer simulations of 800 (N ) 800) enzymes reacting on a 100 × 100 square lattice. K ) 120 was fixed in all calculations. As the initial condition in the simulations, a small number of P particles was distributed randomly on the lattice, and each enzyme occupied an arbitrarily chosen internal phase. We study how the reaction properties are changed when the diffusion rate of P particles is varied. The diffusion rate D was adjusted by altering the number of random steps P particles walk during the time an activated enzyme goes a step ahead, i.e., in the duration of τ/K. K′ ) 60 was fixed in our first series of simulations. To eliminate the transient effects, the simulations have been carried out for a sufficiently long time before the data is analyzed. 3. Results Figure 2 shows the simulation results when the diffusion rate is relatively low with D ) 50. From the histogram of the internal phases of enzymes, one observes that the majority of enzymes is accumulated at the initial state (φ ) 0), waiting to be activated, while the remaining enzymes are distributed over the cycle phases uniformly. The time dependence of the product is maintained at a low level with some fluctuations. In this case, the enzymatic reaction cycles have obviously not been correlated. The mean distance lc passed by a P particle during its
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Figure 3. Simulation results with a high diffusion rate, D ) 500. Other parameters are same as those in Figure 2. The enzymes population splits into two groups.
life-span can be estimated readily according to its decay rate g ) 0.3:
Ic ) xD/g ≈ 12.9
(2)
We see that this characteristic length for correlation radius of the reaction is much smaller than the lattice size (l ) 100), and is insufficient to establish an effective coherence in the system. With the low diffusion rate, most of P particles newly produced could have been annihilated before they could have an opportunity to activate effectively the enzymes at rest. The produce of P particles is mainly contributed by the spontaneous activation. When the diffusion rate is increased to D ) 500, the system undergoes a qualitative behavior change, as shown in Figure 3. The mutual synchronization between the individual enzymatic reaction cycles, as having previously been observed in the stochastic automaton,9 now shows up. The population of enzymes is divided about equally into two groups. Enzymes belonging to the same group have a narrow distribution of cycle phase and evolve synchronously in time. The shift between the central phases of these two groups is found to be half the cycle period. With this phase shift the enzymatic groups oscillate periodically in time. Accordingly, the population of P particles in the system now also varies periodically in time (bottom part, Figure 3). Each spikes relates to a synchronous release of P particle by one of the two groups. The bold line of the plot represents the time dependence of enzymes population in φ ) 0 state. Notice that the maximum of the population of enzymes at initial state coincides with maximum of the number of P particle. Thus, we see a good example of cooperation. At this time, the correlation length lc is much larger:
Ic ) xD/g ≈ 40.8
(3)
The compact radius of the reaction is much extended and is able to establish a full coherence in the system. During the synchronous evolution of the system, enzymes in a certain group seldom changes its allocation. This can be clearly seen in Figure 4. The demonstration depicts the distributions of enzymes on the lattice (left panels) and phase distributions over the turnover cycle (right panels) at two consecutive moments with an interval of half the cycle period. When enzymes in group B finish their
Figure 4. Distributions of enzymes on the lattice (left parts) and histograms of phase distribution over the turnover cycle (right parts). They were sampled at moments t ) 22820 (a), and t ) 22880 (b), with the shift of a half enzymatic cycle. Parameters are the same as those in Figure 3 except N ) 200.
Figure 5. Histogram of enzymatic phases spiking with (a) 3 groups and (b) 4 groups. All parameters are the same as those in Figure 3 except for (a) K′ ) 80, for (b) K′ ) 96, and D ) 250.
turnover and arrive to the initial state φ ) 0, they are immediately activated by P particles released by group A, which is in its maximum production of P. After a half cycle, this scenario takes place again, with enzymes in group A and B exchange its role. Accordingly, circles for enzymes of group A, which are formerly open (Figure 4a), have turned into solid (Figure 4b), and the vice versa for circles of group B. If we fix the diffusion rate of regulatory particles at the high rate, and change the value of K′, i.e., the phase stage at which enzymes release products, the enzyme population can also break into other numbers of groups, enzymes in each group react synchronously. Figure 5 show histograms spiking with three (Figure 5a) and four enzyme groups (Figure 5b). The typical oscillation behaviors are sustained because reaction cycles are highly correlated and enzymatic groups cooperate. Every time
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Figure 6. Complete synchronization of enzymes with a single coherent group, parameters are the same as those in Figure 3 except K′ ) 12.
enzymes in one group reach their final state (φ ) K) and return to the state of φ ) 0, there is always another group that is just passing through the state with φ ) K′, releasing regulative P particles which diffuse swiftly and effectively activate enzymes with phase φ ) 0. Full synchronization of all enzymes as a whole appears when products are released at an early stage of the cycle, i.e., for relatively small values of K′. Figure 6 shows such an example with K′ ) 12. The phase distribution of enzyme spikes with only one group and the entire population sustains a synchronous oscillation in time. In this case the cooperation is established from the inside of the group. The forerunners in the group passes through the phase stage φ ) K′, they release the product molecules that activate the cycles in the rear of the group. In the above simulations, the complicated intramolecular conformational dynamics of enzymes has been simplified as deterministic transitions, proceeding like a clock, between discrete phase stages along a single internal coordinate. To take into account the fluctuations in the conformational dynamics of enzymes in our model, we consider the intramolecular changes along the reaction coordinate as a diffusive process, which has been used previously to approximate the kinetics of conformational transformations of proteins12,13 and enzymes.10,14 The intramolecular dynamics of enzymes is now probabilistic: an activated enzyme has a probability pinc or pdec to increase or decrease by a phase unit. Thus, depending on these probabilities, an enzymatic turnover cycle is not a definite period of time. It fluctuates and therefore introduces extra noises into the dynamics. We studied the effect of intramolecular diffusive noises on the synchronization behavior observed above. The coherent activity was found to be sensitive to the noises. Figure 7 shows the simulation results of the two-group synchronization in Figure 3 with small probabilities of walking backward. The mean turnover time and its statistical dispersion were directly calculated in the simulations to be 124.4 and 2.7 for Figure 7a, and 129.6 and 4.2 for Figure 7b, with a relative dispersion of 2.2% and 3.2%, respectively. One observes that the previous narrow spikes (refer to Figure 3 and Figure 4) have been now greatly smeared out by the weak noises, and that the phase fluctuations within the groups were much enhanced. The coherence between the two groups still persisted, but was considerably weakened. More intensive noises with pinc ) 0.94,
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Figure 7. Effect of intramolecular noises on coherent synchronizations. (a) pinc ) 0.976, pdec ) 0.012, (b) pinc ) 0.95, pdec ) 0.025. Other parameters are the same as those in Figure 3.
pdec ) 0.04, and 4% relative dispersion of the turnover time, were found to have destroyed completely the coherence. 4. Conclusion In summary, we have investigated the coherent synchronization behavior of enzymatic reactions in small volume with the lattice model. It was demonstrated that under conditions of large but limited diffusion rate, thus sufficient correlation length for regulatory particles, strong correlations between individual enzymatic cycles can be established. This correlation generates coherent enzymatic groups that cooperate. All coherent behavior demonstrated with the stochastic automaton9 were revived in the lattice model. Our simulation results suggest that the interesting microscopic self-organization behavior proposed by Mikhailov et al.9,10 can be realizable in real biochemical systems. The requirement for the realization, however, might be severe. Our simulation results revealed that the noises in enzymatic conformational transformation dynamics have strong effects on the self-organization behavior. Small fluctuations in the duration of enzymatic cycles were found to have weakened significantly the coherence. In the model of stochastic automaton,10 the coherent enzymatic groups were found to have survived the relative large statistical dispersion of turnover times (20%), whereas in the lattice model a weak dispersion of 4% were found to be sufficient to ruin entirely the synchronization. While sustained synchronization behaviors, as predicted previously in refs 9 and 10 and here above, have not been reported in experiments, damped oscillations in the population of product molecules following initial synchronization of internal enzymatic states have been observed experimentally15 when the dispersion of turnover durations was relatively weak. Our Simulations showed that the enzyme molecules split into several subpopulation groups when the parameter K′ was changed. This obviously results in the changes of the frequency and amplitude of the temporal oscillations of the product. Because the total enzyme population is definite, a high frequency corresponds to a small amplitude. The experimental signals for the cases of complete synchronization (Figure 6) and the twosubgroup synchronization (Figure 3) will therefore be more easily detected in the laboratory. In our simulations, the enzymes were considered to spatially distribute. Simulation results showed that the coherent synchronization between enzymes’
Microscopic Self-Organization internal dynamics do not necessary bring on ordered spatial distributions. Enzymes belong to different coherent subgroups are mixed up on the lattice without any discernible order (please refer to Figure 4). The model studied here is much simplified, and ordered spatial patterns might be still possible in real experiments. Acknowledgment. This work is partly supported by National Basic Research Foundation of China. References and Notes (1) Schro¨dinger, E. What is life?; Cambridge University Press: Cambridge, 1944. (2) Jaffe, L. F. Proc. Natl. Acad. Sci. U. S. A. 1991, 88, 9883. (3) Lechleiter, J.; Girard, S.; Peralta, E.; Clapham, D. Science 1991, 252, 123.
J. Phys. Chem. B, Vol. 105, No. 29, 2001 7103 (4) Tabony, J. Science 1994, 264, 245. (5) Nicolis, G.; Prigogine, I. Self-Organization in Nonequilibrium Systems; Wiley: New York, 1977. (6) Hess, B.; Mikhailov, A. Science 1994, 264, 223. (7) Albert, B. Cell 1998, 92, 291. (8) Hess, B.; Mihailov, A. J. Theor. Biol. 1995, 176, 181. (9) Mikhailov, A.; Hess, B. J. Phys. Chem. 1996, 100, 19059. (10) Stange, P.; Mikhailov, A.; Hess, B. J. Phys. Chem. B 1998, 102, 6273. (11) Stange, P.; Mikhailov, A.; Hess, B. J. Phys. Chem. B 1999, 103, 6111. (12) Agmon, N.; Hopfield, J. J. J. Chem. Phys. 1983, 78, 6947. (13) Socci, N. D.; Onuchic, J. N.; Wolynes, P. G. J. Chem. Phys. 1996, 104, 5860. (14) Schienbein, M.; Gruler, H. Phys. ReV. E 1997, 56, 7116. (15) Blumenfeld, L.; Pleshanov, P. Biophysics 1986, 31, 826.
