In this section we investigate the dynamics of virus capsid formation for direct and hierarchical assembly and compare them in order to identify their generic differences. To characterize the assembly performance, the yield (i.e. the relative number of successful trajectories within a given simulation time) and the first passage times (FPTs) of selected intermediates are recorded for different model parameters. We systematically compare both assembly mechanisms in a parameter space ranging from ka=3.0 ns−1 to 9.0 ns−1 and from kd=1.5·10−3 ns−1 to 1.95·10−2 ns−1. For the simulations of assembly of T1 capsids we use an initial monomer concentration of c=4.5 mM (60 particles in a cubic box with side length L=28nm). Investigation of T3 is carried out at an initial concentration of c=1.7 mM (nf=180, L=55nm). To classify different assembly regimes we distinguish between three different phases: the early, intermediate and final phases which we define to be delineated by the emergence of cluster sizes 1/3 nf, 2/3 nf and nf, respectively.
T1 direct assembly
Figures a and b show the temporal evolution of the relative population of all cluster sizes for one favorable and one unfavorable set of model parameters, respectively. The average cluster size
(see Eq. 1) is shown as solid line and shows a sigmoidal shape. Starting from a full set of available monomers, we observe subsequent formation of dimers, trimers and then larger intermediate clusters. In the favorable case shown in Figure a, the distribution always stays close to the average and complete capsid formation is achieved. Remarkably, this successful case is also characterized by the relatively long persistence of a monomer pool (inset to Figure a). The persistence of a relatively high number of monomers during the intermediate assembly phase shows the system’s capability to reorganize and enables one dominant cluster to grow. In marked contrast, for the unfavorable case shown in Figure b, the distribution of intermediates considerably broadens. The average does not reach complete capsid formation, and the monomer pool is depleted much earlier. Here the intermediates are more restricted in undergoing recombinations, many trajectories become kinetically trapped and the average does not capture anymore the dynamics of the assembly process. In both cases, the assembly dynamics slow down during the final phase. This can, at least partly, be attributed to monomer starvation as the slow-down occurs when only very few monomers are left. The prominent features found here (sigmoidal kinetics, fast growth after lag time, kinetic trapping, monomer starvation in the final phase) have been found before also with coarse-grained MD-simulations [6
Figure 2 T1 direct assembly. a) and b) show the relative population of different cluster sizes as a function of time for a favorable (ka=5.0 ns−1, kd=13.5·10−3 ns−1) and a unfavorable (ka=8.0 ns−1, kd=1.5·10−3 (more ...)
The main difference between the two parameter sets used in Figure is that the second (unfavorable) case leads to more stable intermediates (higher ka, lower kd). In Figure c we systematically investigate the effects of the bond parameters on direct assembly by comparing yield and assembly speed for different combinations of kaand kd. The upper left corner of the parameter plots represent strong bonds (high ka, low kd), while weak bonds are found in the lower right corner (low ka, high kd). The left plot shows the relative yield averaged over an ensemble of 40 trajectories. Direct assembly of T1 shows a large region of high yield for dissociation rate values above a threshold of around kd=10.5·10−3 ns−1. Below this value almost no successful assembly is observed. This is due to the limited possibilities of the intermediates to reorganize, which results in the occurrence of kinetically trapped structures. For low dissociation rates we also observe a dependency of the yield on the choice of ka. In this region, lowering of the bond breaking rate kdcan at least partly be compensated by lowering of the association rate ka.
In the right plot of Figure c, we show the relative assembly speed as a function of model parameters. The assembly speed is defined as the inverse of the completion time of the capsid, v = 1/FPT(nf). Because this quantity can be obtained only for successful assemblies, here we average only over completed trajectories. In contrast to the relative yield, we see a clear dependence of the assembly speed on the association rate kaacross the whole parameter space. Fastest assembly is observed for relatively low values of ka. The observation that relatively high association rates lead to slower assembly can be explained by the increasing tendency to form more than one large cluster in the early and intermediate phases. Thus, even for high dissociation rates, the necessary rearrangement of the clusters slows down the assembly process considerably. We also record a relatively high assembly speed at low kdvalues where only low yield is observed. Since the relative speed values are obtained by averaging over successful trajectories only, these results show that, if a full capsid is formed, it is completed within a short time.
To conclude, we see that the success of assembly in terms of yield is mostly determined by the choice of the dissociation rate kd
. For low values of kd
the system becomes kinetically trapped, while large values of kd
allow for the reorganization of the clusters. The relative assembly speed of successful trajectories is strongly influenced by the choice of ka
. Here we identify an optimum at ka
, with speed being worse both at larger and smaller values. In agreement with previous studies, we observe that most efficient assembly (i.e. high yield combined with fast capsid completion) occurs at intermediate bond stability and that bond reversibility is an important requirement for successful capsid formation [9
T1 hierarchical assembly
Hierarchical assembly of a T1 virus capsid is analyzed in a similar manner as direct assembly. Figures a and b show the evolution of relative cluster size population and the average cluster size for assembly under favorable and unfavorable conditions, respectively. Due to the imposed hierarchy, clusters above pentamer size adopt only particular size values (multiples of five). Hierarchical assembly under favorable conditions (Figure a) shows a long early phase during which the first pentamers are formed. The following intermediate phase is characterized by addition of newly formed capsomers to one dominant cluster. A striking feature of hierarchical assembly is the dramatic slow-down in the final phase. A majority of trajectories remains in the n=55 state for a long time where all but one pentamer have formed and joined the almost complete capsid. This can be explained by increased monomer starvation. In hierarchical assembly, all small clusters of sizes below five are connected by single bonds only. Since the low monomer concentration in the final phase reduces the frequency of diffusional encounter, it takes a long time before the last pentameric ring can be closed irreversibly. From the inset of Figure a we clearly identify the formation of the last pentamer as the bottleneck of capsid completion in hierarchical assembly. Here the capsid completion time is plotted against the formation time of the last pentamer for several successful trajectories of one exemplary parameter set. We observe that the completion of the last pentamer is almost instantly followed by its integration into the capsid.
Figure 3 T1 hierarchical assembly. a) and b) show the relative population for a favorable (ka=8.0 ns−1, kd=1.5·10−3 ns−1) and unfavorable (ka=5.0 ns−1, kd=1.35·10−2 ns−1) set of parameters, respectively. (more ...)
The assembly dynamics shown in Figure b for an unfavorable parameter combination does not lead to complete assembly within the given simulation time. In contrast to the favorable case (Figure a), the association rate is lower and the dissociation rate is higher, which results in a reduced overall bond stability. This is found to strongly hinder the formation of the late pentamers. Although slower, the overall course of the assembly process is not substantially different from the successful case in Figure a. The main difference between the two parameter combinations becomes clear by looking at the completion times of the pentamers which are shown in the inset of Figure b. We see that the pentamer FPTs of both cases follow the same shape during the early and intermediate phases, but that for low bond stability the completion times in the late phase are delayed. This delay grows with ongoing assembly, so that the final pentamer does not close within the simulation time. Here the negative effect of low monomer concentration on capsomer assembly, which was discussed earlier, is amplified by the low bond stability.
To quantify the effects of different combinations of kaand kd on hierarchical assembly of T1, we again perform a systematic investigation of the bond parameter space as shown in Figure c. Considering the relative yield of complete capsids (Figure c, left image), we observe that only a narrow range of parameters leads to a considerable fraction of successful trajectories. High yield is only observed at high bond stabilities (high ka, low kd) in the upper left corner of the parameter plot. To take into account the critical role of the formation of the last pentamer in our simulations, we also show the yield of almost finished capsids (n=55) at the end of the simulation time (Figure c, right image). The region where we observe almost finished capsid is considerably expanded and a large fraction of trajectories reaches n=55 in the upper left corner of parameter space. The yield decreases along the diagonal from high towards low bond stability values (lower right corner). It becomes clear that the unfavorable parameter combinations do not show kinetically trapped states as they occur in direct assembly, and that most trajectories are close to capsid completion. The high yield of almost finished capsids and the lack of trapped trajectories suggests that the bond hierarchy promotes successful capsid completion, but is vulnerable to monomer starvation.
T1 direct versus hierarchical assembly
The above analysis has revealed a marked difference between direct and hierarchical assembly schemes. From Figures and , it is also clear that the final state of a trajectory is strongly affected by the finite length of the simulation and provides only limited information on the dynamics of assembly. In particular hierarchical assembly depends strongly on the formation of the last pentamer and suffers from monomer starvation in the final phase. To evaluate in more detail the performance of the assembly process in its different phases, we systematically compare the FPTs of certain intermediates for both direct and hierarchical assembly. The results are depicted in Figure in a sequence of phase diagrams. Blue areas are those where direct assembly performs better while red indicates parameter combinations where hierarchical assembly is faster. Points where a clear distinction is not possible are shown in gray (difference of direct and hierarchical FPTs less than 10% of the sum of both FPTs).
Figure 4 Comparison of T1 direct and hierarchical assembly. We evaluate a) FPT(30), b) FPT(40), c) FPT(50) and d) FPT(60) for different parameter combinations (ka, kd). Blue fields indicate points at which the respective FPT for direct assembly is smallest while (more ...)
For the first emergence of intermediates of half the capsid size (FPT(30), Figure a), direct assembly is faster throughout the whole parameter space. This is related to the earlier observation of an extended initial phase of hierarchical assembly when compared to direct assembly (see Figure ). It can be explained by the fact that the monomers in direct assembly exhibit three active binding sites and thus easily form clusters of considerable size. Since hierarchically assembling monomers are designed to form flat pentamer rings, only the two patches forming intra-capsomer bonds are active until full capsomers are formed. Thus the number of fruitful encounters is reduced remarkably, which leads to the observed slow-down of the initial phase.
Looking at the FPTs for the two-third assembled capsid (FPT(40), Figure b), we see a large region in the upper left part of the parameter space (high ka, low kd) where hierarchical assembly is now able to overtake direct assembly. This can be attributed to two effects. Firstly, hierarchical assembly speeds up once a pool of capsomers is available. Secondly, direct assembly is slowed down at high bond stabilities. Since the combination of fast formation of large, stable clusters in the early phase (due to high ka) and slow dissociation of small clusters leads to a small number of free monomers, the dominant cluster grows only slowly. In the region of lower bond stability, direct assembly remains faster. Here the increased ability of un- and rebinding of single proteins allows for fast rearrangement, leading to a sufficiently large supply of free monomers so that the dominant cluster can easily grow beyond n=40. Simultaneously the pentamer rings in the hierarchical setup form slower than at high bond stabilities.
The difference between the two assembly mechanisms becomes even more evident when looking at FPT(50) (Figure c). At low kdvalues, direct assembly experiences kinetic trapping. As a consequence, hierarchical assembly is superior for almost all small kdvalues, also at points where the question of dominance remained undecided for FPT(40). The parameter region of weak bonds where direct assembly is faster than hierarchical one is observed to extend during the step from FPT(40) to FPT(50) (lower right corner of parameter space). Under these conditions the effect of beginning monomer starvation delays the pentamer completion of hierarchical assembly.
For the assembly speed of the complete virus capsid (FPT(60), Figure d), direct assembly dominates again across almost the whole parameter space. Only at very high bond stabilities hierarchical assembly shows lower or comparable FPT values. This is not surprising taking into account the results for the overall yield of hierarchical assembly (Figure c) and underlines the large impact of monomer starvation on hierarchical assembly.
We conclude that hierarchical assembly is not always better than direct assembly. Direct assembly performs better both in the initial and final phases. During the intermediate phase, however, hierarchical assembly is more successful, because it does not suffer from stable bonds preventing structural rearrangements. Due to the limited number of possible interactions, hierarchical assembly is unlikely to get trapped in sub-pentameric units. In general, for hierarchical assembly parameter combinations resulting in high bond stability are favorable. At these values we observe kinetic trapping of most of the directly assembling systems. In addition, the symmetry of the pentamers themselves and the low complexity of their interactions prevent them from getting trapped in large clusters incompatible with the final capsid. For T1, this favors the step-wise build-up of the target structure.
T1 effect of initial number of monomers
Until now we have used exactly as many monomers as needed to form one complete capsid. In experiments, monomers are likely to be present in surplus or to be provided with a certain rate. To study the effect of the limited number of monomers on our simulation, we next increase the initial supply to N=80 and N=120 monomers while keeping the concentration constant by enlarging the simulation box. For the case N=80, a surplus of 20 monomers will be present upon formation of a complete capsid. For the case N=120, two capsids might be formed in parallel and thus the benefit of an increased initial monomer concentration might be shared by them in a complex manner. Figures a-c show a comparison between direct and hierarchical assembly of the FPT(50) for an initial number of N=60, N=80 and N=120 monomers, respectively. As in Figure , blue fields indicate that direct assembly has a lower FPT(50), red fields mark parameter pairs for which hierarchical assembly is faster and for gray fields no clear distinction is possible. We see that the comparison of both mechanisms leads to similar results for all setups. When increasing the number of initial monomers we observe a slightly larger region of the parameter space in which hierarchical assembly becomes favorable. This is not surprising, as we identified monomer starvation to strongly hinder the final capsid completion for hierarchical assembly. However, in general the effect of monomer starvation seems to have relatively little impact on the relative efficiency of the two different assembly schemes for clusters of size N=50.
Figure 5 Effect of initial number of monomers on T1 assembly. Comparison of FPT(50) (a-c) and yield (d-f) for direct and hierarchical assembly with an initial number of N=60, N=80 and N=120 monomers, respectively. Blue fields indicate points at which the FPT for (more ...)
Figures d-f show the yield of the first capsid within simulation time for an initial number of N=60, N=80 and N=120 monomers, respectively. Here again red indicates a higher yield of hierarchical assembly while blue indicates a higher yield of direct assembly. Gray marks parameter pairs with the same yield. In contrast to the FPT(50), we can clearly see that increasing the initial number of monomers results in a largely expanded parameter space in which hierarchical assembly is favorable. This shows that monomer starvation affects the final phase of hierarchical assembly in particular as it has been inferred in the previous section. In fact hierarchical assembly performs well throughout the whole parameter space and shows high yield for intermediate and weak bonds. At very high bond strength we even observe some trapping for hierarchical assembly. However, direct assembly still strongly suffers from kinetic trapping so that the parameter space corresponding to high bond strength remains clearly dominated by hierarchical assembly. We also note that increasing the initial number of monomers from N=80 to N=120 does not lead to a further promotion of hierarchical assembly, presumably because now two capsids form in parallel, each drawing monomers in a similar manner as before for N=60.
To conclude, we find that our main results from the previous section remain valid for an increased number of initial monomers. Hierarchical assembly is favorable at high bond strength due to the decreased possibility of trapping while direct assembly is favorable at low bond strength allowing for fast reorganization of large clusters. In general, we expect that our results also carry over to even larger systems.
T3 direct versus hierarchical assembly
Given the results for T1 virus assembly, we now ask how they carry over to more complicated geometries like T3 viruses. In this section we compare the characteristics of direct and hierarchical assembly of T3 viruses which are composed of nf=180 monomers. Now we place again exactly the number of monomers needed for the formation of one complete capsid into the simulation box. While in the hierarchical assembly of T1 viruses the capsid was built from pentameric subunits only, T3 virus capsids consist of 12 pentameric and 20 hexameric capsomers. Figure shows a model capsid which, in the hierarchical case, assembles from two different subunits. While the pentamers are formed from identical proteins, the hexamers contain two different particle types. Due to the increased complexity of the T3 capsid, we observe only a small range of bond parameters to lead to high yield for direct assembly in our computer simulations. Moreover, we are not able to identify a parameter combination that allows successful hierarchical assembly within the used simulation time. This is caused by the lowered concentration of individual species of monomers which leads to a dramatic slow-down of capsomer formation in the final phase. As hierarchical assembly reaches the largest cluster sizes at a high association rate of ka=9.0 ns−1, we now systematically analyze the effect of different dissociation rates kd(5·10−4 ns−1≤kd≤1.35·10−2 ns−1) while keeping kafixed.
Model for T3 virus capsid. (a) Visualization of the T3 virus capsid and its capsomers of (b) pentameric and (c) hexameric structure. The hexamer is composed of two different protein types.
Figure shows different FPTs (1/3nf, 2/3nf and 5/6nf) for direct (blue color) and hierarchical (red) assembly for ka=9.0 ns−1and varying dissociation rate. The FPTs are complemented with yield histograms showing the relative number of trajectories which reached the corresponding size within the simulation time. From Figure a we immediately see that all trajectories in the investigated parameter interval have grown beyond a cluster size of n=60 at the end of the simulation. Comparison of the FPTs for direct and hierarchical assembly reveals assembly speeds of the same magnitude at low values of kd. With growing dissociation rate the FPTs increase for direct as well as for hierarchical assembly. This is not surprising since a lower bond stability leads to an increased number of dissociation events and a slower cluster growth. The FPTs of direct assembly increase only moderately (about one order of magnitude) compared to those of hierarchical assembly (two orders of magnitude). This extreme sensitivity of hierarchical assembly is caused by the strong impact of the low bond stability on capsomers formation. The effect was already observed in hierarchical assembly of T1 and is amplified here due to the presence of several protein types and the resulting lowered effective initial concentration: The number of fruitful monomer encounters is not only reduced by the smaller number of active patches compared to direct assembly, but also by the limited number of suitable binding partners. As a consequence of the dramatic slow-down of hierarchical assembly with increasing kd, we observe zero yield of intermediates of size n=120 above a threshold around kd=7.5·10−3 ns−1(Figure b). On the contrary, we record a decrease in the yield of direct assembly below this kd value. This can be explained with the occurrence of kinetic trapping which we already encountered in T1 direct assembly. Analysis of the corresponding FPT values of so far successful trajectories reveals that, despite the trapping tendency, the speed of direct assembly is still comparable to that of hierarchical assembly at low kd values. For even larger cluster sizes (FPT(150), Figure c) we see further partitioning of the parameter space. Above a threshold around kd=4.5·10−3 ns−1, no hierarchical assembly is observed, while below this value, only one directly assembling trajectory reaches this size.
Figure 7 Comparison of T3 direct and hierarchical assembly. a), b) and c) show the first passage times FPT(60), FPT(120) and FPT(150) together with the relative yield of the corresponding cluster size for fixed ka=9.0 ns−1. Blue and red boxes show the (more ...)
T3 effect of initial number of monomers
As for the assembly of T1 capsids, we again investigate the role of an increased initial number of monomers on the simulation results for the T3 capsid. We increase the initial number of monomers by 10% and 20% (without changing the concentration) and record the FPTs for these simulations. In Figures a-c the FPT(120) and the yield of clusters of size 120 are shown for an initial number of N=180, N=196 and N=216 monomers, respectively. As in the previous section we explore the effect of varying kd while keeping ka=9.0 ns−1fixed. Comparing the FPT(120) for the different setups we see that above kd=1.5 ns−1, hierarchical assembly becomes faster for an increased initial number of monomers. Direct assembly in contrast is only slightly affected throughout the parameter space. When looking at the yield of clusters of size 120 within simulation time (9·106ns), we clearly see the positive effect of an increased initial number of monomers on hierarchical assembly for weaker bonds (higher kd). However, it remains worse than direct assembly at these bond strengths. These findings are in agreement with the effect observed for T1 when increasing the initial number of monomers. While the dynamics of direct assembly is only weakly affected by the initial number of monomers, hierarchical assembly suffers less from the effect of monomer starvation. Considering the FPT(150) we again see a complete separation of the parameter space into one region in which only direct assembly is observed and another region in which hierarchical assembly dominates. Looking at the yield we see that for an initial number of 180 monomers hierarchical assembly is only observed for kd≤1.5·10−3 ns−1 while this region expands to kd≤4.5·10−3 ns−1for an increased initial number of monomers. It might be possible that the parameter space in which hierarchical assembly is favorable expands further for a larger increase of the initial number of monomers, similar as it was observed for T1 (Figure ). However, it seems that the favorable effect of an increased initial number of monomers is weaker for T3 capsids than for T1 capsids due to the more complex geometry. In the following section we will investigate the role of complexity of the T3 capsid for the hierarchical assembly of a T3 capsid.
Figure 8 Effect of initial number of monomers on T3 assembly. Comparison of T3 direct and hierarchical assembly for an initial number of N=180, N=198 and N=216 monomers. a)-c) show the FPT(120) and d)-f) show the FPT(150) together with the relative yield of the (more ...)
Capsomer formation in T3 hierarchical assembly
In order to further investigate the effects that slow down hierarchical assembly, we now analyze the dynamics of hexamer and pentamer formation both with computer simulations and a master equation approach. To compare the FPTs for pentamer and hexamer formation, we scale these values with the number of monomers per capsomer ring. This linear scaling is based on the assumption that the mean time for a net addition of monomers to small ring-forming clusters is independent of the cluster size. This simplification in particular neglects the increased number of decay paths of hexamers compared to pentamers. However, the assumption seems justified for the present case of high bond stabilities (high ka, low kd), at least for the early and intermediate phase of assembly.
In Figure a the average capsomer formation times from T3 assembly at the most promising parameters identified from Figure are shown (now again for N=120). We find that during T3 capsid assembly hexamers form slower than pentamers (for the same sequential number). The difference between the completion times increases with time (the last hexamer data point, no. 18, is an exception to this rule since its FPT is artificially cut down to lower values by the finite length of the simulation). In order to investigate whether this is caused by the different relative densities of monomers forming pentamers (60/180) and hexamers (120/180) or a result of the increased complexity of the hexamer rings, we perform a separate set of simulations. In this complementary simulation we compare the assembly of hexamers consisting of one type of protein (identical hexamers) and hexamers built from two different types of proteins (T3-like hexamers). To reduce the computational effort we downscale the system to half its size while preserving the concentration (i.e. assembly of 10 hexamers in the presence of 30 pentamer-forming monomers). In Figure a the hexamer-FPTs from the complementary simulation are compared to those at the same relative positions in the assembly process of the full simulations. The FPTs are again scaled with the ring size. While the dynamics of the identical hexamers follow the course of the pentamers in the full simulation, the FPTs of the T3-like hexamers and the T3 hexamers of the full simulation are in good agreement. This observation suggests that the delay in hexamer formation observed in the full simulation is caused by the two-type complexity of the hexamers compared to the uniformly structured pentamers.
Figure 9 Analysis of T3 capsomer assembly. a) FPTs of pentamer and hexamer capsomers emerging during T3 hierarchical assembly are compared to those of T3-like and identical hexamers from the down-scaled simulations. All simulations use ka=9.0 ns−1and (more ...)
To complement this investigation we use an analytical master equation approach to perform a closer analysis of the dynamics of parallel assembly of several hexamers. Here we develop a set of equations which gives analytic results for the number of clusters of size n, νn (t)(1≤n≤nf=6), as a function of association and dissociation rate. The time evolution of the macroscopic quantity νnis the result of reactions between clusters of all sizes k which cause a change of νn. We introduce the association rate a for successful binding of two clusters per unit time and the dissociation rate bnkwhich denotes the rate for decay of a cluster of size n to two daughters of sizes k and (n−k). bnkis composed of the dissociation rate per bond per unit time, b, and a factor dnk which quantifies the probability for the decay of a cluster of size n to a constellation where one of the daughters is of size k. dnk is determined by the ratio of total dissociation probability (proportional to the number of bonds which compose n) and the probability of the decay products to have the required size. The population νiincreases by the decay of clusters with sizes larger than i, so that dji(i<j<nf) is always positive. For these cases we find dji=2 for each pair j, i, since the decay from 2i to two daughters of sizes i accounts for a double increase of νi. The factor dnndenotes the total decay probability of a cluster, it is thus negative and proportional to the cluster size. Here we use dnn=−(n−1) for every n<nf. We account for one-step processes only, which means we focus on transitions where two clusters merge or one cluster falls apart into two daughter clusters. If we assume that the formation of the complete hexamer ring is irreversible and that the total number of particles N is preserved, the complete set of equations describing the time evolution of the system reads
Numerical evaluation with the initial condition ν1
gives the time evolution of all cluster size populations νn
. By fitting the set of equations to the course of all νn
) from the complementary simulation (nf
=60), we obtain parameter combinations a
which reproduce the observed assembly dynamics. Under the constraint that the dissociation rate b
per bond is constant for identical and T3-like hexamers (since the simulations apply the same kd
), we find the following parameters: aid
(identical hexamers), aT3
(T3-like hexamers) and b
. In general, all νn
)are reproduced well. This suggests that the assumption of a constant association rate a
per bond, independent of the sizes of the encountering clusters, is a reasonable approximation for the formation of small rings. The results for ν1
) and ν6
) are displayed in Figure b together with the simulation data points for both types of hexamers. The early phase is the region which exhibits the largest discrepancies between data and ME results, while the final phase of assembly shows a high level of consistency. This can be explained by the fact that the rate equation framework does not include any spatial constraints and is thus not able to reproduce the same sort of lag time before the first protein reactions as was observed in the simulations, where the randomly distributed particles react only after diffusional mixing leads to the first encounter events. This is also the reason why the difference between the cases of T3-like and identical hexamers becomes visible in the simulation data only after a certain time, while the ME results differ from the very first iteration step (see Figure b). Since the rate equations do not contain a diffusional component, the coefficients a
cannot be directly related to the simulation parameters ka
. While ka
determines the rate of transition from encounter to a bound state, a
as well includes the formation of diffusional patch overlap. We estimate a
), where V
is the simulation box volume. Using the initial concentration c
), we find the expression
is thus, as expected, proportional to the initial monomer concentration in the simulation box. Applying this relation to the fit parameters using the effective initial concentration of the protein types, we estimate the overall association rate values to be
. The fact that the association rate aid
for identical hexamers is about twice the value found for aT3
confirms that the difference in assembly dynamics for identical and T3-like hexamers has its origin in a reduced association rate, caused by reduced encounter of matching protein types. Our observations suggest that the association rate decreases linearly with increasing number of bond partners in the system and thus the number of different protein types needed to form a capsomer ring. When comparing our values for the diffusional encounter rate to data from experiments, we see that we overestimate the association rate. In general, the association rate for bimolecular binding reactions is experimentally found to lie between 4·106
]. Absence of long-ranged forces, as it is the case for our simulation framework, is predicted to push the rates below 106
]. The reason for our relatively high estimates for the encounter rate could be the treatment of dissociation as a stochastic event without immediate relocation of the partners. In the present implementation, two patches stay in an encounter after dissociation and their movement is subject to the cluster mobility. We assume this to cause an overestimation of rebinding frequencies which results in an increased association constant. Whereas the association rate constant can be related to other results, there is no such argument for the value of b