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The elementary unit of all of these cellular processes is the bimolecular protein-protein interaction. The strength and specificity of protein-protein association are determined by the integrated effect of different interactions, including shape complementarity, van der Waals interactions, hydrogen bonding, electrostat... |
The main features of the dynamics of protein association can be conceptualized within the framework of the encounter complex Schreiber ( 2002 . To this end, the association is divided into two parts. First, mutual entanglement - the encounter complex - is achieved by the proteins due to a transport process including ma... |
Spaar et al. ( 2006 . If diffusion-controlled, classical continuum approaches can be used to describe this part of the process Berg and van Hippel ( 1985 . To form the final complex, the system then has to overcome a free energy barrier due to local effects like dehydration of the binding interface Ahmad et al. ( 2008 ... |
Thermal fluctuations are an essential element of protein-protein encounter because they allow the two partners to exhaustively search space for access to the binding interface. From the viewpoint of stochastic dynamics, protein-protein association is a first passage time problem which can be addressed mathematically in... |
For many important aspects, analytical approaches are not possible and computer simulations are required. This approach has been used early for protein-protein association Northrup et al. ( 1988 . The importance of electrostatic interactions for long ranged attraction was also emphasized by Brownian dynamics simulation... |
Gabdoulline and Wade ( 1997 . Brownian dynamics have also been used for the simulation of high density solutions, e.g. by Bicout and Field who studied a cellular “soup” containing ribosomes, proteins and tRNA molecules Bicout and Field ( 1996 , or by Elcock and coworkers who simulated a crowded cytosol for 10 [MATH] Mc... |
In order to develop a quantitative framework for modelling the dynamics of protein complexes, it is essential to understand the relative importance of generic principles and molecularly determined features of specific systems of interest. Only a good understanding of this issues will allow us in the future to develop r... |
- cytochrome c peroxidase (Cytc:CCP) complex, charged with [MATH] and [MATH] , respectively, and exhibiting dipoles aligned well with the reactive areas Northrup et al. ( 1988 ); Pelletier and Kraut ( 1992 . Finally, we selected the medically important complex of a peptide fragment of p53 and its inhibitor MDM2, which ... |
In this paper we systematically explore the effect of various coarse-graining procedures on the rate for protein-protein encounter for the three selected model systems. We revisit early approaches based on Langevin equations and combine them with current knowledge on molecular structure. The paper is organized as follo... |
II Models and Methods II.1 Modelling proteins at different levels of detail One aim of this work is to determine how important specific details of the model proteins are with respect to the association properties. Therefore, we considered three different levels of detail as depicted in Fig. for the three chosen systems... |
II.2 Diffusion properties The diffusion of the protein model particles is described by an anisotropic [MATH] diffusion matrix in all versions of our model. In Ref. de la Torre et al. ( 2000 , de la Torre and coworkers present a method to calculate this diffusion matrix from the pdb structure of a protein. This method h... |
). The basic concept is to put spheres of a certain size at the position of any non-H atom. The volume of these spheres effectively models a fixed hydration shell. This construct is then filled up with smaller, densely packed, but non-overlapping spheres. Since the hydrodynamic properties of a rigid body are determined... |
[MATH] Pa s. For simplicity, hydrodynamic interactions were not introduced in our models, because the corresponding effect on the association rates is expected to be well below 10% Antosiewicz et al. ( 1996 |
II.3 Langevin equation and simulation method For the integration of the Langevin equation, which describes the stochastic motion of the particles, we follow an approach which has been recently developed to model cell adhesion via reactive receptor patches Korn and Schwarz ( 2006 2007 . Let [MATH] be a six-dimensional v... |
[EQUATION] Here, [MATH] is a six-dimensional vector containing the force and torque acting on the particle, and [MATH] denotes Gaussian white noise: |
[EQUATION] As explained in App. C of Ref. Korn and Schwarz ( 2007 , the Euler algorithm can be used to solve a discretized version of this equation: |
[EQUATION] For proteins, the typical orders of magnitude are [MATH] cm -1 and [MATH] nm. Therefore, a reasonable choice for the time step is [MATH] ps, as this leads to a mean step length of [MATH] nm. |
The mobility matrix of a particle is defined in a particle-fixed coordinate system. Thus, the whole step has to be calculated in terms of particle-fixed coordinates and then transformed to the laboratory coordinate space. In particular, this transformation implies a rotation [MATH] |
regarding the orientation of the particle. Special attention has to be payed to the force [MATH] , which is typically calculated in the global frame of reference and hence has to be transformed to particle space before Eq. can be evaluated. This back-transformation is achieved by applying [MATH] to [MATH] Since rotatio... |
[EQUATION] As [MATH] and [MATH] are six-dimensional and contain information about torque and rotation, Eq. is only formally correct, as [MATH] acts on both the translational and rotational parts of the respective vectors separately. |
In each step of the simulation, a displacement vector [MATH] is drawn for each particle as described above. If this global displacement leads to any violation of the hardcore repulsion, all suggested displacements are rejected and new [MATH] are calculated. This procedure continues, until an update of all positions and... |
II.4 Anisotropic versus isotropic diffusion As mentioned before, the [MATH] mobility matrix [MATH] represents anisotropic diffusion. For large times, anisotropic diffusion crosses over into isotropic diffusion because the information about the initial orientation gets lost after a certain relaxation time due to the rot... |
[MATH] [MATH] [MATH] . Therefore, the effect of diffusive coupling is [MATH] and [MATH] smaller than rotational and translational diffusion, respectively. Finally, the typical time scale at which the cross-over is expected can be calculated to be |
[MATH] ns. Time steps of this magnitude were rarely used in the simulations (see below), so that for most of the steps, the anisotropicity is well preserved. Therefore we can safely neglect changes in the anisotropicity of the mobility matrix. |
II.5 System size and time step adaption The simulations were performed in a cubic box with periodic boundary conditions. Schreiber and Fersht used concentrations between |
[MATH] M and [MATH] M in their experimental studies of the association rate of the Barnase:Barstar complex Schreiber and Fersht ( 1996 The average volume containing one particle at a concentration [MATH] is |
[MATH] with the Avogadro number [MATH] mol -1 Hence, the edge length of a cubic boundary box representing concentration [MATH] can be calculated from [MATH] . E.g. [MATH] M leads to [MATH] Å for one pair of particles, which is two orders of magnitude larger than the size of the proteins. Due to this low density, the fi... |
following from the anisotropic diffusion matrix. For an isotropic random walk, the displacement probability is given by a Gaussian distribution with spherical symmetry. Thus, large spatial steps are exponentially suppressed, which makes a step of size [MATH] an [MATH] -sigma event. By setting |
[MATH] the smallest effective particle distance in the system, where effective means the distance of the surfaces [MATH] with [MATH] determining the maximal steric interaction radius of particle [MATH] , one can estimate a reasonable time step for which a collision is highly improbable. Van Zon and ten Wolde found that... |
[MATH] ps as explained earlier. Thus, the adapted time step is given by: [EQUATION] In practice, most time steps are in the ps-range, with very few time steps coming up to the ns-range. |
II.6 Electrostatic interactions Electrostatic interactions are known to play an important role in protein association. To study the effect of electrostatics in our generic model, the models [MATH] and [MATH] utilized the dipolar sphere model (DSM), following Refs. Eltis et al. ( 1991 ); Gorba et al. ( 2004 . The DSM ef... |
[EQUATION] Here, [MATH] is the inverse Debye screening length, which typically has a value of [MATH] nm under physiological conditions. We assume a value of [MATH] for the relative static permittivity of the medium, which reflects the properties of water at ambient temperature. [MATH] is a correction to the screening o... |
from the surface of the surrounding protein, it is approximately given by [MATH] . This potential leads to a force of charge [MATH] |
on [MATH] [EQUATION] As our simulation uses periodic boundary conditions, actually an infinite number of copies exists for every charge. However, due to the very fast decay of the screened electrostatic interaction, only the minimum image distance of two charges is considered in the force calculation. Two model particl... |
[MATH] feel the sum of the Coulomb forces [MATH] between all pairs of the three complementary charges mimicking the monopolar and dipolar interactions. Thus the full force between particle [MATH] and [MATH] is [MATH] , where [MATH] [MATH] run over the charges of [MATH] [MATH] respectively. |
As explained earlier, the action of the force on a particle in the Langevin equation is weighted with the mobility matrix [MATH] The HYDROPRO software directly gives the diffusion matrix [MATH] . This means that in our case the force action should be rewritten as [MATH] [MATH] . Considering a time step of [MATH] ps and... |
[MATH] m and [MATH] m for typical distances [MATH] nm and [MATH] nm, respectively. In contrast, the typical step length due to the Brownian motion is [MATH] m. This shows that the magnitude of electrostatic interactions at distances of [MATH] nm is much smaller than thermal energy. Therefore the effect of force is also... |
systematic drift, albeit small, will still lead to an altered encounter behavior. II.7 Parameterization Gabdoulline and Wade Gabdoulline and Wade ( 1997 used several criteria to define contact areas of bimolecular protein complexes. In our studies, we define the contact area to consist of those atoms in the two interac... |
[MATH] , the center of the patch is set to the center of the reactive area. The contact area has a diameter of approximately 10Å to 20Å for the three systems studied here. Following earlier Brownian dynamics simulations with atomistic details Spaar et al. ( 2006 we have performed an in-depth analysis of the free energy... |
As already stated in the beginning, two types of excluded volume structures are taken into account. In the first case, used in [MATH] and [MATH] , the proteins are assumed to have an approximately spherical form. The radius for the model spheres determining the hard core interaction follows as the radius of gyration of... |
The monopole charge is the sum of all elementary charges in a protein and is placed at the center of the respective model particle. The dipole moment [MATH] is obtained by summing over the product of all atomic charges due to the xyz force field and their relative position to the center of mass. In the model, it is rep... |
for the proteins considered here. III Results III.1 Encounter frequency and encounter rate Langevin dynamics simulations were performed for cubic boxes containing two model proteins. Simulations were conducted until the encounter condition was met for the first time (typically after milliseconds). Because in our Langev... |
Therefore, events hitting a particular bin are rare because of the small width of the bins at [MATH] , which then leads to bad statistics in this domain. |
As the encounter process is purely diffusion limited in [MATH] , one would expect the encounter frequency to scale linearly with concentration. Fig. demonstrates for the Barnase:Barstar system that this is indeed the case. Hence, it is reasonable to always scale the encounter frequencies with the inverse concentration,... |
III.2 Finite size effects In most of the simulations, only one instance of the final complex was considered, i.e. one model particle of each kind. Using such small systems could lead to undesired finite size effects. We therefore considered the effect of having many particles in the simulation box. Fig. shows the simul... |
[MATH] ), is therefore: [EQUATION] Thus, the probability that any of the four possible particle pairs reaches encounter before the respective three other pairs do, is |
[MATH] as just calculated, i.e. [MATH] has again a Poisson form like [MATH] and [MATH] . In general, for higher numbers of particle pairs [MATH] , we expect to again find an exponential distribution of the time to first encounter with the encounter frequency [MATH] . This quadratic behavior is nicely confirmed by the d... |
III.3 Alignment during encounter One feature of special interest which we can address with our Langevin equation approach is the pathway through which the encounter is formed. We dissected the encounter process into several parts as visualized in Fig. . At the start of each run, the systems were prepared in the unalign... |
in translational alignment [MATH] before the two model particles separated again. As an example, Fig. shows the distribution of [MATH] for the Barnase:Barstar model system at [MATH] M in the framework of [MATH] . Surprisingly the distribution of the number of contacts has again a Poisson form. Note that the number of u... |
[MATH] ). We also found that the distribution of [MATH] is roughly independent of concentration. This is reasonable, as after the two proteins were in contact once, the further encounter process is guided by returns to state [MATH] and thus should be more or less independent of system size. |
Fig. shows that the return time [MATH] (plotted with the plus-symbol) is not exponentially distributed. Instead, it follows a power law [MATH] and undergoes an exponential cutoff due to the finite size of the boundary box at large [MATH] . Therefore, there is a high probability for very small return times, i.e. situati... |
Pólya ( 1921 ); Hughes ( 1995 . In principal, these two situations are equivalent since the relative motion of the two proteins while unaligned [MATH] can be approximately understood as an isotropic random walk, and the criterion for going over to translational alignment [MATH] reflects an absorbing boundary in the con... |
The distribution of resting times [MATH] (plotted with the cross-symbol in Fig. ) follows the same power law as [MATH] , but the exponential cutoff occurs much earlier. The reason is that here the cutoff is determined by the region in configuration space where the two model proteins are in state [MATH] . As this is muc... |
[MATH] , a random walk in state [MATH] will end earlier. The differences we obtain in the distributions of [MATH] and [MATH] when using the variants [MATH] and [MATH] compared to |
[MATH] are generally very small and unlikely to account for any deviations in the overall encounter rates. Also, the distribution of |
[MATH] is always well described by a single exponential decay. However, the inverse decay length [MATH] significantly varies between the different situations. Therefore, changes in the overall encounter rate are mainly caused by a different probability for reaching state |
[MATH] from state [MATH] . This is reasonable when considering that the interactions are strongly localized and can thus only act while the system is in the aligned state [MATH] |
III.4 Three bimolecular systems with different physico-chemical interface properties So far we have only considered Barnase:Barstar ( [MATH] ) to demonstrate how our computational model works. We now use our setup for a more comprehensive investigation. In particular, we also apply our method to two other systems, cyto... |
[MATH] Å in addition to the initially considered value of [MATH] Å. Tab. lists the encounter rates [MATH] as obtained from these simulations. The rates are all roughly of the same order of magnitude. Yet several interesting qualitative features are readily apparent. First, for decreasing patch sizes, the rates generall... |
The findings for the encounter rate [MATH] are also reflected in the results for [MATH] . As expected, an increase in [MATH] correlates with a decrease in [MATH] . The only exception is Cytc:CCP observed in [MATH] , which is also special in regard to the effect of patch size. Here, the effective Coulombic interaction i... |
[MATH] is larger for smaller patches, as this implies a smaller relative distance. This obviously compensates the fact that afterwards the encounter is formed even quicker, as reflected by the decreasing |
[MATH] The strong correlation between the encounter rate [MATH] and the mean number of contacts [MATH] is also evident from the correlation plot in Fig. . Indeed, [MATH] seems valid for most of the different systems and models. It is noteworthy that the prefactor is very similar in all cases. Basically, this means that... |
in Fig. , which shows that [MATH] . As the average time for one contact will be approximately [MATH] , it is dominated by [MATH] , which is only marginally influenced by the local details of the system and the chosen model. Therefore it can be concluded, that for [MATH] and [MATH] the incorporation of a more detailed m... |
The only exceptions for the clear correlation of [MATH] and [MATH] are [MATH] and [MATH] for the case Cytc:CCP ( [MATH] ), where |
[MATH] is nearly independent of [MATH] because of the strong electrostatic interaction. This is consistent with the earlier finding, that the behavior of Cytc:CCP is qualitatively different |
Northrup et al. ( 1988 , as its electrostatic interactions would facilitate long-lived nonspecific encounters between the proteins that allowed the severe orientational criteria for reaction to be overcome by rotational diffusion. For all three systems studied, in [MATH] the smallest patch size [MATH] Å leads to a some... |
III.5 Size of the reaction patches We next address the dependence of the data on the size of the reaction patches in more detail. This behavior is exemplary studied with the Barnase:Barstar model system. In Fig. , the encounter frequency has been obtained from simulations for Barnase:Barstar-like model particles in the... |
[MATH] . However, at high densities and large [MATH] , the patches span a large part of the simulation box of edge length [MATH] and do immediately encounter for a threshold value of [MATH] , where the sum of the patch diameters [MATH] equals the triagonal. Thus, the encounter frequency must diverge with [MATH] , where... |
[MATH] . This assumption in addition with the Smoluchowski behavior would lead to [MATH] for large [MATH] , which follows the data in Fig. well (black dashed lines). |
As already mentioned it is well known that the electrostatic interaction of proteins can severely increase the association rate. However, under physiological salt conditions, Coulombic interactions are screened by counter ions in the solution on a small length scale of approximately 1 nm. Thus, deviations from case [MA... |
IV Discussion The main goal of this work was to model protein encounter in a generic framework which allows us to include molecular details without making future upscaling to larger complexes impossible. Our model approach incorporates steric, electrostatic and thermal interactions of the proteins considered. These int... |
The biggest advantage of our coarse-grained model is the possibility to extend the simulations to large scales in terms of particle numbers, time and system size. In many of the earlier studies |
Northrup et al. ( 1984 ); Eltis et al. ( 1991 ); Northrup and Erickson ( 1992 ); Zhou and Szabo ( 1996 , the system was prepared already close to encounter and the overall association rate was then calculated via a sophisticated path-integral like procedure. In contrast, our simulations account for the whole process of... |
Being able to directly obtain the first passage times (FPT) of the encounter processes in our model allows to check the validity of several phenomenological assumptions. First of all, the FPT distribution matched very well a Poisson process with a single stochastic rate, as seen in Fig. , which validates the notions of... |
encounter of any of the possible complementary pairs of model particles, the mean first passage time to this event is not only lowered by a factor [MATH] but we show that the expected behavior is an enhancement of the encounter frequency by [MATH] , which is nicely matched by the results of the simulations. Therefore w... |
To test our model against known results we have chosen three well-known bimolecular systems with different characteristics. The Barnase:Barstar complex is the gold standard for protein-protein association and characterized by relatively strong electrostatic steering. The association of Cytochrome c and its peroxidase i... |
When comparing the results for the encounter rates in Tab. with previous studies from the field of bimolecular protein association, several aspects have to be kept in mind. First, throughout this study, we do only consider the |
encounter of our model particles. As explained in the beginning, the complete association of the complex still lacks the step over a final free energy barrier, which is due to effects such as the dehydration of the protein surfaces and thus requires more detailed modelling. In the framework of our approach, this final ... |
In the work on Barnase:Barstar by Schreiber et al. Schreiber and Fersht ( 1996 , the authors reported that the association between Barnase and Barstar is a diffusion-limited reaction. The argument for this is that the association rates at high ionic concentrations in the solution, i.e. for the limit in which the electr... |
[MATH] -1 -1 at physiological salt Schreiber and Fersht ( 1993 . However, the basal association rate, i.e. the rate at high ionic strength, is reported as |
[MATH] -1 -1 from experiments Schreiber and Fersht ( 1996 . Given that the association process of Brn:Brs is diffusion limited, these findings should actually coincide with our values for [MATH] . But as we already discussed in the results section, in our simulations the influence of the effective electrostatics introd... |
In several earlier approaches, similar problems have been addressed by computational and analytical studies. In work by Zhou and coworkers, basal encounter rates for particles with reactive patches have been found to be |
[MATH] -1 -1 Zhou ( 1993 and [MATH] -1 -1 Zhou ( 1997 , that is closer to the basal rates reported by Schreiber and coworkers. It has to be noted that, in both cases, the patches were flat areas above the surface of the spherical model particles, which had a smaller angular extension compared to our cases, and especial... |
Zhou ( 1993 ) to form the encounter. If we expand the graph in Fig. to smaller patch radii like [MATH] Å, we also find basal rates in the order of [MATH] -1 -1 . Also, the deviation between |
[MATH] and [MATH] , i.e. the influence of the effective electrostatics, is more prominent and could enhance the encounter rate by about two orders of magnitude, which is consistent with the findings in the previously cited work. There, the effect of Coulombic interaction is reflected with a Boltzmann factor due to a pa... |
Any model for the reaction patches has to rely on results obtained from more detailed modeling. The surface of a protein is typically densely covered by water molecules due to the hydrophilic nature of its surface. This hydration shell has a thickness of about 3Å and will therefore in principal hinder the approach of t... |
Our approach makes it possible to observe general features of the encounter process. In particular, we dissect the pathway to the encounter complex in several levels of alignment between our model proteins. As we observe the full trajectory to encounter in our simulations, we are able to extract the number of unsuccess... |
[MATH] is small. Finally, longer contact resting times [MATH] also increase the probability of encounter. It is interesting, that all these effects still lead to a simple Poisson distribution of the number of contacts |
[MATH] when averaging over the initial conditions as it is done in this work. Furthermore, we find that the distributions of these resting and return times cannot be described by a Poisson process, but are consistent with the expectations for a spatially constricted random walk in three dimensions. We find that the par... |
In summary, here we have presented a Langevin equation approach to protein-protein association which in principle allows us to combine long simulation times and large systems with molecular details of the involved proteins. This first study has focused on bimolecular reactions and has proven that this approach is capab... |
Acknowledgements. We thank Christian Korn for many helpful discussions. This work was supported by the Volkswagen Foundation through grants I/80469 and I/80470 to V.H. and U.S.S., respectively. J.S. and U.S.S. are supported by the Center for Modelling and Simulation in the Biosciences (BIOMS) at Heidelberg and by the K... |
# Source: arxiv 0810.1307 # Title: The dynamics of cargo driven by molecular motors in the context of asymmetric simple exclusion processes # Sections: all # Downloaded: 2026-03-03T05:14:08.352636+00:00 |
The dynamics of cargo driven by molecular motors in the context of asymmetric simple exclusion processes. (September 8, 2008) Abstract |
We consider the dynamics of cargo driven by a collection of interacting molecular motors in the context of an asymmetric simple exclusion processes (ASEP). The model is formulated to account for i) excluded volume interactions, ii) the observed asymmetry of the stochastic movement of individual motors and iii) interact... |
. Item (iii) is new. It is introduced here as an attempt to describe explicitly the dependence of cargo movement on the dynamics of motors. The steady-state solutions of the model indicate that the system undergoes a phase transition of condensation type as the motor density varies. We study the consequences of this tr... |
PACS 87.16.Nn; 87.10.M Introduction Asymmetric simple exclusion processes (ASEP) are specially convenient for describing general properties of dynamical systems consisting on a collection of many-interacting particles in situations for which the physicochemical characteristics of the components and thus the nature of i... |
. Because of this, ASEP models have been used to study the collective movement of molecular motors that happen at the microtubules within cellular environment |
. These are models that can be defined in one-(spacial)-dimension and incorporate the asymmetry of the motion of individual motors. |
Since the initial proposal pointing out to ASEP as a possibility to describe the collective dynamics of molecular motors , the general interests are mainly focused on the properties of the system at different boundary conditions that allow to make predictions on the stationary currents or on the average motor velocitie... |
. Also, the effects of motor coordination onto the process of pulling on fluid membranes have been studied in the literature in the context of discrete ASEP models with disorder |
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