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Dynamics of protein-protein encounter a Langevin equation approach with reaction patches

Dynamics of protein-protein encounter:a Langevin equation

approach with reaction patches

Jakob Schluttig,1,2Denitsa Alamanova,3Volkhard Helms,3and Ulrich S.Schwarz 1,2,?

1University of Heidelberg,Bioquant 0013,

Im Neuenheimer Feld 267,69120Heidelberg,Germany

2University of Karlsruhe,Theoretical Biophysics Group,Kaiserstrasse 12,76131Karlsruhe,Germany 3Center for Bioinformatics,Saarland University,66041Saarbr¨u cken,Germany (Dated:September 17,2008)

a r X i v :0809.2871v 1 [q -

b i o .B M ] 17 S e p 2008

Abstract

We study the formation of protein-protein encounter complexes with a Langevin equation ap-proach that considers direct,steric and thermal forces.As three model systems with distinctly di?erent properties we consider the pairs barnase:barstar,cytochrome c:cytochrome c peroxidase and p53:MDM2.In each case,proteins are modeled either as spherical particles,as dipolar spheres or as collection of several small beads with one dipole.Spherical reaction patches are placed on the model proteins according to the known experimental structures of the protein complexes.In the computer simulations,concentration is varied by changing box size.Encounter is de?ned as overlap of the reaction patches and the corresponding?rst passage times are recorded together with the number of unsuccessful contacts before encounter.We?nd that encounter frequency scales linearly with protein concentration,thus proving that our microscopic model results in a well-de?ned macroscopic encounter rate.The number of unsuccessful contacts before encounter decreases with increasing encounter rate and ranges from20–9000.For all three models,encounter rates are obtained within one order of magnitude of the experimentally measured association rates. Electrostatic steering enhances association up to50-fold.If di?usional encounter is dominant (p53:MDM2)or similarly important as electrostatic steering(barnase:barstar),then encounter rate decreases with decreasing patch radius.More detailed modeling of protein shapes decreases encounter rates by5–95percent.Our study shows how generic principles of protein-protein associ-ation are modulated by molecular features of the systems under consideration.Moreover it allows us to assess di?erent coarse-graining strategies for the future modelling of the dynamics of large protein complexes.

?E-mail:ulrich.schwarz@bioquant.uni-heidelberg.de

I.INTRODUCTION

Protein-protein interactions play key roles in many cellular processes such as signal trans-duction,bioenergetics,and the immune response[1].Moreover,many proteins function in the context of protein complexes of variable sizes and lifetimes.Examples of such complexes are ribosomes,polymerases,spliceosomes,nuclear pore complexes,cytoskeletal structures like the mitotic spindle or actin stress?bers,adhesion contacts,the anaphase-promoting complex,and the endocytotic complex[2].For yeast,800di?erent core complexes have been identi?ed,suggesting the existence of3000core complexes for humans[3].In addition it has been shown for yeast that most protein complexes are assembled just-in-time during the course of the cell cycle[4].In fact many protein complexes in the cell are highly dynamic, with fast turnover of many components.One can argue that their dynamics,although exper-imentally very hard to access,is biologically more relevant than their equilibrium properties. Therefore a systematic understanding of the dynamics of protein complexes in cells is one of the grand challenges in quantitative biology.

The elementary unit of all of these cellular processes is the bimolecular protein-protein interaction.The strength and speci?city of protein-protein association are determined by the integrated e?ect of di?erent interactions,including shape complementarity,van der Waals interactions,hydrogen bonding,electrostatic interactions and hydrophobic e?ects.For example,the importance of electrostatic interactions has been demonstrated by experimental measurement at di?erent ionic strengths[5].To a?rst approximation,bimolecular reactions are characterized by on-and o?-rates.The equilibrium association constant(or a?nity) then follows as the ratio of the two.From a conceptual point of view,on-and o?-rates are very di?erent.On-rates are commonly believed to be controlled by the di?usion properties as well as by long-ranged electrostatic interactions,whereas o?-rates are rather controlled by short-ranged interactions like hydrogen bonding and van der Waals forces.

The main features of the dynamics of protein association can be conceptualized within the framework of the encounter complex[6].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 mainly di?usion but also electrostatic steering on small length scales[7].If di?usion-controlled,classical continuum approaches can be used to describe this part of the process[8].To form the?nal complex,the system then has to

overcome a free energy barrier due to local e?ects like dehydration of the binding interface [9].Due to the various molecular contributions involved in this step,here the two binding partners essentially have to be modelled at atomic detail.Moreover the solvent may need to be treated explicitly and one might has to account for conformational changes[10].Thus, it appears reasonable to use the encounter complex as a crossover point from a detailed, atomistic treatment to a coarse-grained model and vice versa.

Thermal?uctuations 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?rst passage time prob-lem which can be addressed mathematically in the framework of Langevin equations.The application of Langevin equations to association phenomena goes back to early work in the colloidal sciences[11,12].In these early approaches,the reactants were considered to have small spatial extensions and to be uniformly reactive.For large biomolecules like proteins, the situation is fundamentally di?erent.Typically,proteins and other biomolecules have speci?c sites on their surface,where a particular binding reaction can take place.There-fore,such binding events are subject to intrinsic geometric constraints for every particular protein-protein pair or larger assembly.The standard model for ligand-receptor interaction was introduced by Berg and Purcell in the context of chemoreception based on the idea of using reactive patches to model anisotropic reactivity[13].Due to anisotropic reactivity, also rotational di?usion becomes important.Shoup et al.showed that the e?ect of rotational di?usion can strongly increase the association rate between a receptor with a?at reactive patch and uniformly reactive ligands[14].Later,analytic expressions for the association rate between two spherical particles with both carrying a?at axially symmetric[15]and asym-metric[16,17]reactive patch were derived.Similar concepts were also applied by Schulten and colleagues[18].

For many important aspects,analytical approaches are not possible and computer simu-lations are required.This approach has been used early for protein-protein association[19]. The importance of electrostatic interactions for long ranged attraction was also emphasized by Brownian dynamics simulations of protein-protein encounter[19,20,21,22].If atomic structure is taken into account,then successful encounters are de?ned by simultaneous ful-?llment of two to three distance conditions between opposing residues on the two surfaces [23].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[24],or by Elcock and coworkers who simulated a crowded cytosol for10μs [25].

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 speci?c systems of interest.Only a good understanding of this issues will allow us in the future to develop reasonable coarse-graining strategies to address also large complexes of biological relevance.In this study,we therefore address how general principles guiding the di?usional association of biomolecular pairs are modulated by their particular physicochemical properties.To this end we have selected three molecular systems of interest with di?erent steric and electrostatic properties.One of the best studied bimolecular complexes is the extracellular ribonuclease Barnase and its intracellular inhibitor Barstar.Both proteins carry a net charge of2e and?6e,respectively,which leads to a considerable electrostatic steering[26,27,28,29,30].Considering the structure of the two proteins,Barnase has a bean-like form,matching well on a large reactive area with the nearly spherical Barstar.A classic example of electrostatically-driven protein association is the iso-1-cytochrome c-cytochrome c peroxidase(Cytc:CCP)complex,charged with6e and?13e,respectively,and exhibiting dipoles aligned well with the reactive areas[19,31]. Finally,we selected the medically important complex of a peptide fragment of p53and its inhibitor MDM2,which is used for anticancer drug design.In this system,electrostatic attraction plays a minor role.On the other hand,the steric match of the two surfaces is of particular importance here.It is a perfect example of a key-lock binding interface,where p53is buried deep into a cleft on the MDM2surface.

In this paper we systematically explore the e?ect 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 follows:In Sect.II,we present our di?erent stochastic models and describe the methods we use to parameterize the three considered bimolecular model systems.Sect.III contains the main?ndings of our study,which are discussed and summarized in Sect.IV.

II.MODELS AND METHODS

A.Modelling proteins at di?erent levels of detail

One aim of this work is to determine how important speci?c details of the model proteins are with respect to the association properties.Therefore,we considered three di?erent levels of detail as depicted in Fig.1for the three chosen systems.In the most generic approach(M1),we only considered the steric interaction between spherical particles covered with reaction patches.As a?rst re?nement(M2),an e?ective Coulombic interaction was introduced using the dipolar sphere model(DSM).Finally,since our Langevin equation approach is particularly suited to capture anisotropic transport,we consider a more re?ned version for protein sterics(M3).In this approach the excluded volume of each protein was modeled by8-25smaller beads.M3uses the DSM as well.In Fig.1,we also show the full structures in the bottom row as surface representations,including the locations of the binding interfaces.In the following,the general properties of the simulation model and the di?erent techniques used in this work will be explained.

B.Di?usion properties

The di?usion of the protein model particles is described by an anisotropic6×6di?usion matrix in all versions of our model.In Ref.[32],de la Torre and coworkers present a method to calculate this di?usion matrix from the pdb structure of a protein.This method has been implemented in a software called HYDROPRO which is provided online by the same authors (http://leonardo.fcu.um.es/macromol).The basic concept is to put spheres of a certain size at the position of any non-H atom.The volume of these spheres e?ectively models a ?xed hydration shell.This construct is then?lled up with smaller,densely packed,but non-overlapping spheres.Since the hydrodynamic properties of a rigid body are determined by its outer boundary only,a shell of these small spheres is generated by deleting all spheres which have a maximum number of possible neighbors.To this shell a sophisticated technique is applied,which has been developed by de la Torre and colleagues over the years,to calculate the di?usion matrix of such a cluster of non-overlapping spheres(see references in[32]). Several system properties are implicitely contained in the mobility matrix,such as ambient temperature T a=293K,as well as the density and dynamic viscosity of the solvent,where

we chose the respective parameters of water,ρ=1g/cm3andμ=10?3Pa s.For simplicity, hydrodynamic interactions were not introduced in our models,because the corresponding e?ect on the association rates is expected to be well below10%[33].

https://www.wendangku.net/doc/6816413870.html,ngevin 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[34,35].Let X t be a six-dimensional vector describing position and orientation of a particle at time t.Since the noise due to Brownian motion is additive (which means that it does not depend on X t due to a constant mobility matrix M),the Langevin equation is given by:

?t X t=M F+g t.(1)

Here,F is a six-dimensional vector containing the force and torque acting on the particle, and g t denotes Gaussian white noise:

g t =0, g t g t =2k B T a Mδ(t?t ).(2) As explained in App.C of Ref.[35],the Euler algorithm can be used to solve a discretized version of this equation:

X(t+?t)=X(t)+M F(t)?t+g(?t)+O(?t2).(3)

For proteins,the typical orders of magnitude are D=10?6cm2s?1and R=1nm.Therefore, a reasonable choice for the time step is?t=1ps,as this leads to a mean step length of √D?t=0.01nm.

The mobility matrix of a particle is de?ned in a particle-?xed coordinate system.Thus, the whole step has to be calculated in terms of particle-?xed coordinates and then trans-formed to the laboratory coordinate space.In particular,this transformation implies a rotation R regarding the orientation of the particle.Special attention has to be payed to the force F,which is typically calculated in the global frame of reference and hence has to be transformed to particle space before Eq.3can be evaluated.This back-transformation

is achieved by applying R?1to F.Since rotation matrices R simply consist of a list of or-thonormal vectors,their inverse is equal to the transposed matrix R?1=R T.Thus Eq.3 can be rewritten as

X(t+?t)=X(t)+R

R T F(t)

?t+g(?t)

+O(?t2).(4)

As F and g are six-dimensional and contain information about torque and rotation,Eq. 4is only formally correct,as R acts on both the translational and rotational parts of the respective vectors separately.

In each step of the simulation,a displacement vector?X(t)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?X(t)are calculated.This procedure continues,until an update of all positions and orientations is found which does not lead to any overlap.In this way,the constraint according to the excluded volume e?ect is included in the stochastic motion.The spherical reactive patches are not taken into account for the steric interactions,i.e.they may not only overlap pairwise but also with the model particles. One would expect that our procedure leads to errors of order?t if two particles are in close proximity of order

√D?t.However,it has been shown for a di?erent system[36]that in practise the deviation from the expected behavior is very small and thus the approach is reasonable.

D.Anisotropic versus isotropic di?usion

As mentioned before,the6×6mobility matrix M represents anisotropic di?usion.For large times,anisotropic di?usion crosses over into isotropic di?usion because the information about the initial orientation gets lost after a certain relaxation time due to the rotational di?usion[37].In general,translational and rotational di?usion are coupled so that large time steps cannot be used.However,for the particular systems studied here,we found that the di?usive coupling is a very small e?ect.In particular,the major entries in the di?usion matrix of the proteins used here according to HYDROPRO multiplied with di?erent powers of the Stokes radius R～10?7cm to make the dimensions comparable are D tt/R2～108s?1, D rr～107s?1,D tr/R～105s?1.Therefore,the e?ect of di?usive coupling is10?2and10?3 smaller than rotational and translational di?usion,respectively.Finally,the typical time

scale at which the cross-over is expected can be calculated to be 1/(6D rr )≈10ns.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.

E.System size and time step adaption

The simulations were performed in a cubic box with periodic boundary conditions.Schreiber and Fersht used concentrations between 0.125μM and 0.5μM in their experi-mental studies of the association rate of the Barnase:Barstar complex [5].The average volume containing one particle at a concentration c is 1/cN A with the Avogadro number N A =6·1023mol ?1.Hence,the edge length of a cubic boundary box representing concentra-tion c can be calculated from L =3√V =1/3√cN A .E.g.c =0.125μM leads to L ≈2370?A for one pair of particles,which is two orders of magnitude larger than the size of the proteins.Due to this low density,the ?rst passage times (FPT)for encounter can be expected to be much longer than the chosen time step.For computational e?ciency,we therefore used a variable time step in our simulations.Van Zon and ten Wolde suggested a method to avoid unwanted collisions when they introduced their Green’s function reaction dynamics (GFRD)

[38].In contrast to our work,however,this method is based on isotropic di?usion.Gener-alizing the GFRD to anisotropic di?usion is out of the scope of our work and we therefore used the following scheme.We ?rst note that in GFRD each time step is chosen such that it includes the next reaction.In our case,we also want to investigate the stochastic dynamics before the next encounter event takes place.Thus a large time step is not chosen to include the next encounter,but to bring the system to such a con?guration that encounter becomes more likely.This step can be well represented by isotropic di?usion with an overall di?usion constant D =(D 11+D 22+D 33)/3following from the anisotropic di?usion matrix.

For an isotropic random walk,the displacement probability is given by a Gaussian distribution with spherical symmetry.Thus,large spatial steps are exponentially sup-pressed,which makes a step of size ?r H max ≥H √6D ?t an H -sigma event.By setting

?r H max =min {r e?ij }the smallest e?ective particle distance in the system,where e?ective

means the distance of the surfaces r e?ij =|r i ?r j |?R i ?R j with R i determining the maximal steric interaction radius of particle i ,one can estimate a reasonable time step for which a

collision is highly improbable.Van Zon and ten Wolde found that the choice H=3provides good results,combined with the fact that wrongly sampling a collision event would need a certain direction of the displacement in addition to the length.As the particles reach close proximity min{r e?ij}→0,the estimated?t vanishes and thus the simulation would be slowed

down in?nitely.Therefore,there has to be some lower boundary for the time step?t min, which is generally chosen to be?t min=1ps as explained earlier.Thus,the adapted time step is given by:

?t ad=min

min{r e?

ij}

;?t min

.(5)

In practice,most time steps are in the ps-range,with very few time steps coming up to the ns-range.

F.Electrostatic interactions

Electrostatic interactions are known to play an important role in protein association.To study the e?ect of electrostatics in our generic model,the models M2and M3utilized the dipolar sphere model(DSM),following Refs.[39,40].The DSM e?ectively models a monopole and dipole interaction by summing over the interactions of three charges,one posi-tioned in the center of each particle,and two close to its surface in opposite positions.Taking into account the Debye screening function due to the presence of counter ions in solution, the electrostatic interaction energy between two charges q i/j at positions r i/j respectively with distance r ij=|r ij|=|r j?r i|is:

W ij=

4πε0εr

q i q j

e?κ(r ij?B ij)

(1+κB ij)r ij

.(6)

Here,κ=l?1

is the inverse Debye screening length,which typically has a value of≈1nm under physiological conditions.We assume a value ofεr=78for the relative static permittivity of the medium,which re?ects the properties of water at ambient temperature.

B ij is a correction to the screening of charges which are placed in an object like a protein which has no free charges inside.Taking b i/j as the closest distance of q i/j from the surface of the surrounding protein,it is approximately given by B ij=b i+b j.This potential leads

to a force of charge q j on q i :

F ij =??r i W ij =??W ij ?r ij

·(?r i r ij )=?14πε0εr q i q j e ?κ(r ij ?B ij )(1+κr ij )(1+κB ij )r 2ij r ij r ij .(7)

As our simulation uses periodic boundary conditions,actually an in?nite 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 particles m and n feel the sum of the Coulomb forces F ij between all pairs of the three complementary charges mimicking the monopolar and dipolar interactions.Thus the full force between particle m and n is F mn = 3

i =1 3j =1F ij ,where i /j run over

the charges of m /n respectively.

As explained earlier,the action of the force on a particle in the Langevin equation is weighted with the mobility matrix M .The HYDROPRO software directly gives the di?usion matrix D =k B T a M .This means that in our case the force action should be rewritten as M F ?t =D F ?t/k B T a .Considering a time step of ?t =10ps and a typical di?usion constant D =10?10m 2/s,we have D |F (1nm)|?t/k B T a ～10?13m and D |F (4nm)|?t/k B T a ～10?15m for typical distances r ij =1nm and r ij =4nm,respectively.In contrast,the typical step length due to the Brownian motion is √D ?t ～10?11m.This shows that the magnitude of electrostatic interactions at distances of 1nm is much smaller than thermal energy.Therefore the e?ect of force is also not considered in our adaptive scheme for the time steps.However,it can be expected that the systematic drift,albeit small,will still lead to an altered encounter behavior.

G.Parameterization

Gabdoulline and Wade [23]used several criteria to de?ne contact areas of bimolecular protein complexes.In our studies,we de?ne the contact area to consist of those atoms in the two interacting proteins that are at 5?A or less distance from an atom of the complementary protein.The center of mass of these atoms is considered as the center of the reactive area.For M 1and M 2,the reactive patch is centered at the surface of the sphere modeling the excluded volume such that it has the same relative direction from the center of mass as

obtained by the method described.In the case of M3,the center of the patch is set to the center of the reactive area.

The contact area has a diameter of approximately10?A to20?A for the three systems studied here.Following earlier Brownian dynamics simulations with atomistic details[7]we have performed an in-depth analysis of the free energy landscape and the encounter state of the protein complexes considered in this work(unpublished results).This showed that the encounter complex is typically located at relative separations of the two protein surfaces of about10?A compared to their positions in the?nal complex.As the spherical reactive patches used in this study simultaneously determine both the size of the contact area on the surface and the distance above their surface at which an encounter will be possible,values in the range of5?A to10?A seem to be reasonable.Note that as long as physical considerations do not dictate non-spherical reaction patches,the spherical choice is highly favorable for computational e?ciency.

As already stated in the beginning,two types of excluded volume structures are taken into account.In the?rst case,used in M1and M2,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 protein,which is also calculated by the HYDROPRO software.The underlying data in the more detailed approach M3is obtained using the AtoB bead modeling software[41,42].In this way,the three-dimensional structure of the proteins is modeled with a comparably small number of8to25spheres of di?erent sizes.

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 p is obtained by summing over the product of all atomic charges due to the xyz force?eld and their relative position to the center of mass.In the model,it is represented by two opposing charges which are positioned along the direction of p and at a distance r p=R gyr?4?A.The magnitude p is

chosen such that|p|=2p r p.We found that the particular choice of r p does not have a noticeable in?uence on the results.The resulting parameterization is given in Tab.I for the proteins considered here.

III.RESULTS

A.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?rst time(typically after milliseconds).Because in our Langevin simulations we measure the distribution of?rst passage times(FPT)to encounter,from which we can deduce the mean ?rst passage time(MFPT) T ,the encounter frequency is de?ned as k=1/ T .This choice is motivated by the fact that for a Poisson-like process,the distribution of?rst passage times is given by f(T)=ke?kT and therefore the encounter frequency k indeed satis?es k=1/ T . As the preparation of a comparable experiment would never allow knowing the particular initial positions and orientations of the unbound proteins,it makes sense to average over the possible initial con?gurations in the computer simulations.Therefore,we started a large number of runs(typically104to105)with random initial positions and orientations for all involved model particles,under the constraint that the initial pairwise distance is at least large enough to prevent an immediate encounter.The“?rst passage”is de?ned as the?rst overlap of two complementary reactive patches.Interestingly,due to this averaging the ?rst passage process becomes Poisson-like,see Fig.2.The data show a clear exponential behavior.This means that it is justi?ed to use the notion of an“encounter frequency”,as the FPT distribution is indeed represented by a single stochastic rate.The?nite probability at small FPT is due to the possibility that the two model particles are already in close proximity when the simulation is started.The large errors in the histogram at T→0are caused by the fact that exponentially sized histogram bins were used to sample the behavior for small T.Therefore,events hitting a particular bin are rare because of the small width of the bins at T→0,which then leads to bad statistics in this domain.

As the encounter process is purely di?usion limited in M1,one would expect the en-counter frequency to scale linearly with concentration.Fig.3demonstrates for the Bar-nase:Barstar system that this is indeed the case.Hence,it is reasonable to always scale the encounter frequencies with the inverse concentration,as will be done for the rest of this work.We will denote these rescaled quantities as encounter rate,i.e.the encounter rates have the dimension M?1s?1.In summary,we have demonstrated here that our microscopic

model leads to a well-de?ned macroscopic encounter rate.

B.Finite size e?ects

In most of the simulations,only one instance of the?nal complex was considered,i.e.one model particle of each https://www.wendangku.net/doc/6816413870.html,ing such small systems could lead to undesired?nite size e?ects.We therefore considered the e?ect of having many particles in the simulation box. Fig.4shows the simulation results for the encounter frequency k for an increasing number of Barnase:Barstar pairs,while keeping the size of the boundary box constant.In order to understand the expected e?ects,consider a system with molecules of Barnase(A and A ) and two molecules of Barstar(B and B )randomly distributed over the boundary box.The relative alignment of any pair of A s and B s is therefore random again.For a particular pair the distribution of times to?rst encounter will thus look very similar to the case with a single pair in the box,which is a simple exponential decay with respect to the encounter frequency k1:f1(T)=k1exp[?k1T].The probability that,e.g.,the particular pair A?B reaches encounter at a certain time t before the three other possible pairs(A ?B,A?B ,

A ?

B ),is therefore:

p(t)=

∞

d t1

∞

d t2

∞

d t3

∞

d t4δ(t1?t)

i=1

k1e?k1t i=k1e?4k1t.(8)

Thus,the probability that any of the four possible particle pairs reaches encounter before the respective three other pairs do,is4×p(t)as just calculated,i.e.f2(T)has again a Poisson form like f1(T)and k2=4k1.In general,for higher numbers of particle pairs N,we expect to again?nd an exponential distribution of the time to?rst encounter with the encounter frequency k N=N2k1.This quadratic behavior is nicely con?rmed by the data shown in Fig.4,which suggests that even for small systems with only two particles,no severe?nite size e?ects have to be expected.In particular,this rules out that larger numbers of particles lead to noticeable three-body interactions or hindering of the encounter process.

C.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.5.At the start of each run,the systems were prepared in the unaligned state A1,as described earlier.A state of close approach which however does not allow for binding is called A2.The two model proteins will switch between states A1and A2a number of times N,until they?nally reach the encounter complex A3due

to a favorable combination of translational and rotational di?usion.In the following,each occurance of A2will be termed a contact.Thus N counts the number of unsuccessful contacts before the encounter is?nally formed.A separate set of simulations was performed to measure the distribution of N.Furthermore we analyzed the distribution of return times T o?.This is the time it takes for two model proteins to get into contact again(A1→A2) after having lost translational alignment(A2→A1),i.e.after they were in close proximity. Finally,we determined the distribution of resting times T on in translational alignment A2 before the two model particles separated again.

As an example,Fig.6shows the distribution of N for the Barnase:Barstar model system at c=0.5μM in the framework of M1.Surprisingly the distribution of the number of contacts has again a Poisson form.Note that the number of unsuccessful trials in state A2can be rather large(up to104).We also found that the distribution of N 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 A2and thus should be more or less independent of system size.

Fig.7shows that the return time T o?(plotted with the plus-symbol)is not exponentially

and undergoes an exponential distributed.Instead,it follows a power law p(T o?)～T?3/2

cuto?due to the?nite size of the boundary box at large T o?.Therefore,there is a high probability for very small return times,i.e.situations,where the two model proteins do not really separate,but immediately after loosing translational alignment(A2→A1)get closer again(A1→A2).The power law behavior of the return time is consistent with the problem of a random walk to an absorber in three dimensions[43,44].In principal,these two situations are equivalent since the relative motion of the two proteins while unaligned A1 can be approximately understood as an isotropic random walk,and the criterion for going over to translational alignment A2re?ects an absorbing boundary in the con?guration space of relative positions.

The distribution of resting times T on(plotted with the cross-symbol in Fig.7)follows the same power law as f(T o?),but the exponential cuto?occurs much earlier.The reason is

that here the cuto?is determined by the region in con?guration space where the two model proteins are in state A2.As this is much smaller than the whole volume of the boundary box,in which they are unaligned and therefore in state A1,a random walk in state A2will end earlier.

The di?erences we obtain in the distributions of T on and T o?when using the variants M2and M3compared to M1are generally very small and unlikely to account for any

deviations in the overall encounter rates.Also,the distribution of N is always well described by a single exponential decay.However,the inverse decay length N signi?cantly varies between the di?erent situations.Therefore,changes in the overall encounter rate are mainly caused by a di?erent probability for reaching state A3from state A2.This is reasonable when considering that the interactions are strongly localized and can thus only act while the system is in the aligned state A2.

D.Three bimolecular systems with di?erent physico-chemical interface properties

So far we have only considered Barnase:Barstar(S1)to demonstrate how our compu-tational model works.We now use our setup for a more comprehensive investigation.In particular,we also apply our method to two other systems,cytochrome c and its peroxi-dase(S2)as well as the p53:MDM2complex(S3).Those represent systems with di?erent interface characteristics and where the role of electrostatics is either much stronger(S2)or much weaker(S3)than for S1.To this end,all the previously described quantities were measured for8di?erent concentrations c={125,250,500,750,1250,2500,5000,7500}pM. Furthermore,to?nd out how the choice of the radius of the reactive patch a?ects the re-sults,we used patch radii of r=6?A and r=3?A in addition to the initially considered value of r=10?A.

Tab.II lists the encounter rates k 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 generally decrease.Second, this e?ect is weaker for M2compared to M1,which basically means that the electrostatic attraction and orientation due to the dipole interaction are indeed enhancing the encounter. The strongest e?ect of the electrostatic interaction is obtained for Cytc:CCP,which is the system with the largest monopole and dipole and the best alignment of the directions of

the dipoles and the reactive patches.On the other hand p53:MDM2is nearly una?ected by the e?ective charges,due to its weak monopole charges and,additionally,an unfavorable alignment of the dipolar interaction and the reactive surface area.Furthermore,regarding the results with detailed steric structure M3,the e?ect on the rate is correlated with the deviations of the protein forms from the spherical excluded volume approach in M1and M2.This deviation is smallest for Cytc:CCP and largest for p53:MDM2.

The?ndings for the encounter rate k are also re?ected in the results for N .As expected, an increase in k correlates with a decrease in N .The only exception is Cytc:CCP observed in M2,which is also special in regard to the e?ect of patch size.Here,the e?ective Coulombic interaction is strongest and the dipole moment is best aligned with the reactive patches. Therefore,having reached state A2once,the proteins do systematically orient towards A3, while they are additionally strongly steered back towards A2when loosing their translational alignment.This behavior is the stronger the closer the model proteins have approached once –i.e.for the case of small patch sizes,where state A2implies the smallest distance.While this only explains the inversion in the N behavior as a function of patch size r,k is obviously still slightly decreasing with smaller patch sizes.This can be explained by the fact that the time to the?rst approach of state A2is 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 re?ected by the decreasing N .

The strong correlation between the encounter rate k and the mean number of contacts N is also evident from the correlation plot in Fig.8.Indeed,k～ N ?1seems valid for most of the di?erent systems and models.It is noteworthy that the prefactor is very similar in all cases.Basically,this means that one unsuccessful contact takes the same amount of time on average,no matter what the local details of the system are.This gets more obvious recalling the distributions of the resting and return times T on and T o?in Fig.7, which shows that T o? > T on .As the average time for one contact will be approximately T on + T o? ,it is dominated by T o?,which is only marginally in?uenced by the local details of the system and the chosen model.Therefore it can be concluded,that for S1and S3 the incorporation of a more detailed modeling approach in?uences k and N ,but not the overall characteristics of the encounter process.

The only exceptions for the clear correlation of k and N are M2and M3for the case Cytc:CCP(S2),where k is nearly independent of N because of the strong electrostatic

interaction.This is consistent with the earlier?nding,that the behavior of Cytc:CCP is qualitatively di?erent[19],as its electrostatic interactions would facilitate long-lived non-speci?c encounters between the proteins that allowed the severe orientational criteria for reaction to be overcome by rotational di?usion.For all three systems studied,in M3the smallest patch size r=3?A leads to a somewhat arti?cial slowing down,because in this case an overlap of the patches is rather hindered by the beads modeling the protein structure.

E.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.9,the encounter frequency has been obtained from simulations for Barnase:Barstar-like model particles in the framework of M1at several concentrations c0={5μM,125nM,2.5nM,125pM}and varying patch sizes r.All values in the?gure have been scaled with the concentration,which leads to data collapse.It is obvious that as r gets larger than2R at around r=40?A,the reactive patch covers the whole model particle and we therefore cross over to the Smoluchowski limit of isotropic reactivity,where k～r.However, at high densities and large r,the patches span a large part of the simulation box of edge

√3/4,where length L,and do immediately encounter for a threshold value of r=r max=L

the sum of the patch diameters4r equals the triagonal.Thus,the encounter frequency must diverge with～1/(r max?r)α,where we supposeα=3,as the volume of con?gurational space without immediate encounter is decreasing with r3.This assumption in addition with the Smoluchowski behavior would lead to k～r/(r max?r)3for large r,which follows the data in Fig.9well(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,Coulom-bic interactions are screened by counter ions in the solution on a small length scale of ap-proximately1nm.Thus,deviations from case M1without e?ective charges will only arise for small r.Fig.10shows the results of respective simulations for M2compared to the results for M1,as considered before.Indeed,for large patch radii r,the results are similar, while for smaller r,the encounter rates in M2are clearly higher compared to M1.However, the crossover to a power law behavior with roughly～r9/4can be detected for very small r,

but at a prefactor of about50times larger than for M1.

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 interactions are thought to be the major factors governing protein encounter.Not included are conformational changes of the proteins upon association, related entropic terms,and the molecular nature of the surrounding solvent that becomes relevant at close distances.The model parameters are extracted from the atomic structures available in the protein data bank by generally applicable protocols as described in Sect.II. In principle,these methods of data extraction can be fully automatized.

The biggest advantage of our coarse-grained model is the possibility to extend the sim-ulations to large scales in terms of particle numbers,time and system size.In many of the earlier studies[39,45,46,47],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 di?usional encounter and is thus rather general,allowing for spanning large time scales via our adaptive time step al-gorithm.In particular,each set of simulations consists of104to105runs of lengths up to the order of seconds and could be performed on a standard CPU within hours of computer time.

Being able to directly obtain the?rst 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.2,which validates the notions of encounter and association.Moreover,our approach provides two ways of controlling the particle density and for both cases the results corresponded well to the expected scaling.First,the concentration is inversely correlated with the size of the periodic boundary simulation box.We show that the encounter rate grows linearly with the particle concentration.Furthermore,leaving the box size constant, the concentration can also be varied by adding a higher number N of particles.Considering only the?rst encounter of any of the possible complementary pairs of model particles,the

mean?rst passage time to this event is not only lowered by a factor N but we show that the expected behavior is an enhancement of the encounter frequency by N2,which is nicely matched by the results of the simulations.Therefore we can conclude that the computational model studied here satis?es the general requirement of stochastic bimolecular association processes that describe binding by a single rate constant.

To test our model against known results we have chosen three well-known bimolecular systems with di?erent 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 is even more strongly a?ected by Coulombic attraction.Here,both proteins have a rather spherical form.Finally,the p53:MDM2com-plex has a di?erent characteristic with a very small net charge and a deep cleft perfectly matching the small peptide p53,whose reactive area is therefore nearly spanning over its whole surface.These model systems were purposely chosen to check whether our e?ective representations of the protein properties would lead to reasonable and signi?cantly distin-guishable results.Indeed,this is the case as the discussion of the results in Tab.II in the respective section shows.

When comparing the results for the encounter rates in Tab.II with previous studies from the?eld 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?nal free energy barrier,which is due to e?ects such as the dehydration of the protein surfaces and thus requires more detailed modelling.In the framework of our approach,this ?nal step could be modelled by a stochastic rate criterion,where the rate can be obtained by transition state theory from the energy landscapes characterized in atomistic calculations. In any case,any additional process to be included can only lower the values found in our study.

In the work on Barnase:Barstar by Schreiber et al.[5],the authors reported that the association between Barnase and Barstar is a di?usion-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 electrostatic steering gets negligible,are clearly lowered by the addition of glycerol,which will lead to slower di?usion.Assuming di?usion control,the reactive step over the?nal barrier should be kinetically unimportant,as generally discussed in Ref.

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微软MSDN官方（简体）中文操作系统全下载这不知是哪位大侠收集的，太全了，从DOS到Windows，从小型系统到大型系统，从桌面系统到专用服务器系统，从最初的Windows3.1到目前的Windows8，以及Windows2008，从16位到32位，再到64位系统，应有尽有。全部提供微软官方的校验文件，这些文件都可以在微软官方MSDN订阅中得到验证，完全正确！下载链接电驴下载，可以使用用迅雷下载，建议还是使用电驴下载。你可以根据需要在下载链接那里找到你需要的文件进行下载！太强大了！！！产品名称: Windows 3.1 (16-bit) 名称: Windows 3.1 (Simplified Chinese) 文件名: SC_Windows31.exe 文件大小: 8,472,384 SHA1: 65BC761CEFFD6280DA3F7677D6F3DDA2BAEC1E19 邮寄日期(UTC): 2001-03-06 19:19:00 ed2k://|file|SC_Windows31.exe|8472384|84037137FFF3932707F286EC852F2ABC|/ 产品名称: Windows 3.2 (16-bit) 名称: Windows 3.2.12 (Simplified Chinese) 文件名: SC_Windows32_12.exe 文件大小: 12,832,984 SHA1: 1D91AC9EB3CBC1F9C409CF891415BB71E8F594F7 邮寄日期(UTC): 2001-03-06 19:21:00 ed2k://|file|SC_Windows32_12.exe|12832984|A76EB68E35CD62F8B40ECD3E6F5E213F|/ 产品名称: Windows 3.2 (16-bit) 名称: Windows 3.2.144 (Simplified Chinese) 文件名: SC_Windows32_144.exe 文件大小: 12,835,440 SHA1: 363C2A9B8CAA2CC6798DAA80CC9217EF237FDD10 邮寄日期(UTC): 2001-03-06 19:21:00 ed2k://|file|SC_Windows32_144.exe|12835440|782F5AF8A1405D518C181F057FCC4287|/ 产品名称: Windows 98 名称: Windows 98 Second Edition (Simplified Chinese) 文件名: SC_WIN98SE.exe 文件大小: 278,540,368 SHA1: 9014AC7B67FC7697DEA597846F980DB9B3C43CD4 邮寄日期(UTC): 1999-11-04 00:45:00 ed2k://|file|SC_WIN98SE.exe|278540368|939909E688963174901F822123E55F7E|/ 产品名称: Windows Me 名称: Windows? Millennium Edition (Simplified Chinese) 文件名: SC_WINME.exe

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工作效率以及其绩效评价。 ③工程项目立项：根据信息分析部门所提供的分析与认证报告，确定信息的处理方式，并上报公司决策层予以决策。公司决策层通过信息分析部门的信息分析报告结合公司的经营状况，对信息进行确定是否立项，一旦立项，就要分析会有哪些承约商参加投标，各自的优势以及他们同客户的关系。主要考虑的因素包括自身的技术能力、项目风险、资源配置能力及其它因素。同时也可对信息分析部门的工作效率以及其绩效评价。申请书填写项目申请书：当决定参加投标竞争的时候上，就需要完成一份项目的申请书或称为投标书，一份完整的申请书一般包括三个部分的内容，即技术、管理、成本三个方面。如果是一份较复杂的申请书，这三部分可能是三个独立的册子：技术部分的目的是让客户认识到：承约商对其需求和问题的理解，并且能够提供风险最低且收益最大的解决方案。管理部分的目的是使客户确信，承约商能够做好项目所提出的工作，并且收到预期的结果。成本部分的目的是使客户确信，承约商申请项目所提出的价格是现实的、合理的。这一部分任务将由公司的技术支持部门根据市场信息部门的有关报告完成，同样也可以通过其工作效率及质量对其进行绩效评价。申请书完成后在项目申请书完成的同时，市场信息部门的所有部门都应密切注视该项目的进展情况，及时更新项目的最新状况，并通报各有关部门特别是技术支持部门，使该部门能根据项目的最新情况调整项目申请书。以增大我们在项目中的竞争能力。在合同的签订即项目确定之后，项目管理又可划分为项目准备阶段、项目实施阶段、竣工验收阶段及系统运行维护阶段等。各阶段的工作内容的不同，其实施与管理也应各异。 ①项目准备阶段：其项目实施管理方式的确定(即项目组织)，各种资源的配备与落实，以及具体项目实施方案的进一步确定。即根据项目的特点，对项目作业进行分解，确定其阶段性成果验收，以及必要的监督反馈，这样就能够很好地解决项目组织与客户的分歧，增加项目风险的可控性。 ②项目实施阶段：根据项目实施的具体方案，并以各阶段性成果按其技术要求、质量保证进行验收。这样，在每个阶段完成后，客户和项目

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附件2 内蒙古建筑职业技术学院院长科研基金课题申请·评审书课题名称：高职院校校园数字化建设系统研究主持人：所在部门：）联系电话：申请日期：

填报说明一、课题（项目）主持人必须从事教学或管理工作五年以上并具有中级以上职称。课题主持人必须是该课题的实际主持者和指导者，并在课题研究中担负实质性的任务。二、课题（项目）主持人原则上不超过55岁。鼓励35岁以下优秀青年教师和科技人员特别是博士、硕士学位的年轻教师申报。申报项目的课题组必须具有良好的政治思想素质、独立开展和组织教育教学研究工作的能力，有较充分的前期准备和相应合理的学术梯队。三、课题论证应尽量充分。研究计划和阶段成果应尽量明确。四、申请书各项内容，要实事求是，逐条认真填写。表达要明确、严谨，字迹要清晰易辩。外来语要同时用原文和中文表达。第一次出现的缩写词，须写出全称。五、申请书复印时用A4复印纸，于左侧装订成册。各栏空格不够时，请自行加页。一式三份，由所在部门签署意见后，报高职教育研究所。六、每一个课题的主要成员(含主持人)只允许申报一个项目。七、本申请书与任务批准书同时作为立项依据。

课题主持人承诺我确认本申请书及附件内容真实、准确。如果获得资助，我将认真履行课题负责人职责，积极组织开展研究工作，合理安排研究经费，按时报送有关材料并接受检查。若申请书及附件内容失实或在课题执行过程中违反科研项目管理的有关规定，本人将承担全部责任。负责人签字： 20 年月日课题依托部门承诺本部门已按照课题申报要求对课题主持人的资格及课题申请书内容进行了审核，课题主持人及课题组主要成员具备课题研究的素质与水平，能够保证课题研究计划实施所需的人才、物力及工作时间，课题如获资助，本部门将根据学院要求和课题申请书内容，落实课题研究所需其他条件，并按照学院科研课题管理有关规定，认真履行课题依托部门的管理职责。部门公章负责人签章 20 年月日

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