of fundamental interest 
 to biology is the extent to which electrostatic 
 interactions enhance the rate of protein  protein association . an essential baseline for determining the magnitude of these rate 
 enhancements is the  basal kon , which is the rate constant for association in 
 the absence of electrostatic interactions . in principle , the basal kon should be measured in the same solvent environment using the 
 hydrophobic isosteres  that is , hypothetical mutants with molecular 
 shapes that are identical to those of the wild - type proteins , but 
 are entirely uncharged . however , due to the difficulty of engineering 
 hydrophobic isosteres , experimental studies have instead estimated 
 the basal kon by measuring 
 the kon for the wild - type 
 proteins at various salt concentrations and extrapolating to the limit 
 of infinite salt concentration where electrostatic interactions would 
 be completely screened . an alternative 
 approach is to construct the exact hydrophobic isosteres in 
 silico by setting all partial charges of the wild - type 
 proteins to zero and directly computing the basal kon by simulating the association of the 
 hydrophobic isosteres . ideally , such simulations would involve the 
 use of flexible molecular models in order to capture conformational 
 changes during the association process . however , since the weak associations 
 of completely hydrophobic proteins are beyond the milliseconds time 
 scale , it has only been feasible to directly compute the basal kon using rigid , models with 
 atomically detailed simulations . theoretical 
 estimates of the basal kon have also been made using spherical models with orientational constraints and applications of transition - rate theory to rigid , atomistic models . here , for the first time , we directly computed 
 the basal kon for a protein we focused on barnase and barstar , which are among the most rapidly 
 associating proteins by virtue of their complementary electrostatic 
 surfaces . our simulations employed flexible , 
 pseudoatomic protein models of barnase and barstar that were designed 
 by frembgen - kesner and elcock to retain 
 the molecular shapes , electrostatic potentials , and diffusion properties 
 of the corresponding atomistic models at the experimental ionic strength 
 ( 50 mm ) . the same authors have demonstrated 
 that the use of these models with standard  brute force  
 simulations can reproduce the experimental kon values of both the wild - type and mutant 
 protein pairs . however , they were unable to carry out such simulations 
 to obtain a statistically robust estimate of the kon for the hydrophobic isosteres ( i.e. , 
 the basal kon ) due to 
 the large computational cost . a 
 critical feature of our study is the application of the weighted 
 ensemble ( we ) strategy to enhance the 
 sampling of rare events , e.g. the slow association of completely hydrophobic , 
 uncharged proteins . although the we strategy has been previously applied 
 to protein binding processes using brownian dynamics ( bd ) simulations , these studies were carried out without the inclusion of his ) between , 
 and within , the diffusing proteins . in the absence of explicit solvent , 
 it has been demonstrated that the translational and rotational diffusion 
 coefficients of flexible protein models are drastically underestimated 
 unless intramolecular his are included in the simulations . in addition , the neglect of intermolecular his 
 in previous bd studies of protein binding processes is likely to have contributed to their consistent 
 overestimation of association rate constants . importantly , our simulations were validated by computing the kon values for both wild - type 
 barnase and its r59a mutant , which associates more slowly than wild - type 
 barnase with barstar , and comparing the 
 computed values to experiment . the wild - type 
 and mutant pairs of barnase and barstar were represented using flexible , 
 pseudoatomic models developed by frembgen - kesner and elcock . briefly , the 
 generation of these models began with all - atom models of the wild - type 
 proteins , which were based on the crystal structure of the barnase barstar 
 complex ( pdb code : 1brs ) ; the same models were used for both 
 the unbound and bound states . approximately one pseudoatom was then 
 used to represent every three amino acid residues ( 33 pseudoatoms 
 for the 110 residues of barnase and 27 pseudoatoms for the 89 residues 
 of barstar ) . for the wild - type proteins and r59a mutant barnase , the 
 effective charge method was used to derive 
 effective charges for the pseudoatomic models such that the electrostatic 
 potentials of the corresponding all - atom models were reproduced . boltzmann 
 equation under experimental conditions ( ph 8 , 25 c , and ionic 
 strength of 50 mm ) . pseudoatoms were then 
 positioned and sized to replicate the electron density envelope of 
 the all - atom model . to generate models of the exact hydrophobic isosteres 
 of barnase and barstar , we started with the pseudoatomic models of 
 the wild - type proteins and set all effective charges to zero . the energy function consisted of a single intramolecular term involving 
 flexible , harmonic bonds between the pseudoatoms and intermolecular 
 terms for electrostatic and nonelectrostatic interactions . to maintain 
 the molecular shapes of the proteins , three bonds per pseudoatom nonelectrostatic 
 interactions were calculated using a very weak go-type 
 potential energy function with a shallow well depth (  = 0.1 
 kcal / mol ) . thus , native contacts were only slightly rewarded by a 
 weakly attractive lennard - jones - like potential and nonnative contacts 
 were penalized by a purely repulsive potential . the well - depth was kept at a minimal value in order to avoid implicitly 
 double counting the attractive electrostatic interactions , which are 
 assumed to be a primary driving force for the formation of the barnase - barstar 
 complex . two pseudoatoms were considered 
 to form a native contact if any non - hydrogen atoms of the residues 
 in the all - atom model were within 5.5  of each other in the 
 crystal structure of the native complex , yielding a total of 34 intermolecular 
 native contacts . all simulations 
 were carried out using the we path sampling strategy , as implemented in the westpa software package ( https://westpa.github.io/westpa ) . in this strategy , a large number 
 of simulations , or trajectory  walkers  , are started 
 in parallel from the initial state and iteratively evaluated at fixed 
 time intervals  for resampling in which walkers are either 
 replicated or combined to maintain a similar number of walkers per 
 bin along a progress coordinate toward the target state . rigorous 
 management of the statistical weights associated with each walker 
 ensures that no bias is introduced into the dynamics . in this 
 study , the we strategy was applied using steady - state simulations 
 within the framework of the northrup this framework involves 
 the definition of two concentric spherical surfaces with radii b and q that correspond to center - to - center 
 separation distances for barnase and barstar . the inner sphere , or b surface , represents the initial , unbound state , and the 
 outer sphere , or q surface , is an absorbing surface 
 that is positioned at a much larger separation distance ( q  b ) to avoid wasting computational effort 
 sampling the indefinite drifting apart of the proteins . each we simulation 
 was started from 24 configurations of the unbound state in which barnase 
 and barstar were randomly oriented at a center - to - center separation 
 distance of b. a walker was continued until the pair 
 of proteins either exceeded a separation distance q or satisfied the criterion for the target state for successful association , 
 i.e. reaching a threshold value , qrxn , 
 in the fraction of native intermolecular contacts , q , that reproduces the experimental kon for the wild - type proteins . consistent with previous 
 brute force simulations , b and q were set to 100 and 500  , respectively . 
 upon reaching the q surface , a walker was  recycled  
 by starting a new walker from the unbound state with the same statistical 
 weight thereby maintaining a steady state and enforcing a constant 
 effective protein concentration ( 3.2 m ) . upon reaching a particular qrxn value , a walker was effectively recycled 
 after completing the we simulation by removing the walker and its 
 replicas prior to calculating the kon . for each barnase - barstar pair , five independent we 
 simulations 
 were performed with different initial random seeds for bd propagation . 
 in each simulation , the configurational space of the protein pairs was divided into 760 bins along a progress coordinate that was intended 
 to capture the slowest protein motions of the association process . we used a progress coordinate that consisted of three zones : ( a ) a 
  far  zone involving the distance between barnase and 
 barstar , ( b ) an  intermediate  zone involving the rms 
 deviation of barstar from its bound - state position following alignment 
 of barnase , and ( c ) a  near  zone involving the same 
 rms deviation metric as in part b and the fraction of native contacts 
 between barnase and barstar . simulations were evaluated for resampling 
 at fixed time intervals  ( or iterations ) of 2 ns to maintain 
 24 walkers per bin . each simulation was carried out for 1000 iterations , 
 or a molecular time of 2 s ( defined as n 
 where n is the number of iterations ) . the dynamics of our we simulations 
 were propagated using the uiowa - bd software , which is the same bd engine that was used for the brute force simulations 
 by frembgen - kesner and elcock . consistent 
 with these simulations , our we simulations were performed at a constant 
 temperature of 25 c using a standard bd algorithm with the inclusion 
 of hydrodynamic interactions ( his ) via calculation of the diffusion 
 tensor using the equations of rotne and prager and yamakawa ; the same values 
 were used for the hydrodynamic radii of the pseudoatoms to reproduce 
 the translational diffusion coefficients of the corresponding all - atom 
 protein models by the hydrodynamics program hydropro ; and a time step of 0.25 ps was used throughout the simulations . for each barnase - barstar pair , the kon value was computed from each of five 
 independent we simulations using conformations that were sampled every 
 20 ps once a steady state was achieved ( figure s1 , supporting information ) . all we simulations were sufficiently long to yield relative percent 
 uncertainties in the average kon of < 20% ( figure s2 , supporting information ) . the kon from each we simulation was calculated 
 using the nam method according to the following equation : where k(b ) and k(q ) are the diffusion rate 
 constants for achieving separation distances of b and q , respectively , and  is the probability 
 of successful collisions , i.e. that a simulation starting from the 
 unbound state with a separation distance of b ( 100 
  ) reaches the bound state before drifting apart to a separation 
 distance of q ( 500  ) . assuming that the motions 
 of the binding partners are isotropic , k(b ) and k(q ) are given 
 by the smoluchowski result ; k(r ) 
 = 4dr , where d is the relative translational 
 diffusion coefficient of the two proteins ( i.e. , the sum of their 
 corresponding diffusion coefficients ) . as done for the brute force 
 simulations by frembgen - kesner and elcock , we used the estimate from hydropro for d ( 2.672  10 ps ) . the  value was calculated using the following 
 equation : where fssbound is the steady - state flux 
 into the bound state and fssqsurf is the steady - state flux 
 into the q surface . as evident in the above equations , 
 the influence of his is considered in our calculation of the probability 
 of successful collisions (  ) , but only approximately on the 
 diffusion of the two proteins by using the sum of their diffusion 
 coefficients ( d ) . for each barnase barstar 
 pair , we determined the efficiency of a single we simulation relative 
 to brute force simulation in computing the kon for each of five independent we simulations ; 
 these efficiencies were then averaged and uncertainties in the efficiencies 
 were determined by calculating the 95% confidence intervals . the efficiency 
 of each we simulation was calculated using the following equation : where tbf and twe are 
 the wall - clock times required by brute force simulation and the we 
 simulation , respectively , to generate the same number of independent 
 ( uncorrelated ) association events using the same computing resource 
 ( i.e. , 256 cpu cores of 2.3 ghz amd interlagos processors ) . association 
 events were considered independent if , within the period between the 
 event and one correlation time before the event , their corresponding 
 trajectories did not share a common simulation segment . the correlation 
 time was determined by monitoring autocorrelation of the flux into 
 the bound state as a function of the lag time and identifying the 
 first lag time that results in zero autocorrelation ( within a 95% 
 confidence interval ; see figure s2 , supporting information ) . since it was not practical to directly obtain 
 tbf for all of the barnase - barstar pairs ( i.e. , the hydrophobic 
 isosteres ) , we estimated tbf in a consistent manner for 
 each pair using the following equation : where mbf is the number of trajectories in a 
 brute force simulation 
 to generate the same number of independent association events observed 
 in a we simulation  given that the brute force trajectories 
 are terminated when the proteins either associate or reach a separation 
 distance of q according to the nam method ; 0.02 days / trajectory / core 
 is the average wall - clock time that would be required to complete 
 a single brute force trajectory before the proteins reach a separation 
 distance of q ; and  ( as defined above ) is 
 the probability calculated by we for a single brute force trajectory 
 to generate a successful association event before dissociating to 
 a separation distance of q. our general strategy for computing kon values from our simulations was to first identify 
 a criterion for successful association that reproduces the experimental kon for wild - type barnase and 
 barstar . next , we validated the simulations by using this criterion 
 to calculate the kon for 
 r59a mutant barnase and wild - type barstar , which associates 9-fold 
 more slowly than the wild - type proteins , and comparing the calculated kon to the experimental value . finally , we used this criterion 
 to estimate the basal kon , i.e. , the kon for 
 the hydrophobic isosteres in which all effective charges of the wild - type 
 proteins are set to zero . following the brute force simulations by 
 frembgen - kesner and elcock , our criterion 
 for successful association was to reach a threshold value , qrxn , in the fraction of native intermolecular 
 contacts , q ; dynamics were propagated using the same 
 bd engine with the inclusion of intramolecular his to achieve realistic 
 diffusive properties of the individual proteins ; and kon values were calculated according to 
 the nam method ( see methods ) . figure 1 shows the computed kon as a function of qrxn for all five independent we simulations 
 of each barnase  barstar pair . the experimental kon for wild - type barnase and barstar 
 ( 2.86  10 m s ) was reproduced when using qrxn values of 0.27 and 0.56 for simulations with and without 
 intermolecular his , respectively . these values differ slightly from 
 those determined by frembgen - kesner and elcock using brute force simulations 
 and the same protein models ( 0.32 and 0.47 , respectively ) due to more frequent monitoring of the reaction 
 criterion ( every 20 ps instead of 100 ps ) ; thus , our we simulations 
 are less likely to have missed conformations that satisfy the reaction 
 criterion . importantly , using the qrxn values that we have identified , the computed kon values for r59a barnase and wild - type barstar 
 are in excellent agreement with experiment , regardless of whether 
 or not intermolecular his were included ( figure 2 ; see also table s1 , supporting information ) . the reproduction of experimental kon values for both wild - type 
 and mutant pairs of barnase and barstar is consistent with results 
 from brute force simulations , providing 
 validation of our we simulation protocol . relative to the basal kon , our computed kon values for wild - type barnase and barstar 
 are 53- and 103-fold faster with and without intermolecular his , respectively . these rate enhancements are solely due to the electrostatic interactions 
 between the wild - type proteins given the omission of intramolecular 
 electrostatic interactions in our simulations . computed kon values 
 for each barnase - barstar pair from each of five independent we simulations 
 as a function of the fraction of intermolecular native contacts qrxn . results from simulations without and with 
 the inclusion of intermolecular hi are shown in the left and right 
 panels , respectively . the vertical gray line in each panel indicates 
 the qrxn value that reproduces the experimental kon for the wild - type pair for 
 simulations without and with hi ( 0.56 and 0.27 , respectively ) and 
 was used for calculating kon values for the mutant pairs in that panel . comparison of computed and experimentalkon values 
 on a log 
 scale . reported values from simulation ( with and without intermolecular 
 hi ) are averages from five independent we simulations with the error 
 bars representing 95% confidence intervals . the pseudoatomic protein 
 model of barnase ( gray ) and barstar ( cyan ) is shown in the upper right 
 corner . the basal kon computed from our simulations 
 with and without intermolecular 
 his are ( 2.85  1.30 )  10 m s and ( 5.79  0.17 )  10 m s , respectively . at the 
 effective protein concentration maintained in our simulations ( 3.2 
 m ) , our computed basal kon values are similar to those using less computationally 
 intensive strategies ; in particular , the use of spherical models with 
 orientational constraints has provided estimates in the range of 1010 m s and the use of rigid , atomistic models in either the application 
 of transition - rate theory or direct bd 
 simulation of protein  protein association has yielded estimates of 1  10 m s. the similarity of our 
 estimates to these previous estimates suggests that flexible models 
 may not be essential for obtaining realistic estimates of kon values for proteins such 
 as barnase and barstar that do not undergo significant conformational 
 changes upon binding ( the c rms deviation between 
 the crystal structures of the unbound and bound conformations is only 0.5  for both barnase 
 and barstar ) . however , it has not been possible to directly estimate 
 the basal kon with uncertainties 
 of < 100% using standard bd simulations with rigid , atomistic models 
 since the association events were much slower in the absence of electrostatic 
 forces . on the other hand , our we simulations 
 with flexible molecular models enable significantly more precise calculations 
 of the kon ( uncertainties 
 of 2246% ) and could therefore be used for even more complicated 
 binding processes , including ones that involve large conformational 
 changes . notably , our computed kon values are significantly lower than that obtained by experiment 
 from extrapolation to infinite salt concentration ( 1.4  10 m s ) , suggesting that the favorable electrostatic interactions 
 between the proteins are not completely eliminated at high salt concentrations . although the inclusion of intermolecular his has no effect on the 
 ability of the simulation model to reproduce the effects of mutation 
 on the kon , for a fixed 
 value of qrxn , the inclusion of intermolecular 
 his significantly slows down the rate of association for all three 
 pairs of the barnase - barstar system ( figures 2 and 3 ) . surprisingly , 
 the extent to which kon decreases is essentially the same for wild - type 
 and r59a mutant pairs ( e.g. , by 5-fold at qrxn = 0.27 ) . in contrast , the impact of intermolecular 
 his in the brute force simulations by frembgen - kesner and elcock was 
 more pronounced for slower associating mutants of barnase such as 
 r59a in which the electrostatic interactions with barstar are diminished . on the basis of these results , it was predicted 
 that the impact would be the most pronounced for the hydrophobic isosteres 
 of barnase and barstar . however , the enhanced sampling provided by 
 the we strategy reveals no statistical difference between the impact 
 of the intermolecular his on the kon for the wild - type and r59a mutant pairs . for the hydrophobic 
 isosteres , our results are inconclusive . although it was possible 
 to obtain statistically robust estimates of the basal kon which was the primary goal 
 of this work  our simulations did not reach the level of precision 
 in the ratio of the kon values with and without intermolecular his that 
 would be required to determine the effect of his on the association 
 kinetics relative to the wild - type pair ( note the large confidence 
 intervals in figure 3 ) . for future studies of this effect , significantly greater sampling 
 using a larger number of simulations and/or longer simulations would 
 be required to achieve a sufficient level of precision in the computed kon values , particularly in 
 the absence of intermolecular his . ratio of association rate constants konwith  hi / konwithout  hi computed from simulations 
 with and without intermolecular 
 his as a function of the fraction of intermolecular contacts qrxn . the shaded regions represent 95% confidence 
 intervals for averages ( filled circles ) from five independent we simulations . finally , 
 it would not have 
 been practical to obtain converged estimates of the basal kon without the use of the we 
 strategy . in addition , a highly scalable , parallel implementation 
 of this strategy was essential since it would have otherwise required 
 > 2 years to carry out the simulations using a serial implementation . 
 to determine the efficiency of parallelized we vs brute force simulation 
 in estimating the kon , we compared the wall - clock time that would be required of we vs 
 brute force simulation ( both using the nam framework ) to generate 
 the same number of independent ( uncorrelated ) association events using 
 the same computing resource ( 256 cpu cores of 2.3 ghz amd interlagos 
 processors ) . figure 4 shows the efficiencies of we simulations relative to brute force 
 simulations for each barnase - barstar pair ( see also table s2 , supporting information ) . for the wild - type pair , a we simulation was 6-fold more efficient than brute force simulation 
 with the inclusion of intermolecular hi . this efficiency increased 
 to 46-fold for the r59a mutant pair and ultimately 131-fold for the 
 hydrophobic isosteres . in the latter case , brute force simulation 
 using the same flexible protein models would be highly impractical , 
 requiring 386 days in wall - clock time to generate the same number 
 of association events ( > 1000 ) as a single we simulation , which 
 required 
 only 3 days . the greater efficiency of we observed for the slower 
 processes ( i.e. , increasing with the barrier height ) is consistent 
 with previous we studies of other rare events . protein association 
 process for the first time using flexible models with molecular simulations . 
 in particular , we computed the basal kon for the barnase - barstar system using highly efficient 
 we simulations . our computed basal kon is significantly lower than that obtained by experiment 
 from extrapolation to infinite salt concentration , suggesting that 
 the electrostatic interactions are not completely eliminated at high 
 salt concentrations . this result underscores the importance of directly 
 computing the basal kon using the true hydrophobic isosteres of the proteins under regular 
 salt concentrations  a goal that can only be achieved by molecular 
 simulation . relative to our basal kon , the electrostatic interactions of the wild - type proteins 
 enhance the rate of association by > 130-fold . as demonstrated by 
 frembgen - kesner 
 and elcock using brute force simulations , the inclusion of intermolecular his significantly decreases the 
 computed kon values for 
 both wild - type and mutant pairs . however , the extensive sampling provided 
 by our we simulations has revealed that the extent by which the kon is reduced is the same for 
 both the wild - type and r59a mutant pairs . for the hydrophobic isosteres , 
 the relative extent to which the kon was affected by the intermolecular his was inconclusive due 
 to insufficient precision in the ratio of the kon with and without intermolecular hi . finally , 
 our results demonstrate that we simulations are orders of magnitude 
 more efficient than brute force simulation in providing converged 
 estimates of rate constants for the slow associations of proteins 
 in the complete absence of electrostatic interactions . the computation 
 of such rate constants is otherwise impractical when using flexible 
 protein models  even when these models are coarse - grained . given 
 its high efficiency , the simulation strategy used in this study would 
 be useful for even more complicated systems , including those that 
 undergo large conformational changes upon binding .