assembly of biomolecules into supramolecular complexes is at the heart of many biological processes and the dynamic interplay of the different components leads to biological functionality .the most important type of self - assembling biomolecules are proteins .although they often have a globular shape , protein assemblies can be strongly anisotropic due to the localized binding interactions between the proteins . like biological systems in general , protein assembliesare operative on many different scales .their size ranges from the nanometer - scale ( for example for two - component complexes like barnase and barstar ) through tens of nanometers ( for example for viruses , which typically consist of small multiples of 60 components ) up to the micrometer scale ( for example the mitotic spindle ) . even in steady statemost biological complexes remain highly dynamic , with association events being balanced by dissociation events .prominent examples are nuclear pore complexes or focal adhesions .another important example for the dynamic nature of biological complexes are actin filaments , which are called _ living polymers _ due to their continuous exchange dynamics . similar to the broad range of length scales , association rate constants observed in biological systems span a wide spectrum ranging from up to . the highly dynamic nature of biological assemblies as well as the large range of involved spatial and temporal scales renders this physical problem challenging but fascinating .interestingly , recent advances in the fabrication of functionalized colloids with directional interactions ( _ patchy particles _ ) make it possible to design elementary building blocks of micrometer sizes which can be used to self - assemble complexes with new functionality .experimental techniques ranging from dna - mediated self - assembly to entropic depletion interactions have been rapidly advancing during the last decade , thus providing a plethora of possibilities to fabricate colloidal particles whose shape and interactions can be controlled in detail . moreover external stimuli such as temperature , light or ph can be used to control the inter - particle interactions during the assembly process . these techniques allow for a state- or time - dependent switching of the interactions and can be used to steer the assembly process . controllingthe particle interactions during the assembly process can prevent kinetic trapping resulting in a higher yield of the desired structure , as has recently been shown in a computational study for virus assembly . to fully exploit the potential of these techniques , a detailed understanding of the dynamics of the assembly process ( distribution of intermediates , relevant time scales ) is of crucial importance .  understanding the mechanisms governing chemical reactions has a long tradition in theoretical physics and chemistry .the most powerful analytical technique in this context is the fokker - planck or smoluchowski equation , which has been used early to study bimolecular association based on diffusive motion .this approach was pioneered by smoluchowski who first calculated the maximum diffusion - limited reaction rate for a fully reactive spherical particle . an important generalization of this result was derived by collins and kimball who studied the effect of finite reactivity by introducing a radiation boundary condition relating the concentration at contact to the reactive flux .another essential extension is the case of particles with anisotropic reactivity ( _ patchy particles _ ) . however , most of these generalizations focus on the calculation of association rates and only a few aim at describing reversible reactions .  to study the full dynamics of protein assembly one needs to consider both association and dissociation processes . moreover , even for globular proteins the intermediates formed during the assembly process are often of highly non - spherical geometry , thus rendering an analytical treatment of assembly in the framework of the fokker - planck equation very difficult . in this casecomputer simulations provide a valuable alternative .while detailed molecular dynamics ( md ) simulations have been successfully used to investigate the behavior of a single molecule in great detail , studying the dynamics of large protein complexes on biologically relevant time and length scales is prohibited by the high computational costs of these simulations .therefore coarse - grained models are required for this case . browniandynamics ( bd ) simulations , the numerical counterpart to the fokker - planck equation , have been extensively used to study bimolecular association kinetics with realistic protein shapes . these studies have been focused mainly on a detailed calculation of association rates in bimolecular reactions based on realistic protein shapes and binding interactions . to study reaction dynamics in a large system consisting of many proteins ,various simulation frameworks have been developed . they range from space- and time - continuous bd simulations as for example in the smoldyn framework through event - driven tools like greens functions reaction dynamics ( gfrd ) , which is based on the analytical solution of the fokker - planck equation , up to a space - discretized version of gillespie s chemical master equation approach , as for example used in mesord . while these simulation frameworks have been successfully used to study reaction kinetics on a large scale , they can not be used to study the details of assembly processes as all of these frameworks lack a detailed description of protein anisotropy , including their shape or directional interactions . to study the assembly of virus capsids as a paradigm of a biological self - assembly process for which these elements are essential , coarse - grained md simulations and monte carlo ( mc ) techniques have been used .similarly the interaction of colloidal particles has been investigated on various scales with different simulation techniques , including md simulations , mc simulations and bd studies .  herewe introduce a simulation framework which allows us to study the spatial and stochastic aspects of protein assembly to large detail and nevertheless is computationally relatively cheap . in our approach , we combine bd of realistic protein shapes with stochastic reactivity for both , association and dissociation .proteins are described as assemblies of spheres equipped with reactive patches .their motion in real and orientation space is based on the overdamped langevin equation with an anisotropic diffusion tensor .inspired by the notion of an encounter complex , we decompose the reaction process into a diffusion and a reaction part . upon the diffusive formation of an encounter, two particles can stochastically react with a microscopic reaction rate and thus form a bond .similarly an existing bond can be disrupted with a microscopic dissociation rate .we first show how these microscopic reaction rates can be related to macroscopic , experimentally measurable rates .we then verify that our algorithm correctly reproduces the macroscopically expected equilibrium behavior for bimolecular reactions .finally we investigate the assembly of a pentameric ring structure and compare our simulation results to a macroscopic rate equation approach . herewe again find excellent agreement between the stochastic simulations and the macroscopic description if the macroscopic rates are calculated without any free parameter in the correct way that includes both the diffusive and reactive contributions .these results show the importance of including spatial and orientational effects as well as realistic diffusion properties to understand the dynamics governing protein assembly .model definition .a ) a dumbbell - shaped protein modeled by two hard spheres . the protein is covered with a patch of radius whose center coincides with the center of one of the steric spheres and is thus shifted by relative to the center of the protein ( contact point of the spheres ) .the patch has an opening angle of around the orientation vector pointing along the long axis of the dumbbell .b ) the encounter between two of the proteins of a ) . here is the center - to - center(ctc)-vector between the proteins and is the center - to - center vector between the patches .c ) illustration of the reorientation processes upon reaction . in a first stepthe ctc vectors of the proteins are aligned with the desired ctc - vectors . in a next stepthe projection of the torsion vectors and on a plane perpendicular to are aligned .finally the clusters are shifted so that the desired relative distance between the proteins is achieved ., width=321 ]   our simulation approach is based on the concept of the encounter complex . in this concept a bimolecular reaction is decomposed into two steps : the undirected diffusive motion of the two binding partners and until they stochastically reach the encounter state and the reaction from the encounter state to the bound complex . in terms of the free energy landscape the encounter complex represents a barrier separating the flat diffusive energy landscape from the reaction funnel .thus , it is a transient state which can be characterized by the onset of highly correlated motion between the binding partners . a barrier in the free energy landscape might arise for example due to a necessary rearrangement of the binding site or due to a reorganization of the water layer . in some cases theremight not exist an explicit barrier , but even then the encounter complex is a helpful theoretical concept .considering the encounter complex as the `` watershed '' between diffusion and reaction processes , the bimolecular reaction can be split according to the following reaction scheme : here and represent the overall kinetic rates of the reaction . in eq .encounter ] the binding and unbinding process is split into a diffusive part represented by the rates and and a reaction part represented by and , respectively . separating the binding process in free diffusive motion without steering forces andlocalized binding requires short - ranged interactions .thus , our approach for the simulation of protein association corresponds to a regime of high salt concentration screening long - ranged electrostatic forces .indeed this is a reasonable assumption for physiological salt conditions .moreover , our approach can be used to study the assembly of - sized colloids functionalized with reactive patches where the size of the reactive region is small compared to the particle size .as we consider a situation in which the interaction between the proteins is short - ranged and only affects the reaction probabilities in the encounter state , individual clusters undergo free diffusive motion without an additional drift term .moreover , here we do not consider hydrodynamic interactions which also might influence the binding kinetics .our simulation system can be decomposed into four elementary structures : proteins , patches , bonds , clusters .clusters are rigid objects that can consist of a single or of multiple proteins held together by bonds .two clusters can react with each other by bond formation between adhesive patches .a cluster consisting of multiple proteins can decay into smaller clusters by bond dissociation .proteins are described by sets of non - overlapping , hard spheres approximating their shape .the proteins are equipped with reactive patches which reflect the localized binding sites .a patch is defined starting from a sphere of radius whose center ( relative to the protein ) can be chosen independent of the center of the protein .in addition , each patch is described by an orientation vector . as a simple example for a non - globular protein , in fig .1a we schematically show a dumbbell - shaped protein consisting of two hard spheres in contact .the protein is equipped with one reactive patch of radius whose center coincides with the center of one of the steric spheres and is thus shifted by from the protein center located at the contact point of the steric spheres .the orientation vector points along the long axis of the dumbbell . using the above definitions we formulate the following set of equations to define the encounter between two clusters mediated by a pair of patches , compare fig .1b : eq .encounter1 ] means that the distance between the centers of the two patches has to be sufficiently small for an encounter to occur and can be implemented easily in a simulation .[ eq . encounter2 ] is more complex .it involves the center - to - center vector and not only reflects that the patches are anisotropic , but also assumes that an encounter only occurs if the two partners have a favorable orientation relatively to each other which is defined by the two parameters and . in generalan anisotropic protein can be described by multiple spheres and the center of the patches can be chosen independent of the steric spheres , allowing for a variable description of proteins and their localized interactions .rejection methods .the distance distribution of two hard spheres of radius nm in a periodic box of volume is shown for two different rejection methods . in the first rejection method ( green )new positions of the particles are chosen ( starting from their positions before collision ) until no overlap is observed . in the second rejection method ( red )the spheres are set back to their position before the collision .the simulations were performed at a time - step resolution of . , width=321 ]  in our approach particle motion is described by the six - dimensional , overdamped langevin equation ( brownian motion ) describing translational and rotational diffusion of an arbitrarily shaped but rigid object : here is a six - dimensional position vector describing the position and orientations at time t and is gaussian white noise . is the mobility matrix which is calculated on - the - fly for any cluster shape following the method of de la torre . because intermediates continuously grow and shrink due to association and dissociation , the diffusive properties of the involved clusters are continuously changing during a simulation .protein particles are modeled as hard spheres and their collision requires special attention . in a bd framework ,particle velocities are not defined and thus a ballistic reflection scheme is not appropriate .here we treat steric collisions by a rejection method similar to mc simulations .if a steric overlap between two clusters is created during a move step , this move step is rejected .one method to deal with this situation would be to repeatedly choose a new position of the two clusters , starting from their original location and orientation , until the move step is accepted .this method leads to incorrect simulation results as is demonstrated for the example of two hard spheres in a periodic box in fig . 2 . herea histogram of the distance distribution between the two spheres , scaled by the volume of a spherical shell , is shown in a range in which boundary effects are not observed . by choosing a new position until successful acceptance of the move step ( green histogram ) ,an effective repulsion between the two spheres is introduced , leading to a depletion zone at small particle distances instead of the expected uniform distribution . if , instead of choosing a new position , we set the two clusters involved in a steric collision back to their original position ( and orientation ) before the move step , we recover the expected uniform equilibrium distribution ( shown in red in fig . 2 ) .thus , this method ensures that the correct equilibrium distribution is realized .   in our approach , clusters are rigid assemblies consisting of one or multiple proteins . assuming that the assembly geometry is determined by unique local interactions , a set of local rules is necessary to describe the new relative orientation and position of two reacting clusters. these local rules are encoded in the bond that is formed between the clusters .each bond carries the information of the desired center - to - center ( ctc ) vector of the two proteins directly involved in the binding process as well as their relative orientation ( torsion vectors ) . upon reactionthe two clusters instantaneously flip into the desired relative configuration .this rule follows from the assumption that the short - ranged forces involved in the final binding step lead to a fast rearrangement on the time scale of our bd simulation .the reorientation process is schematically shown in fig .in a first step the clusters are shifted and rotated so that the predefined ctc - vector matches the real ctc - vector .this procedure enforces the correct position of the two merging clusters relative to each other . in a next step the relative orientation of the clustersis corrected by aligning the projection of the two torsion vectors on a plane perpendicular to the ctc - vector . the necessary rotation and translation of the clustersis distributed between them according to their diffusive weights .this means that for a small and a large partner , essentially only the small partner is moved , as one expects for physical reasons ; for two similarly sized partners , both are moved to a similar extent .since all interactions are local ( e.g. local patches or constraints on orientation / torsion ) , the encounter complex corresponds to a region in configuration space surrounding the desired relative position and orientation .if this region is sufficiently small , the instantaneous flipping into the `` correct '' configuration is comparable to the resolution of the bd simulation .reorientation of the clusters during the binding process can lead to a steric overlap either with another cluster or between the merging clusters .if this is the case , the reaction is rejected and the old positions and orientations of the clusters before the move step are resumed .the steric collision of two merging clusters reflects that during the assembly process incompatible fragments can prevent the formation of the desired structure .for example if the desired structure encoded by the local bonds is a ring consisting of five proteins , two ring fragments each containing three proteins can not bind with each other .illustration of the reactive volume .the configuration space is defined by the relative position of the particles and the dimensional orientation vector .a ) in the case of one reactive patch corresponds to the region in configuration space in which two particles are considered to be in encounter ( eq .encounter1 ] and eq .[ eq . encounter2 ] ) .b ) for particles with multiple reactive patches different regions in configuration space correspond to an encounter mediated by one particular patch combination , .for identical microscopic rates and non - overlapping the total reactive volume between the two particles is given by the sum over the encounter volumes associated with each patch combination.,width=321 ]  in order to implement full reversibility , both reaction directions are treated as stochastic reactions with two corresponding microscopic rates .if the reactive patches of two clusters form an encounter , specified by the constraints in eq .encounter1 ] and eq .encounter2 ] , they can react with a bond - specific rate .similarly an existing bond can dissociate with a bond - specific rate . herewe assume that the waiting times for association or dissociation of a bond between two clusters in encounter are poisson - distributed , , .in this case the probability that no association or dissociation has occurred after a timestep is given by and hence the probabilities for bond dissociation or bond formation are given by :  the approximations used in eq.[eq .p_bond ] are valid if and , which is the case throughout this work .we now show how the microscopic reaction rates and can be related to macroscopic reaction rates . in a macroscopic frameworkthe reaction scheme in eq .encounter ] can be interpreted as a system of ordinary differential equations describing the changes in the concentrations ,, and of the different species involved in the reaction . in this framework the encounter is understood as a single , intermediate state connecting the bound complex and the unbound and particles .the encounter can either react to the final complex with the first order rate or decay into two separated particles with the first order rate .an encounter can be formed by the first order decay of cluster with a rate or by the diffusive association of an and particle described by the second order rate . here has the dimension if we consider particle concentrations in 3 dimensions . using a steady state approximation of the encounter complex, the following relations are obtained for the overall forward and backward rates and the equilibrium constant : in the case of a diffusion - limited reaction . in this case the overall forward and reverse rates simplify into the purely diffusion - limited forward rate and the backward rate becomes . in the casethat the reaction is reaction - limited ( ) , the overall forward rate becomes and the overall backward rate is .  in the framework of the fokker - planck equation two spherical particlesare considered to be in an encounter if they are in contact . finite reactivity for particles in encounter was introduced by collins and kimball in the form of a radiation boundary condition in which the rate ( termed in ref . and in ref . ) relates the concentration at contact to the reactive flux . by comparing the escape probabilities from the encounter in the fokker - planck framework and the macroscopic rate equation framework, shoup and szabo showed that can be related to macroscopic rates by identifying and . agmon and szabo derived the same relation for the equilibrium constant ( ) by considering an isolated pair of reactive spheres using a back - reaction boundary condition .  here we show how the microscopic reaction rate used to describe reactions in our simulation approach can be related to the reaction scheme in eq .. encounter ] and to the equilibrium constant ( eq .[ eq keq2 ] ) . inspired by the work of shoup and szabo calculate the equilibrium constant for the formation of the encounter . in our approachthe encounter is defined as a region in configuration space ( eq . [ eq .encounter1 ] and eq .encounter2 ] ) around the desired relative position in the bound complex instead of a boundary as commonly used in the fokker - planck picture .the configuration space of two rigid unbound particles in a periodic box of volume can be described by their relative position vector , their center - of - mass vector and angular coordinates and . assuming free diffusive motion without reactions all configurations in the two particle configuration space are equally probable and they only interact by steric repulsion . by using the thermodynamic extremum principle for the gibbs free energy, we can relate the equilibrium constant for the formation of an encounter to the partition sums of the free molecules and and to the partition sum of the encounter complex  : for the single molecules all positions and orientations ( neglecting the excluded volume ) are accessible , resulting in a factor of from the integration over the translational degrees of freedom and a factor of from the orientational degrees of freedom of the rigid molecules . to determine the partition sum of the encounter complex , we first integrate out the center - of - mass coordinate of the two molecules resulting in a factor of .the boundaries of the remaining integrals over the relative position coordinate and the orientation coordinates and are determined by the patch definition and the particle geometries .they follow from eq .encounter1 ] and eq .encounter2 ] and have to be calculated for every specific case , as will be discussed in more detail below .the resulting reactive volume is the central concept to relate microscopic and macroscopic rates . from eq .[ keq encounter 2 ] we see that it equals the equilibrium constant .if we now allow for reactions , not every configuration in the encounter region is equally populated as some particles will already react before they explore the inner region of . however ,if on average the encounter region is well explored before an association , we can approximate the real distribution within by a uniform distribution .this approximation is equivalent to the underlying assumption in the work of pogson et al and klann et al . who introduced a reactive volume ( and an intrinsic association rate ) based on the assumption that the particles are randomly distributed in the box at all times .  alsallaq and zhou also used thermodynamic arguments to determine the equilibrium constant for the bound complex . however , in contrast to this work we consider all configurations within the generalized reactive volume as encounter configurations . by introducing finite reactivity with the microscopic reaction rate which a bound complex can be formed from the encounter region ( see fig .3a ) , our approach can account for reactions which are reaction- or diffusion - limited , as will be shown below in the results section . to study assembly , this is of great importance as intermediates emerging during the assembly process can have very different diffusion properties and a reaction that was reaction - limited for small clusters can become diffusion - limited for larger clusters .thus , by identifying the equilibrium constant for the formation of the encounter with the reactive volume ( eq . [ keq encounter 2 ] ) and relegating bond association and dissociation to the reaction rates , we can identify the reaction rates and in the microscopic framework with the corresponding reaction rates and in the macroscopic framework ( eq . [ eq keq2 ] ) and the equilibrium constant is given by : in accordance with the work by shoup and szabo the rate is given by the product of the equilibrium constant for the formation of the encounter complex and the reaction rate for the formation of the final complex .this rate can be identified with the reaction rate used by morelli and ten wolde ( in ref . ) .  while the reactive volume is sufficient to predict the correct equilibrium constant ( eq .ka_prime ] ) , the kinetics of the reactions depend on the absolute values of the diffusive rates and . thus in order to determine the macroscopic rates and defined in eq .[ eq k+ k- ] , we need to evaluate . this can be done within our simulation approach based on an algorithm by zhou . in this algorithm be calculated from the survival probability of two particles starting in encounter by : given the diffusive on - rate , the diffusive off - rate can be simply calculated by ( eq . [ keq encounter 1 ] ) .  any simulation algorithm with underlying reversible dynamics needs to satisfy detailed balance . to establish detailed balance we follow the approach by morelli and ten wolde who inferred from a detailed balance consideration that the relative position distribution of two spherical particles prior to a reaction has to be identical ( after renormalization ) to the position distribution of the two particles after dissociation . for two hard spheres covered with a reactive shell, it has been argued that detailed balance can be introduced by placing the two hard spheres according to a radial uniform distribution within the reactive shell followed by one additional move step .here we generalize this idea as follows : upon the dissociation of two clusters , a new configuration is chosen uniformly within the encounter region followed by a diffusive move step of the two clusters . to establish the uniform distribution in exploit the time invariance symmetry of the diffusion propagator .this symmetry ensures that in a confined volume of the configuration space a uniform distribution is established as the steady state distribution .as describes a configuration around the desired relative configuration of the proteins , we can generate a uniform distribution without prior knowledge of the exact shape of the reactive volume by performing various `` pseudo - diffusion '' steps of the two clusters starting from their predefined relative configuration .these `` pseudo - diffusion '' steps are confined to the region in the relative accessible configuration space of the two partners .we call this procedure `` pseudo - diffusion '' as the two dissociating clusters are propagated diffusively within , however , the timestep used to establish the uniform distribution has no physical meaning . this procedure for generating a uniform distribution within the reactive area does not depend on the particle or patch geometry .thus it is especially suited to study assembly dynamics as parts of the reactive volume might become sterically blocked during the assembly process . these steric effects are automatically taken into account with our method . the effect of steric blocking and its consequences for assembly will be discussed in the results section . moreover, this method would also remain valid when including long range electrostatic interactions . in this casethe distribution within the encounter complex in equilibrium would not be uniform .however , the `` pseudo - diffusive '' motion step would lead to the expected distribution within .  in general the encounter volume has to be understood as the volume of all two particle configurations resulting in an encounter between two clusters ( see fig .this means that for clusters with multiple patches the total encounter volume where is the encounter volume between a pair of patches . in the case of a dissociation of clusters with multiple patches a uniform configuration in the subvolume of the patches that formed the bondis realized .if the different patches have a different microscopic reactivity , the subvolumes have to be treated separately and the equilibrium constant for the complex formation is given by the sum of the different equilibrium constants for different bonds .proteins with multiple reactive patches can assemble in structures containing loops . in this casean already connected cluster can contain open bonds with two patches being in permanent encounter due to the geometry of the cluster .these patches can bind with the internal association rate , where we again assume poisson - distributed waiting times for the bond formation .the bond formation in this case is fundamentally different from the above discussed process as no diffusion is involved . since clusters are rigid objects , internal bond formation does not change the shape of the cluster .thus it can be considered as an internal process stabilizing an already existing structure .given a dissociation rate of for a specific bond , we can relate the internal association rate to the free energy of bond formation . for a closed loop containing bonds ,there are different configurations with one open bond in the loop and the fraction of the open and closed loop in equilibrium is given by : thus , the internal association regulates the fraction of open and closed loop configurations . in a more realistic scenario ,the dissociation of one bond in a loop will enable an enhanced internal movement .this will be reflected in an increase in the entropy which would shift the ratio of open and closed ring structures towards the open state ( smaller ) . in principleit should be possible to calculate the change in energy and entropy upon dissociation of a bond using detailed md simulations .such simulations could be used to specify the internal association rate .   in this partwe will describe the work flow of our algorithm .once a starting configuration is seeded the following steps are repeated iteratively .firstly every particle is translated and rotated according to its diffusive properties in the framework of free brownian motion within a timestep . with the new positions and orientationsall clusters with steric overlaps as well as all clusters participating in a reaction are tagged .a reaction between two clusters occurs with the bond specific probability if they are in encounter . in a next step allsteric overlaps ( including box overlaps if the box is non - periodic ) are corrected by setting the involved particles back to their original position before the move step .this might lead to further steric overlaps as a higher order effect .these overlaps are then also corrected . in thisstep clusters tagged for a reaction click into their predefined relative positions and orientations ( see fig .if a reaction leads to a steric overlap either between the two merging clusters or with other clusters , the reaction is rejected and the particles reassume their configuration before the move step . collisions with other clusters after the reorientations step is a higher order effect in the concentration of the particles . in dilute systems , especially for reactive configuration volumes around the desired relative configuration , this happens rarely . in a next step open ,internal bonds can be closed with probability .finally each existing bond can dissociate with the bond specific probability .if this leads to two unconnected cluster fragments , the diffusive properties of the fragments are calculated .subsequently a uniform distribution within the encounter volume is realized by `` pseudo - diffusive '' motion preserving the encounter followed by one unconstrained diffusion step of the two clusters .all simulations have been performed at room temperature and with the viscosity of water . for two identical hard spheres of radius equipped with one polar patch of radius nm as a function of the patch angle .the analytical result from eq . [ eq . v kern - frenkel ] is shown with a dashed line while the mc estimates ( according to eq .mc ] ) are represented by red points with error bars.,width=321 ]  as discussed in the previous section , we can relate the microscopic rates to the macroscopic equilibrium constant using the concept of a generalized reactive volume in configuration space ( eq . [ eq .ka_prime ] ) . herewe introduce a model system in which it is possible to analytically calculate and show that the results from our mc scheme ( eq . [ eq .mc ] ) agree well with the analytical results . in this model systema protein is described by one hard sphere of radius r equipped with one polar patch as depicted in fig . 3a .the patch is centered at the origin of the protein .it has a radius of and an opening angle of 12 & 12#1212_12%12[1][0] * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * , ( ) * * ,  ( ) link:\doibase 10.1016/j.tcb.2008.07.009 [ * * ,  ( ) ] * * ,  ( ) * * , ( ) * * ,  ( ) * * ,  ( ) * * , ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * , ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * , ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * , ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) link:\doibase 10.1126/science.1220869 [ * * ,  ( ) ] * * ,  ( ) * * , ( ) * * ,  ( ) _ _ , edited by and , , vol .( , ) pp .  * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) _ _ ( , )  * * ,  ( ) * * ,  ( ) _ _ ( , ) * * ,  ( ) link:\doibase 10.1529/biophysj.108.132563 [ * * ,  ( ) ] link:\doibase 10.1016/j.cell.2009.04.010 [ * * ,  ( ) ] * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) in  _ _ , vol .( , ) pp.  link:\doibase 10.1016/j.cell.2011.01.008 [ * * , ( ) ] * * ,  ( ) link:\doibase 10.1016/j.cub.2013.06.061 [ * * ,  ( ) ] * * ,  ( ) * * ,  ( )

Abstract: assembly of protein complexes like virus shells , the centriole , the nuclear pore complex or the actin cytoskeleton is strongly determined by their spatial structure . moreover it is becoming increasingly clear that the reversible nature of protein assembly is also an essential element for their biological function . here we introduce a computational approach for the brownian dynamics of patchy particles with anisotropic assemblies and fully reversible reactions . different particles stochastically associate and dissociate with microscopic reaction rates depending on their relative spatial positions . the translational and rotational diffusive properties of all protein complexes are evaluated on - the - fly . because we focus on reversible assembly , we introduce a scheme which ensures detailed balance for patchy particles . we then show how the macroscopic rates follow from the microscopic ones . as an instructive example , we study the assembly of a pentameric ring structure , for which we find excellent agreement between simulation results and a macroscopic kinetic description without any adjustable parameters . this demonstrates that our approach correctly accounts for both the diffusive and reactive processes involved in protein assembly .


media streaming is going to be one of the major applications in wireless networks . for example , it is envisioned that a significant portion of the capacity demand in future wireless systems will come from media streaming applications . in this paper, we consider media streaming in mimo interference networks whereby multiple bss simultaneously deliver media to their associated users using fixed data rates .the performance of this system is fundamentally limited by the inter - cell interference from the cross - links . in traditional cellular networks ,the inter - cell interference is mitigated using frequency planing techniques such as frequency reuse or fractional frequency reuse . to further improve the spectrum efficiency ,more advanced techniques such as cooperative mimo ( comp ) and coordinated mimo have been proposed for future wireless systems .the comp technique can transform the cross - link interference into spatial multiplexing gain by sharing both real - time channel state information ( csi ) and payload data among the concerned bss .however , it requires high capacity backhaul for payload exchange between bss , which is a cost bottleneck especially in dense small cell networks . on the other hand ,the coordinated mimo is a more cost effective technique as it only requires the exchange of real - time csis among the bss to perform joint precoding .many mimo precoding optimization algorithms have been proposed for coordinated mimo .for example , in , a wmmse algorithm is proposed to find a stationary point of the weighted sum - rate maximization problem for multi - cell downlink systems . in , the authors proposed _ polite water - filling _ method for precoding optimization in b - mac interference networks based on the duality principle of interference networks .although the coordinated mimo requires smaller backhaul capacity , the overall performance is usually much lower than that of comp .recently , there have been some works conducted on multi - cell coordination with consideration of backhaul limitation . in , a distributed and hierarchical solution of joint beamforming and power allocationwas proposed to maximize the worst - user sinr in time - division - duplex ( tdd ) multicell downlink systems where only limited inter - cell information exchange is permitted . in , random matrix theoryis leveraged to design a distributed joint beamforming and power control algorithm that only requires statistical information .such design reduces the amount of control signaling over the backhaul .an interesting question is , can we achieve the comp gain with reduced backhaul bandwidth consumption ?we show that this is possible for media streaming applications by using a novel _ cache - induced opportunistic comp _ scheme proposed in this paper .specifically , we can opportunistically transform the interference network into a comp broadcast channel by caching a portion of the media files at the bss . as a result , there are two transmission modes at the physical layer , namely , the _ comp mode _ and the _ coordinated mimo mode _ , depending on the cache state at the bss .if the payload data accessed by each user exists in the cache of the bss , the bss can engage in comp and therefore , enjoy a large performance gain without consuming the backhaul bandwidth .otherwise , coordinated mimo is employed at the bss to serve the users .hence , there is a cache - induced topology change in the physical layer ( dynamic comp opportunity ) of the mimo interference network . as such, a mimo interference network employing the cache - induced opportunistic comp is called a _ cached mimo interference network _ in this paper . with high capacity caches at the bss and a proper caching strategy , the opportunity of comp in the cached mimo interference network can be very large and thus the proposed solution will have a significant gain over the coordinated mimo scheme with even smaller backhaul consumption .note that in the proposed solution , the reduced backhaul consumption is due to the reduced payload data transmission over the backhaul .the payload data transmission consumes much more backhaul bandwidth than the exchange of control signaling because the former needs to be done on a per - symbol basis but the latter needs to be done on a per frame basis .hence the backhaul saving of the proposed solution is much more significant compared to those only reduce the control signaling in the backhaul . since the cost of hard disks is much lower than the cost of optical fiber backhaul , the proposed solution is very cost effective .the performance of the proposed solution depends heavily on the dynamic caching strategy ( which affects the opportunity of comp ) and the mimo precoding design .we study a mixed - timescale joint optimization of mimo precoding and cache control in cached mimo interference networks to minimize the average sum transmit power subject to fixed data rate constraints for all users .the role of cache control is to create more comp opportunities and is adaptive to long - term popularity of the media files ( long - term control ) .the role of mimo precoding optimization is to exploit the comp opportunities ( induced by the cache ) to guarantee the individual rate constraints for each user . as such, it is adaptive to the instantaneous csi and the _ cache state _ at the bss .there are several first order technical challenges to be addressed .  ** limited cache size * : the performance gain of the proposed scheme depends heavily on the comp opportunity , which in turn depends on the cache size and cache strategy .the bss usually do not have enough cache to store all the media files . as will be shown in example [ naive - cahce - scheme ] ,when brute force caching is used , even if a significant portion of the media files are cached at bss , the comp opportunity can still be very small and this is highly undesirable .* * non - convex stochastic optimization * : the mixed - timescale joint optimization of mimo precoding and cache control is a non - convex stochastic optimization problem and the complexity of finding the optimal solution is extremely high .for example , the short - term mimo precoding optimization in the interference networks is well known to be a difficult non - convex problem .furthermore , the objective function for long - term cache control has no closed form expression because the short - term precoding problem has no closed form solution and the popularity of the media files is in general unknown . * * complex coupling between cache control and precoding * * optimization * : caching has been widely used in fixed line p2p systems and content distribution networks ( cdns ) . in ,a femtocaching scheme has also been proposed for wireless systems . however, these schemes do not consider cache - induced opportunistic comp among the bss .hence , the cache control in the above works is independent of the physical layer and is fundamentally different from our case where the cache control and physical layer are coupled together . in our case , the cache control will affect the physical layer dynamics seen by * * precoding * * optimization due to different comp opportunities .on the other hand , the short - term precoding strategy adopted in the physical layer will also affect the cache control due to a different cost - reward dynamic .  to address the above challenges ,we first propose a novel cache data structure called _ mds - coded random cache _ which can significantly improve the probability of comp .we then exploit the timescale separations of the optimization variables to decompose the stochastic optimization problem into a _ short - term precoding problem _ and a _ long - term stochastic cache control problem_. we generalize the wmmse approach in to find a stationary point for the short - term precoding problem . to solve the long - term cache control problem ,we first show that despite the non - convexity in the short - term precoding problem , there is a hidden convexity in the long - term stochastic cache control problem .we propose a stochastic - subgradient - like iterative solution and show that it converges to the optimal solution of this long - term stochastic optimization problem .the proposed solution has low complexity and does not require explicit knowledge of the popularity of the media files .finally , we illustrate with simulations that the proposed solution achieves significant gain over various baselines under the consideration of overhead in the backhaul ._ notations _ : the superscript denotes hermitian . the notation denote the indication function such that if the event is true and otherwise .the notation {i , j} ] denote a vector whose -th element is .in this section , we introduce the architecture of the cached mimo interference networks , the physical layer ( opportunistic comp ) and the mds - coded random cache scheme that supports opportunistic comp .the architecture of the cached mimo interference network is illustrated in fig .[ fig : system_model ] .there are media files on the media server .the size of the -th media file is bits and the streaming rate is denoted by ( bits / s ) .there are users streaming media files from the media server via a radio access network ( ran ) consisting of bss and each bs is associated with one user .each bs is equipped with antennas and each user is equipped with antennas .the index of the media file requested by the -th user is denoted by .define as the user request profile ( urp ) .we have the following assumption on urp .[ asm : urp]the urp is a slow ergodic random process ( i.e. , remains constant for a large number of time slots ) according to a general distribution .the media packets ( payload data ) requested by the -th user are delivered to the -th bs from the media gateway via backhaul as illustrated in fig .[ fig : system_model ] .moreover , each bs is equipped with a cache of size bits . in this paper , the time is partitioned into time slots indexed by with duration .the ran is the performance bottleneck of the system . without caching at each bs , the ran forms a mimo interference network and the performanceis limited by the inter - cell interference between the bss . in this section ,we propose a cache - induced opportunistic comp which can opportunistically use the cached media packets at the bss to transform an interference network into a comp broadcast channel as illustrated in fig .[ fig : cocamimo]-(b ) .the impact of caching at bs on the physical layer is summarized by the _ cache state _ defined as , where means that , the current payload data requested by user is in the cache of all the bss and thus it is possible for the bss to cooperatively transmit the payload data to the users ; and means that the -th user can only be served by the -th bs . hence , there are two transmission modes depending on the cache state s , namely the comp mode ( ) and the coordinated mimo mode ( ) . fig .[ fig : cocamimo ] illustrates two examples of the data flows under different cache states ( or transmission modes ) . as illustrated in fig .[ fig : cocamimo ] , when , the ran can enjoy significant spatial multiplexing gain due to cache - induced comp transmission and the gain is achieved without expensive backhauls .we consider an ofdm based system where the wireless link between each bs and user consists of orthogonal subcarriers .let denote the channel matrix between user and bs on subcarrier .we have the following assumption on the global csi .  [asm : fory] remains constant within a time slot but is i.i.d .time slot index .specifically , has i.i.d .complex gaussian entries of zero mean and variance .  the variance is usually used to model the path gain between bs and user .note that we do not require that the channel is i.i.d .subcarrier index .we consider a centralized optimization scheme in which a central node computes all the control variables and then transmits them to the bss .we assume that the central node has the knowledge of the global csi .then we elaborate the two transmission modes in the proposed cache - induced opportunistic comp .  * coordinated mimo mode * : if , the -th user can only be served by the -th bs .we consider linear precoding and mmse receiving for inter - cell interference cancellation .the received signal for user on subcarrier can be expressed as : where and are respectively the data vector and the number of data streams for user on subcarrier ; is the precoding matrix for user on subcarrier ; and is the awgn noise vector .the mmse receiver for user on subcarrier is given bywhere is the interference - plus - noise covariance matrix of user on subcarrier .then for given csi , cache state and precoding matrices , the data rate ( bps ) of user is where is the bandwidth of the system ; and the corresponding sum transmit power is   * comp mode * : if , the users are served using comp between the bss .similarly , we consider linear precoding and mmse receiving for interference cancellation .the received signal for user on subcarrier can be expressed as : where \in\mathbb{c}^{n_{r}\times kn_{t}} ] .we first use an example to show that , with a naive cache scheme , the comp opportunity (  ] and urp , let .then conditioned on a given and , the cache state is generated by a random cache state generator at the central node ( as illustrated in fig .[ fig : moduleconn ] ) using the following method.first , time is divided into frames where each frame contains time slots .then the central node randomly generates an index set such that .note that for given and , is only generated for once and it remains constant until and changes to a new value .suppose that the current time slot is the -th time slot in the current frame .then if , we let ; and otherwise , we let . finally , the central node broadcasts the cache state to the bss .[ fig : gens ] gives an example of how to generate the cache state for each time slot .it can be seen that for any time interval of consecutive time slots , there are time slots with and time slots with , as illustrated in fig .[ fig : gens ] .the bss decide when to do mimo cooperation according to the generated cache state .if , the bss employ comp to jointly transmit the cached parity bits to the users without consuming the backhaul bandwidth as illustrated in fig .[ fig : mds - cache]-(d ) . otherwise , bs obtains the parity bits requested by user from the backhaul and transmits them to user using the coordinated mimo transmission mode as illustrated in fig .[ fig : mds - cache]-(c ) .   at user , the process of receiving and decoding a segment of the -th media file , where , is summarized as follows .there is a playback buffer and a media decoder at the user terminal .the media decoder has a _ reassembling buffer _ as illustrated in fig .[ fig : rasbuff ] . at each time slot , if , user receives cached parity bits of current segment from all bss using comp and stores them in the playback buffer . if , user receives uncached parity bits of current segment from bs using coordinated mimo and stores them in the playback buffer .user keeps receiving parity bits from the ran until the total number of received parity bits for current segment is equal to .then , user starts to receive the next segment from the ran . on the other hand ,the media decoder keeps reading parity bits from the playback buffer at a constant playback rate ( which is equal to ) and storing them in the _ reassembling buffer _ until the total number of parity bits for current segment at the reassembling buffer is equal to .then the whole segment is decoded and the reassembling buffer is cleared so that the media decoder can read the next segment from the playback buffer .clearly , it takes time slots for user to receive all the parity bits and the number of time slots with is .hence , user receives a total number of cached parity bits for each segment , which is feasible ( i.e. , there is no bs cache underflow ) since each bs stores parity bits for every segment of the -th media file .the following example illustrates the advantage of mds - coded random cache scheme .[ advantage of mds - coded random cache][advantage - of - mds - coded]consider the setup in example [ naive - cahce - scheme ] .we have and thus the probability of is , which is much larger than that of the brute - force caching scheme in example 1 ( ) .  compared with the brute - force caching scheme in example [ naive - cahce - scheme ] , the probability of comp transmission ( ) under the proposed mds - coded random cache is versus are in the cache of all the bss is . for ,whether the packets requested by user is in the bs caches is independent of whether the packets requested by user is in the bs caches .hence , the probability of is * *. ] .this represents a first order improvement in the opportunity of comp gain . yet, there is a fundamental tradeoff between the performance gain and the bs cache size .intuitively , the more popular the media file is , the larger portion of its parity bits should be stored in the bs cache to increase the comp probability .hence , the value of must be carefully controlled to achieve the best tradeoff among performance and the bs cache size .as such , the cache control variable is parameterized by the vector ^{t} ] .this relaxation has little effect on the performance when . in ( ) , ( ) is the instantaneous rate constraint for user under urp and cache state ( ) . in media streaming applications ,the playback process at user can be modeled by a playback queue with random arrival ( from the bs via the ran ) and deterministic departure as illustrated in fig .[ fig : rasbuff ] .the media streaming qos can be represented by _playback interruption probability _ , which is the same as the probability of playback buffer being empty as indicated in fig .[ fig : plabackqueue ] . since the departure process is deterministic with constant rate , the instantaneous rate constraint ( when ) and ( when ) in ( [ eq : fset ] ) essentially guarantee that the playback process is free from interruption .  then the joint cache and power control problemis formulated as : \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\nonumber \\ \textrm{s.t}.\ : v\left(\pi,\mathbf{h}\right)\in\mathcal{d}_{\mathbf{v}}\left(\pi,\mathbf{h}\right),\widetilde{\mathbf{v}}\left(\pi,\mathbf{h}\right)\in\mathcal{d}_{\widetilde{\mathbf{v}}}\left(\pi,\mathbf{h}\right),\textrm{w.p.1},\label{eq : ratecon}\end{aligned}\ ] ] where the expectation is taken w.r.t .the distribution of . note that in constraint ( [ eq : ratecon ] ) , the instantaneous rate requirement is satisfied with probability one because it is impossible to guarantee a fixed data rate for all realizations of .  in general , problem may not even be feasible ( i.e.  ] is bounded.  please refer to appendix [ sub : proof - of - proposition - pf ] for the proof .problem is a non - convex stochastic optimization problem .we first decompose it into simpler subproblems . for convenience ,define according to the analysis in appendix [ sub : proof - of - proposition - pf ] , for any and , there exists such that and are bounded .moreover , we have =1 ] is the lagrange multiplier vector .the dual function of problem ( [ eq : pmse ] ) with fixed is the minimization problem in ( [ eq : dualfun ] ) can be decomposed into independent problems as for fixed , problem ( [ eq : maxlm ] ) has a closed - form solution given by since problem ( [ eq : pmse ] ) with fixed is a convex quadratic optimization problem , the optimal solution is given by , where is the optimal solution of the dual problem the dual function is concave and it can be verified thatis a subgradient of , where is obtained from ( [ eq : emk ] ) with .hence , the standard subgradient based methods such as the subgradient algorithm in or the ellipsoid method in can be used to solve the optimal solution of the dual problem in ( [ eq : mingfun ] ) .   the overall algorithm ( named algorithm sp ) for solving summarized in table [ tab : algps ] .note that using the matrix inversion lemma , it can be shown that if is the mmse receiver given in the step 1 of algorithm sp . hence in step 2 ,we let .the following theorem shows that algorithm sp converges to a stationary point of .[ convergence of alg .sp][thm : convergence - of - algps]for given , any limit point of the iterates generated by algorithm sp is a stationary point of problem ( [ eq : pmse ] ) , and the corresponding is a stationary point of .  please refer to appendix [ sub : proof - of - theoremconvps ] for the proof .problem is a sum power minimization problem under individual rate constraints in parallel broadcast channel , which can be viewed as a parallel interference network with the cross link channel equal to the direct link channel .hence , algorithm sp can also be used to find a stationary point of by replacing , , , respectively with , , , and using the following initial point : {i , j}=\begin{cases } \left[\mathbf{v}_{m , k}^{*}\left(\pi,\mathbf{h}\right)\right]_{i-\left(k-1\right)n_{t},j } , & i\in\mathcal{i}_{k},j\in\left[1,d\right]\\ 0 , & \textrm{otherwise } \end{cases}\label{eq : initvta}\ ] ] for all , where  ] and changes at the boundary of each time interval .then the following lemma gives a noisy unbiased subgradient of the objective function .[ noisy unbiased subgradient of ][lem : subconvcond]let denote the urp for the -th time interval  ] , is given by where is any index satisfying ; are any two non - empty sets of time slot index in  ] and :\ : s\left(t\right)=1\right\ } ] . for the bs cache update process, each bs needs to obtain a total number of parity bits from the backhaul .however , these cache updates can be done offline with much smaller average backhaul consumption compared with the backhaul consumption due to online streaming because the popularity of media files changes very slowly ( e.g. new movies are usually posted on a weekly or monthly timescale ) .let denote the interval of the cache update process , then the average backhaul consumption due to the offline bs cache update is .finally , the overall average backhaul consumption ( bps ) is given by +k\sum_{l=1}^{l}q_{l}f_{l}/t_{c}.\label{eq : onlinebc}\ ] ] table .[ tab : cacheupdateload ] compares the average backhaul consumption of different schemes .impressively , the average backhaul consumption is much smaller than the coordinated mimo for only moderately large bs cache size ..[tab : cacheupdateload ] [ cols= " < , < , < " , ]consider a media streaming system with media files and bs - user pairs .the bss are arranged as in fig .[ fig : hetnet - topo ] with inter - site distance of m .we consider two different user placements . in the _ normal user placement _ , each user is uniformly distributed within its cell under the restriction that the distance between the user and the serving bs must be larger thanm , even the path gain of the strongest cross link is about 28db smaller than the path gain of the direct link .such users are essentially free from the inter - cell interference and we have excluded this uninteresting degenerate case in the simulations . ] m . in the _ edge user placement _ , each user is uniformly distributed within its cell under the restriction that the distance between the user and the serving bs must be larger than m .each bs has antennas and each user has antennas .the path gains s are generated using the path loss model ( urban macro nlos model ) in .the size of each media file is m bytes and the steaming rate is .assume that each user independently accesses the -th media file with probability , and we set =\left[0.6,0.3,0.08,0.01,0.005,0.005\right] ] is bounded for all .consider a simple precoding scheme where and \in\mathbb{c}^{n_{t}\times d},\forall k = m ] , it follows that with probability one .moreover , we have \\  & \leq & d\sum_{k=1}^{k}\left(e^{\frac{m\mu_{\pi_{k}}\ln2}{b_{w}}}-1\right)\textrm{e}\left[\textrm{tr}\left(\mathbf{h}_{k , k , k}\mathbf{h}_{k , k , k}^{\dagger}\right)^{-1}\right]\\  & < & \infty,\end{aligned}\ ] ] where the last inequality holds because ( after proper normalization ) has a chi - squared distribution with degrees of freedom .this completes the proof .[ [ proof - of - theorem - thmequivalence - between - psandpmsesubproof - of - theoremequips11 ] ] proof of theorem [ thm : equivalence - between - psandpmse][sub : proof - of - theoremequips11 ] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~  theorem [ thm : equivalence - between - psandpmse ] can be proved by contradiction .suppose that is not the optimal solution of .then there exists such that .let and , where is the mmse receiver corresponding to and .then it can be verified that   [ [ proof - of - theorem - thmconvergence - of - algpssubproof - of - theoremconvps ] ] proof of theorem [ thm : convergence - of - algps][sub : proof - of - theoremconvps ] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~  following similar analysis as in the proof of ( * ? ? ?* theorem 3 ) , it can be shown that any limit point is a stationary point of problem ( [ eq : pmse ] ) . as a result , andthe corresponding lagrange multiplier vector satisfies the following kkt conditions and , where moreover , the step 1 and step 2 of algorithm sp ensure that is the mmse receiver corresponding to and , from which it follows that using chain rule , it can be shown that ( please refer to ( * ? ? ?* appendix c ) for the detailed derivation ) it follows from ( [ eq : kktps ] ) , ( [ eq : rkbd ] ) and ( [ eq : drbar ] ) that where is the lagrange function of : hence , is a stationary point of with as the corresponding lagrange multiplier vector .the initial point given in ( [ eq : initvta ] ) ensures that the initial objective value of is equal to . in each iteration of algorithm sp , the objective value is strictly decreased before converging to a stationary point of .hence , we must have and \leq\textrm{e}\left[\left.p\left(\mathbf{v}^{*}\left(\pi,\mathbf{h}\right)\right)\right|\pi\right] ] that is a decreasing linear function of for ,\forall l ] , where is a convex function of .it is easy to see that for each and , \end{aligned}\ ] ] is a subgradient of .hence , for each and any , we have taking expectation for both sides of ( [ eq : faisub ] ) , we have ^{t}\left(\mathbf{q}^{'}-\mathbf{q}\right),\ ] ] which implies that $ ] is a subgradient of . on the other hand, we have & = & \textrm{e}\bigg[1\left(l=\pi_{k^{*}}\right)\bigg(\textrm{e}\left[\left.\widetilde{p}\left(\widetilde{\mathbf{v}}^{*}\left(\pi,\mathbf{h}\right)\right)\right|\pi\right]\\  &  & -\textrm{e}\left[\left.p\left(\mathbf{v}^{*}\left(\pi,\mathbf{h}\right)\right)\right|\pi\right]\bigg)\bigg]=\overline{g}.\end{aligned}\ ] ] hence , is a noisy unbiased subgradient of .r. ghaffar and r. knopp , `` fractional frequency reuse and interference suppression for ofdma networks , '' in _ proceedings of the 8th international symposium on modeling and optimization in mobile , ad hoc and wireless networks _ , 2010 , pp. 273277 .o. somekh , o. simeone , y. bar - 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Abstract: consider media streaming in mimo interference networks whereby multiple base stations ( bs ) simultaneously deliver media to their associated users using fixed data rates . the performance is fundamentally limited by the cross - link interference . we propose a _ cache - induced opportunistic _ _ cooperative mimo _ ( comp ) for interference mitigation . by caching a portion of the media files , the bss opportunistically employ comp to transform the cross - link interference into spatial multiplexing gain . we study a mixed - timescale optimization of mimo precoding and cache control to minimize the transmit power under the rate constraint . the cache control is to create more comp opportunities and is adaptive to the long - term popularity of the media files . the precoding is to guarantee the rate requirement and is adaptive to the channel state information and _ cache state _ at the bss . the joint stochastic optimization problem is decomposed into a _ short - term precoding _ and a _ long - term cache control problem_. we propose a precoding algorithm which converges to a stationary point of the short - term problem . based on this , we exploit the hidden convexity of the long - term problem and propose a low complexity and robust solution using stochastic subgradient . the solution has significant gains over various baselines and does not require explicit knowledge of the media popularity .   wireless media streaming , dynamic cache control , opportunistic comp , mimo precoding


as energy consumption is becoming an environmental and therefore social and economic issue , green information and communication technology ( ict ) has attracted significant research attention recently .it was reported that internet consumes as much as of the total electricity in broadband enabled countries , and currently and in the medium term future , the majority of the energy of internet is consumed by access networks owing to the large quantity of access nodes .energy consumptions of access networks depend on the access technologies . among various access technologies includingwimax , fttn , and point to point optical access networks , passive optical networks ( pons ) consume the smallest energy per transmission bit attributed to the proximity of optical fibers to the end users and the passive nature of the remote node .however , as pon is deployed worldwide , it still consumes a significant amount of energy .it is desirable to reduce the energy consumption of pons since every single watt saving will end up with overall terawatt and even larger power saving .reducing energy consumption of pons becomes even more important as the current pon systems evolve into next - generation pons with increased data rate provisioning .  in pons ,energy is consumed by optical line terminal ( olt ) and optical network units ( onus ) .owing the large quantity , onus consume a large portion of the overall pon energy .although olt consumes a less amount of power than the total aggregated onus , one olt line card does consume a much larger amount of power than one onu .reducing energy consumption of olt is as important as reducing energy consumption of onus especially from the operators and home users perspectives .for the network operators , decreasing the energy consumption of olt can significantly reduce the energy consumption of the central office , while decreasing the energy consumption of onus has small and likely negligible impacts on that of home users who have many other electrical appliances with much higher energy consumption .formerly , sleep mode and adaptive line rate have been proposed to efficiently reduce the power consumption of onus by taking advantages of the bursty nature of the traffic at the user side .it is , however , challenging to introduce `` sleep '' mode into olt to reduce its energy consumption for the following reasons . in pons, olt serves as the central access node which controls the network resource access of onus .putting olt into sleep can easily result in service disruption of onus in communicating with the olt .thus , a proper scheme is needed to reduce the energy consumption of olt without degrading services of end users .  in this paper , we propose a novel energy - efficient olt structure which can adapt its power - on olt line cards according to the real - time arrival traffic . to avoid service degradation during the process of powering on / off olt line cards ,proper devices are added into the legacy olt chassis to facilitate all onus communicate with power - on line cards . to the best of our knowledge ,this is the first work focusing on reducing the energy consumption of olt .in the central office , one olt chassis typically comprises of multiple olt line cards that transmit downstream signals and receive upstream signals at different wavelengths .each line card communicates with a number of onus .two wavelengths for the uplink and the downlink are assigned to each onu . in the currently deployed epon and gpon systems, one olt line card usually communicates with either or onus and such an arrangement is referred to as a pon segment . to avoid service disruptions of onus connected to the central office , all these olt line cards in the olt chassis are usually power - on all the time . to reduce the energy consumption of olt ,our main idea is to adapt the number of power - on olt line cards in the olt chassis to the real - time incoming traffic .there are two types of subscribers that each network serves ; business subscribers and residential subscribers .business and residential areas are usually disjunct .it is more likely that each pon segment serves either business customers or residential customers .these two types of customers have different traffic profiles .business users demand high bandwidth during the day and low bandwidth at night while residential customers request high bandwidth in the evening and low bandwidth during the day .  during the day time, residential segments are lightly loaded .therefore , one olt line card can serve several residential segments . in the similar way, the traffic from the business segments can combined to traverse a smaller number of line cards in the evening .business and residential segments usually have low bandwidth demands during the midnight . in these situations ,the whole network is lightly loaded . in order to save energy ,the number of line cards can be reduced based on the traffic volume .parameters of the proposed model are notated below :   : data rate of one olt line card in the upstream direction . : data rate of one olt line card in the downstream direction . : total number of line cards ( pon segments ) . : number of onus connected to pon segment . : fixed traffic cycle in tdm pon . : arrival upstream traffic rate from onu of pon segment at time . : arrival downstream traffic rate to onu of pon segment at time . : smallest number of required olt line cards at time . by powering on all the olt line cards , the overall upstream data rate and downstream data rate accommodated by the olt chassis equal to and , respectively . ( or ) may be greater than the real - time upstream ( or downstream ) traffic.  the traffic rate of each segment can not be more than the provisioned capacity of the dedicated fiber .therefore , the following constraints have to be satisfied for any segment :  the real time incoming upstream and downstream traffics are defined as and , respectively . then ,   our ultimate objective is to * power on only olt line cards to serve all onus * at a given time instead of powering on all line cards .however , powering off olt line cards may result in service disruptions of onus communicating with these olt line cards . to avoid service disruption , power - on olt line cards should be able to provision bandwidth to all onus connected to the olt chassis . to address this issue, we propose several modifications over the legacy olt chassis to realize the dynamic configuration of olt as will be presented next .to dynamically configure the communications between olt line cards and onus , one scheme we propose is to place an optical switch in front of all olt line cards as shown in fig.[fig : engolt ] ( a ) .the function of the optical switch is to dynamically configure the connections between olt line cards and onus . when the network is heavily loaded , the switches can be configured such that each pon system communicates with one olt line card .  as discussed in section [ sec : i ] , when the network is lightly loaded , the switches can be configured such that multiple pon systems communicate with one line card .then , some olt line cards can be powered off , thus reducing energy consumption .assume the energy consumption of the optical switch is negligible .as compared to the scheme of always powering on all line cards , the scheme of powering on only line cards at time can achieve relative energy saving as large as  then , the average energy saving over time span equals to   we define the traffic load as the maximum of upstream and downstream traffic loads to determine the required number of line cards : process of changing the switch configuration is time consuming , frequent change of each line card s status may degrade the onu performance .since the traffic of the onus is also bursty and changes dynamically , it is more efficient to monitor the traffic for an observation period ( ) before changing the switch configuration .if the traffic load of the network remains below a threshold for , the number of line cards will be adjusted accordingly . bydynamically configuring switches , the number of power - on olt line cards is reduced from to when the traffic load falls between and for a period of .thus , a significant amount of power can be potentially saved .[ fig : engolt ] ( b)-(e ) illustrates the configuration of switches for the case that one olt chassis contains four olt line cards .the number of power - on olt line cards is reduced from four to three , two , and one when the traffic load falls within the range  ] , and $ ] , respectively .  by equipping the olt chassis with proper optical switches ,the communications between olt line cards and onus can be dynamically configured .the new olt structure appears to be promising and cost - effective as compared to the wdm based solutions .configuration time of the optical switches and power consumption of each switch are two main specifications that should be considered in the analysis .switches take time to change configurations. the switch reconfiguration time may affect the onu performances when powering on / off olt line cards .we investigate the impacts of the switch reconfiguration time on epon and gpon , respectively .* epon : we argue that services of epon onus are not affected when the switch configuration time is as large as ms .+ in epon , the upstream bandwidth allocation is controlled by olt .onus transmit the upstream traffic using the allocated time durations stated in the gate message .ieee 802.3ah specifies that onus need to send gate messages every ms to maintain registration even if they do not have traffic to transmit .thus , the disrupted servicing time can be transparent to epon onus when the switch reconfiguration time is as large as ms , which can be satisfied by most optical switches .* gpon : we argue that services of gpon onus are not affected when the switching time is no greater than .+ for gpon , itu - t g. 984.3 specifies a fixed frame length of 125 .an onu receives downstream control messages and sends its upstream data or control traffic every gpon frame , i.e. , 125 .therefore , services of gpon onus are not affected when the switch configuration time is no greater than .so far , we have not considered the impact of the power consumption of optical switches on the saved energy of olt chassis .optical switches consume some power , and the nonzero power consumption of optical switches may reduce the saved energy of the olt chassis .denote and as the power consumption of optical switch and one olt line card , respectively . by incorporating considering the energy consumption of the optical switch ,the power consumption of the proposed olt chassis at any given time equals to hence , the energy saving is reduced to the proposed olt chassis can save some energy as compared to the legacy olt chassis when the power consumption of the optical switch satisfy the following condition the power consumption of optical switch differs from the switch size as well as the manufacturing technology . it is reported that one optomechanic switch consumes around w , while an olt line card consumes around w . then , for the case , energy can be saved as long as the average number of power - on olt line cards is less than .this condition can be easily satisfied .therefore , even considering the power consumption of optical switches , the proposed olt structure can still achieve a significant amount of power saving as compared to the legacy olt .we employ semi - markov chains to analyze the system .different observation periods have been considered for decreasing and increasing the number of line cards . is the observation time period for decreasing the number of active line cards . is defined as an observation period for increasing the number of line cards . in order to ease the analysis , from now on ,the formulations are based on the downstream traffic only .poisson processes with the average rate of packets per second is assumed for downlink packet arrivals .a fixed traffic scheduling cycle is assumed in the performance analysis .consider as the probability that downstream packets arrive at the olt chassis during traffic scheduling cycle : the same process can be considered for uplink packet arrival in the case of having downstream and upstream traffic simultaneously.   in the defined markov chain model , the total number of active line cards vary from to .therefore , there are possible states of active line cards . represents active line cards .transition from each active state to another active state should be done through the listening states .there is no direct transition between active states .there are two different types of listening states ; ( listening states for decreasing the number of line cards ) and ( listening states for increasing the number of line cards ) .figure [ fig : mrkv ] shows the state transitions . refers to the state that the total traffic load of the olt chassis remains below for cycles .the number of active line cards during time cycles remains as line cards .if the traffic load goes higher than amount during a traffic scheduling cycle , the transition to occurs . otherwise , the transition from to happens .whenever the number of listening cycles reaches to , the olt chassis switches to the next lower active state ( ) . refers to the state that the total traffic load of the olt chassis remains above for time cycles .the number of active line cards during remains as line cards .the excess amount of the traffic will be buffered during .if the traffic load becomes lower than amount during a traffic scheduling cycle , the transition to occurs .otherwise , the transition from to happens .if the total traffic load stays beyond the defined threshold ( ) for traffic scheduling cycles , the olt chassis switches to the next upper active state ( ) .* state transitions from to , to , or from to   happen when the traffic load in a traffic scheduling time is less than .therefore , the probability that the number of arrival packets is smaller than equals to : .* state transitions from to   happen when the number of arrival packets in a traffic scheduling time is smaller than , which is equals to : * state transitions from to , from to , or from to   happen when the number of arrival packets in a traffic scheduling time is greater than .the probability equals to .* state transitions from to   happen when the number of arrival packets in a traffic scheduling time is greater than , which is equals to : * state transition from to itself happens when the number of arrival packets in a traffic scheduling time remains between and .the probability is equal to .denote ,  , and  ; as the probability of the olt state when the network is at its steady state .therefore , the following constraints are satisfied .steady state probabilities of all the active states except and are as follows : \nonumber\\ & = [ \sum_{k=1}^{\infty}p(i_{i , i+1}(k))pr\{i_{i , i+1}(k ) \rightarrow a_i\}]\nonumber\\ & + p(i_{i-1,i}(n))pr\{i_{i-1,i}(n ) \rightarrow a_i\}\nonumber\\ & + \sum_{j=1}^{m}p(d_{i , i-1}(j))pr\{d_{i , i-1}(j ) \rightarrow a_i\ } \nonumber\\ & + p(d-{i+1,i}(m))pr\{d_{i+1,i}(m ) \rightarrow a_i\ } \nonumber\\ & ( i=2, ... ,l-1)\end{aligned}\ ] ]  steady state probability of is calculated as follows : steady state probability of is acheived as follows :   steady state probability of listening states except the first and last states are as follows : \nonumber\\ & = \negmedspace p(d_{i , i-1}(j-1))pr\ { \negmedspace d_{i , i-1}(j-1 ) \negmedspace \rightarrow \negmedspace d_{i , i-1}(j ) \negmedspace \ } \nonumber\\ & ( i=1, ... ,l ) \ & ( j=2, ... ,m-1)\end{aligned}\ ] ] steady state probability of equals to : \nonumber\\ & = p(a_i)pr{a_i \rightarrow d_{i , i-1}(1 ) } ( i=1, ... ,l)\end{aligned}\ ] ] steady state probability of is as follows : \nonumber\\ & = \negmedspace p(d_{i , i-1}(m \negmedspace - \negmedspace 1))pr\ { \negmedspace d_{i , i-1}(m \negmedspace - \negmedspace 1 ) \negmedspace \negmedspace \rightarrow \negmedspace d_{i , i-1}(m)\negmedspace \}\end{aligned}\ ] ] the following is the steady state probability of the listening states except the first and last states : \nonumber\\ & = p(i_{i , i+1}(k-1))pr\{i_{i , i+1}(k-1 ) \rightarrow i_{i , i+1}(k)\ } \nonumber\\ & ( i=1, ... ,l ) \ & ( k=2, ... n-1)\end{aligned}\ ] ] steady state probability of is as follows : \nonumber\\ & = p(a_i)pr\{a_i \rightarrow i{i , i+1}(1)\ } ( i=1, ... ,l)\end{aligned}\ ] ] steady state probability of is obtained as follows : \nonumber\\ & = p(i_{i , i+1}(l-1))pr\{i_{i , i+1}(l-1 ) \rightarrow i{i , i+1}(l)\}\end{aligned}\ ] ]  moreover , the sum of the probabilities of all the states is equal to , i.e. :   therefore , the steady state probabilities of all the state can be calculated by solving the above equations [ eq : first]-[eq : last ] .denote as the power consumption of one olt line card and ,  , and  ; as the probability of the olt state when the network is at its steady state . therefore , the average power consumption equals to : \ ] ]another problem with optical switches is their high prices .the prices of optical switches vary from their manufacturing techniques . at present , there are generally four kinds of optical switches : opto - mechanical switches , micro - electro - mechanical system ( mems ) , electro - optic switches , and semiconductor optical amplifier switches. currently , the opto - mechanic switches are less expensive than the other three kinds . simply because of their low prices , opto - mechanic switches are generally the adopted choices in designing energy - efficient olt .  for opto - mechanic switches ,an important constraint is their limited port counts .popular sizes of opto - mechanic switches are and . considering the port count constraints , we further propose the cascaded switches structure to achieve the dynamic configuration of olt . more specifically , to replace an switch , the cascaded switch contains stages and  switches . fig .[ fig : casc ] ( a ) illustrates the proposed cascaded switches . in the switch , the stage contains switches .[ fig : casc ] ( b ) shows a two - stage cascaded switches to replace a switch . as illustrated in fig .[ fig : casc](c)-(e ) , when the traffic load is greater than , one pon system is connected with one olt line card ; when the traffic load is between and , two pon systems are connected with one olt line card ; when the traffic load is less than , all pon sytems are connected with a single olt line card .  here , we analyze the saved energy of the proposed olt equipped with cascaded switches . assume the traffic is uniform among all onus .then , when the traffic load is between and , all olt line cards need be power - on ; when the traffic load is between and , half of the olt line cards are powered on . generally , when the traffic load is between and , of the olt line cards are power - on . therefore ,  the saved energy equals to as compared to the olt with an switch , the olt with cascaded switches saves a less amount of energy .  notethat the typical switching speed of the opto - mechanical switch is around ms . as discussed before , it does not affect the performances of users in epon , but may have impacts on the performances of users in gpon .another scheme of avoiding the significant cost increase is to use electrical switches instead . however , before aggregating traffic using electrical switches , the optical transceivers are required to convert the optical signals into electrical signals .thus , only the energy consumption of the electrical part in an olt line card can be saved .the energy saving is limited as compared to the scheme of using optical switches .let be the energy consumption of the electrical part of an olt line card .then , the average energy saving of the olt equipped with an electrical switch equals to : the efficiency of energy saving of this scheme depends on the ratio .in this section , we study the performance of the sleep control scheme of the olt line cards for poisson and non - poison traffic . in the markov chain model , we consider poisson traffic arrival for the ease of analysis .the total number of line cards is assumed to be with the capacity of for each line card .the time duration of each traffic scheduling cycle is considered to be .figure [ fig : numerical ] depicts the numerical results of the markov model . in figure[ fig : loadnum ] , the energy saving performance under different traffic arrival rates is illustrated .we assume that the maximum number of `` d '' listen cycles ( ) as well as the maximum number of `` i '' listen cycles ( ) equal to .the arrival traffic is increased up to . with the increase of traffic arrival , the energy saving decreases as the olt chassis need to have more active line cards to support the arrival traffic . when the arrival traffic is less than , one line card is sufficient to satisfy the traffic and the other line cards are shut down. therefore , the maximum energy saving is achieved .before decreasing the number of olt line cards , the olt chassis needs to monitor the traffic for the maximum of listen cycles .figure [ fig : dlistennum ] shows the effect of changing the number of m on energy saving . with the increase of m , the olt chassis needs to keep a larger number of line cards active for the time duration of .thus , the energy saving decreases by increaing . is the maximum number of traffic cycles during which the olt chassis needs to monitor the traffic before increasing the number of line cards . with the increase of , the energy saving increases as the number of active line cards is smaller for the total duration of .we also study the performance of the proposed scheme for non - poisson traffic .since the actual network traffic with bursts is self - similar , we conduct our simulation for self - similar traffic with the hurst parameter of .the packet length is uniformly distributed between bytes and bytes .as it can be seen in figure [ fig : loadsim ] , the trend of energy saving performance by increasing the traffic load for self - similar traffic is similar to that of the markov model for poisson traffic .figure [ fig : dlistensim ] shows the impact of different listen cycles on the system performance . similar to the theoretical analysis of poisson traffic , energy saving decreases as the number of listen cycles increases .however , increasing the cycles helps saving more energy . as mentioned earlier , during `` i ''listen cycles , the olt chassis receives traffic load more than the available line cards . before increasing the number of line cards , the olt chassis needs to monitor the traffic for cycles . during theses cycles ,the number of line cards stay the same , and the olt buffers the extra traffic .the buffered packets encounter some delay as depicted in figure [ fig : ilistensim ] .therefore , proper setting of needs to be considered to save energy without impairing the qos of the users .we have proposed a novel energy - efficient olt structure which adapts its power - on line cards to the real - time arrival traffic .specifically , we have added a switch into the legacy olt chassis to dynamically configure the connection between olt line cards and onus .we first describe an olt equipped with an switch , and investigate the impacts of the power consumption of the optical switch and switch configuration time on the saved energy of the whole olt chassis . then , to avoid a dramatic cost increase , we advocate the use of opto - mechanical switches among all currently commercially - available switches , and further propose to use a cascaded switch structure .our analysis demonstrates that the proposed olt achieves significant power savings as compared to the legacy olt .m. pickavet , _ et al ._ , `` worldwide energy needs for ict : the rise of power - aware networking , '' in _2nd international symposium on advanced networks and telecommunication systems_.1em plus 0.5em minus 0.4emieee , 2008 , pp .j. baliga , r. ayre , w. sorin , k. hinton , and r. tucker , `` energy consumption in access networks , '' in _ optical fiber communication conference and exposition and the national fiber optic engineers conference _ , 2008 .j. zhang , n. ansari , y. luo , f. effenberger , and f. ye , `` next - generation pons : a performance investigation of candidate architectures for next - generation access stage 1 , '' _ ieee communications magazine _ , vol .47 , no . 8 , pp . 4957 , august 2009 .j. zhang , m. taheri , and n. ansari , `` standards - compliant epon sleep control for energy efficiency : design and analysis , '' _ journal of optical communications and networking _ , vol . 5 , no . 7 ,pp . 677685 , 2013 .r. kubo , j. kani , h. ujikawa , t. sakamoto , y. fujimoto , n. yoshimoto , and h. hadama , `` study and demonstration of sleep and adaptive llink rate control mechanisms for energy efficient 10g - epon , '' _ ieee / osa journal of optical communications and networking _ , vol . 2 , no .9 , pp . 716729 , 2010 ., + http://www.unitedoptronics.com/datasheet/opticalswitch/osw2x2.pdf .w. vereecken , _ et al ._ , `` power consumption in telecommunication networks : overview and reduction strategies , '' _ communications magazine , ieee _ , vol .49 , no . 6 ,pp.6269 , 2011 .

Abstract: this paper proposes a novel scheme which can efficiently reduce the energy consumption of optical line terminals ( olts ) in time division multiplexing ( tdm ) passive optical networks ( pons ) such as epon and gpon . currently , olts consume a significant amount of energy in pon , which is one of the major fttx technologies . to be environmentally friendly , it is desirable to reduce energy consumption of olt as much as possible ; such requirement becomes even more urgent as olt keeps increasing its provisioning data rate , and higher data rate provisioning usually implies higher energy consumption . in this paper , we propose a novel energy - efficient olt structure which guarantees services of end users with the smallest number of power - on olt line cards . more specifically , we adapt the number of power - on olt line cards to the real - time incoming traffic . also , in order to avoid service disruption resulted by powering off olt line cards , proper optical switches are equipped in olt to dynamically configure the communications between olt line cards and onus .  [ cols= " < , > " , ]   design and analysis of green optical line terminal for tdm passive optical networks + mina taheri + nirwan ansari + tr - anl-2015 - 004 +  + advanced networking laboratory + department of electrical and computer engineering + new jersy institute of technology +