the action of enzymes is essential for nearly all processes in living cells .often these enzymes are organized into large multi - molecular complexes associated with specific functional tasks , and this organization can be crucial to the successful operation of the enzymatic system .these `` molecular factory '' assemblies , in which the product of one enzymatic reaction becomes the substrate for the next , are common in metabolic pathways of both prokaryotes and eukaryotes .examples include the cellulosome , the pyruvate dehydrogenase complex and glycolytic enzymes . in some cases , such as the cellulosome ,enzymes are arranged on an inert scaffold in a specific way . in others , such as tryptophan synthase complexes , direct enzyme - enzyme interactions lead to self - assembly into a complex .  despite the ubiquity of these multi - enzyme complexes , we still lack a deep understanding of the consequences of particular arrangements of enzymes for metabolic pathway operation .many advantages of co - localization have been proposed , particularly via the direct transfer or `` channeling '' of substrates from one enzyme to another .for example , reducing the transit time of pathway intermediates between enzymes can minimize the loss of unstable intermediates or the interference of competing pathways . channelingcould also potentially enhance the local density of substrates in the vicinity of the enzymes and reduce exposure to toxic intermediates ; however , whether or not these effects can actually occur has been disputed . on the other hand , compartmentalization of metabolic enzymes can also increase the flux of biosynthetic pathways , indicating that the pathway kinetics can be influenced by localization even in the absence of direct channeling .similar questions about the role of co - localization also arise in the context of protein signaling cascades .for example , there the clustering of enzymes can generate a greater amplification of the signal than distributing enzymes .however , differing functional criteria between signaling scenarios , where discrimination between different inputs is crucial , and metabolic systems , where maintaining a specific flux may be more desirable , mean that these systems are likely subject to different design pressures .more generally , little is known about the effects of the placement of enzymes beyond simple co - localization or clustering scenarios .recently there has been a growing focus on the experimental study of colocalized enzymes .techniques have been developed that allow for the attachment of enzymes to a scaffold , which was shown to significantly increase the yield of the mevalonate production pathway . the `` single - molecule cut - and - paste '' technique allows for the positioning of enzymes on a surface with nanometer precision .dna origami permits the highly - controlled production of three - dimensional structures , enabling the quantitative study of the effects of more complex spatial arrangements of enzymes .over the last few years much progress has been made in engineering of artificial enzymatic pathways on dna and rna assemblies , even _ in vivo_. in particular , a distance - dependence of the activity of a pathway consisting of glucose oxidase ( gox ) and the horseradish peroxidase ( hrp ) was demonstrated : when the enzymes are brought closer together , the efficiency of the two enzyme complex increases .here we study theoretically the impact of enzyme positioning on the flux of pathways .it has been demonstrated previously that in a simple linear reaction - diffusion model , in different parameter regimes co - localization can increase or decrease the pathway flux compared to the uniform distribution of enzymes . in this paper , we extend these results to a range of reaction - diffusion systems . in particular ,we consider also nonlinear reactions and different spatial dimensions .we demonstrate that the qualitative features of these diverse models are similar . in general, a transition occurs as a function of the effective reaction rate between regimes in which clustering or distributing enzymes in space generates a higher pathway efficiency .we calculate the optimal enzyme distribution that maximizes the efficiency of the pathway. the universal nature of our results in these diverse systems shows that the observed transitions arise from general properties of reactions and diffusion , and highlights the applicability of the observed behavior to diverse biochemical pathways .we consider a simple model reaction pathway consisting of two enzymatic reaction steps . in the first reaction step ,an enzyme converts a substrate into an intermediate ; subsequently , a second enzyme converts into the final product of the pathway , .we are interested in how the spatial organization of the enzymes affects the efficiency of the pathway in converting substrate to product . to this endwe assume that the enzymes are fixed in position , and examine the impact of the location of the enzymes relative to . in this scenario , enzymes act as a source of intermediate , with a total production rate . in order to additionally include possible undesirable non - specific competition for the intermediate by secondary pathways , or decay in the case that is unstable , we also allow for the conversion of into an alternative waste product . under the assumption that these processes are independent of the spatial arrangement of enzymes , they are simply modeled as a first - order reaction with a constant , position - independent , rate .the density of intermediate , can then be modeled by the reaction - diffusion equation where is the diffusion constant of and is the ( static ) density of enzymes .the model is illustrated in fig .[ fig : model ] . in writing eq .[ 3d_reaction_diffusion ] we have assumed that the conversion of to by the enzyme can be described by standard michaelis - menten kinetics with catalytic rate and michaelis constant . we implement the production of intermediate by enzymes through boundary conditions to eq .[ 3d_reaction_diffusion ] . in this workwe restrict ourselves to the case of a uniform source at the inner boundary , , where is the unit vector normal to the boundary and is the area of the inner boundary ; thus the total influx integrated over the boundary equals the production rate . for the outer boundary we limit ourselves to reflective ( ) or absorbing ( ) boundary conditions .the former could represent the confinement of the intermediate reaction product by the membranes of a cell or organelle , while the latter might describe an intermediate that can easily cross the membrane and be lost to the extracellular environment .we note that our treatment could easily be generalized to mixed boundary conditions representing partial confinement .enzymes at the center of the system acts as a source of intermediate with total production rate .intermediates diffuse from this center , and can either be converted to the desirable product by the enzymes ( red circles ) , via a michaelis - menten reaction with catalytic rate , or can be lost to a competing pathway with the spatially - uniform rate .we coarse - grain the positions of in space into the continuous distribution , and the local density of intermediates as . while a two - dimensional system with a hard - wall outer boundary is shown here for clarity , we consider systems of all dimension as well as different outer boundary conditions . ]  in the following we will be concerned only with the steady - state flux through the reaction pathway . at steady - state form , eq . [ 3d_reaction_diffusion ] can be recast into the dimensionless form where denotes that the spatial coordinate has been rescaled by a characteristic length - scale of the system , , which we will take to be the system size ; and indicates that the density has further been rescaled such that the total production rate of intermediate is equal to .additionally , we have defined the rescaled enzyme density with the average enzyme density over the system volume .the dimensionless parameters and respectively capture the relative timescales of reactions with and with secondary pathway enzymes , compared to the typical time to diffuse a distance .the parameter , where is the spatial dimension , represents the rate of influx of intermediate relative to the level at which enzymes become saturated , and includes the effect of varying the activity of enzymes via the intermediate production rate . in the followingwe drop the prime notation and work exclusively with the dimensionless system ; this should not be a source of confusion .  integrating eq .[ general_dimensionless ] and applying the boundary conditions leads to the flux - conservation equation on the left hand side we have the ( rescaled ) production of intermediate by .this must be balanced by the flux of reactions by enzymes , , plus the loss of intermediate .this loss can occur to secondary pathways ( the second term on the right hand side of eq . [ general_dimensionless ] ) , and via escape at the boundaries of the system ( the third term , where is the outer boundary of the system that is not a source of intermediate ) .assuming that the efficiency of the conversion of substrate to intermediate by is independent of the localization of enzymes , such that is constant , the efficiency of the system can be described by the ratio , the fraction of intermediates that are converted into the correct product . in this workwe will examine how changing affects the pathway efficiency . to compare different enzyme profiles on an equal footing ,the total amount of is held constant via the condition .  in the remainder of this paperwe systematically characterize the effects of varying localization in different regimes of eq .[ general_dimensionless ] . in the section ` linear reaction models ' we will focus on the low density limit of the intermediate product , in which the rate of reaction with becomes linear in .it was previously shown that in an open one - dimensional system where intermediate is lost at an absorbing boundary , different parameter regimes exist in which the optimal enzyme profile consists of either co - localization of with , or a configuration wherein only a fraction of enzymes are co - localized and the remainder are distributed over a finite region .here we extend these results to consider closed systems where the loss of intermediate occurs only via position - independent secondary reactions , and to three- and two - dimensional systems . finally , in the section` non - linear reactions ' we consider also the full non - linear reaction model . in all caseswe observe a transition in the optimal profile from co - localized to distributed as a function of the system parameters , analogous to that reported in the specific minimal model of , demonstrating the generality of the underlying physics .however , we also highlight qualitative differences in the phenomenology of these different regimes .to understand the impact of different enzyme configurations on the overall pathway flux , the concept of integrated `` enzyme exposure '' has proven to be useful .it allows for the decomposition of the reaction flux of linear systems into two factors , one that depends only on the enzyme distribution and describes the diffusive dynamics of the system , and another that is independent of but captures the reaction dynamics .the concept is best explained with the help of a thought experiment where we first consider the dynamics of individual intermediate molecules in the absence of any enzymes , also shown schematically in fig .[ fig : ecartoon ] .we introduce a single intermediate molecule at at the source , and track its stochastic path until it leaves the system , either through the boundary of the system or via a reaction with a competing pathway .we denote the time at which the trajectory ends , either by escaping at the system boundary or through a competing reaction , as . by repeatedly applying this procedure for many such molecules, we can generate an ensemble of trajectories through the system that is independent of the distribution of enzymes .-dependent enzyme exposure distribution , and the -dependent reaction probability , according to .broader exposure distributions , which maximize the likelihood of large- trajectories , are preferable when is small ( middle column ) .narrower distributions , which minimize the likelihood of small values of , are favored when is large ( right ) . ]next , we suppose that we were to re - introduce enzymes according to the distribution . for each of the diffusive intermediate trajectories generated above , the instantaneous propensity of reaction with an enzyme is given by ( in the linear reaction regime ) .for each trajectory , the survival probability that no reaction has occurred up to the time follows the differential equation .we can therefore straightforwardly calculate the probability that a reaction would have occurred at some point along the trajectory as  ] , which depends on the reaction parameter but crucially not on the distribution itself . the overall probability of reactionis recovered by the expression which ensures a proper weighting of the likelihood of a particular trajectory occurring ( see the supplementary material for a derivation showing the equivalence of eq . [eq : efficiency_traj ] with the expression for defined in eq .[ conservation_equation ] ) .thus , as depicted schematically in fig .[ fig : reactioncartoon ] , we have decomposed the reaction - diffusion dynamics of eq .[ general_dimensionless ] into a diffusion- and -dependent component , and a reaction - dependent component , with the efficiency of the reaction pathway determined by the product of these two distributions .interestingly , using eq .[ conservation_equation ] the reaction efficiency can be rewritten as wherein we see that the fraction of intermediate molecules lost via the reaction or through the boundary , , takes the form of the laplace transform of , with transform variable .it is generally more straightforward to calculate as a function of for a given profile and to compute by performing an inverse laplace transformation , than it is to calculate directly from considering individual diffusive trajectories .we now consider the case where intermediates are unable to cross a cell membrane and therefore can not escape via the boundaries of the system , but can be lost to a competing , spatially - uniform , reaction pathway .that is , all intermediate molecules ultimately end up as either the desirable product or undesirable product . for a one - dimensional system in the linear , low - density , regime of the michaelis - menten reaction , we have the reaction - diffusion equation with a source boundary condition at the left edge , , and a reflecting boundary at the right edge , .the parameters and , defined above , reflect the relative reactivities of intermediate with and competing pathway enzymes respectively , in units of the typical time to diffuse a distance of the system size .consequently , and can also be interpreted as describing the system size in units of the typical distance from the source at that an intermediate molecule will diffuse before leaving the system via reaction with and via the competing pathway , respectively .when , intermediate molecules will typically be able to explore the entire system , and therefore we should expect that the spatial arrangement of enzymes will have little effect on the reaction flux , since the intermediate will be exposed to each enzyme regardless of where it is placed .in contrast , when very few intermediate molecules will diffuse far from the source and we should expect that the amount of enzyme located close to the source will have a strong influence on the pathway efficiency .we begin by examining the case where enzymes have the uniform density throughout the domain  ] . from this expression the reaction efficiencycan be calculated using the definitions in eq.[conservation_equation ] , and is given by next we suppose that all enzymes are clustered at , co - localized with .this is represented by the distribution , which leads to the solution /\cosh\sqrt{\beta} ] , accounting for the production of intermediate , and .this system is a three - dimensional analog of that discussed previously in .we once again begin by exploring the configurations in which are either placed on a shell of radius , , or uniformly distributed throughout the spherical volume , ( recall that these distributions are scaled by the average density such that ) . the reaction efficiency is then given by for the enzyme shell , and for uniformly distributed enzymes .  as the shell radius approaches zero , the efficiency of reaction of intermediate with approaches one .however , this situation is not physically realistic , as enzymes have a finite size and there will be a maximum packing density which limits the potential shell radii .if is taken to be small but finite , for small the clustering of enzymes into a tightly - packed shell configuration still achieves a higher reaction flux ( see fig .[ fig:3d_current]a ) than the uniformly - distributed configuration . however above a critical ( -dependent ) value , the uniform enzyme arrangement is able to achieve a higher reaction flux .this transition is analogous to that seen in the equivalent one - dimensional system .however , as can be seen in fig .[ fig:3d_current]a the region in which the uniform enzyme distribution is favored is shifted to much higher values , from in the one - dimensional system up to for the three - dimensional system with .since the transition occurs at extremely large values , for which almost all intermediate particles react , the difference between the efficiencies of the two profiles is extremely small .furthermore , in the low- domain the advantage provided by clustering of enzymes is much more significant than in the one - dimension system , with an increase in the reaction flux by more than a factor of four .thus clustering of enzymes is more strongly favored in three - dimensional than in one - dimensional systems .  ) and distributed ( thin blue line ) profiles .the clustered distribution achieves a significantly higher efficiency for small .a transition is observed at , above which the distributed profile reaches a marginally higher efficiency .( b ) enzyme exposure distributions for the clustered ( thick red , ) and the distributed ( thin blue ) profiles .the enzyme exposure distribution for the distributed profile is sharply peaked at , whereas the distribution for the clustered profile extends to large values of . ]  [ fig:3d_enzymeexposure ]  calculating the corresponding enzyme exposure distributions ,    we find that is sharply peaked around a small but finite value of ( see fig .[ fig:3d_enzymeexposure]b ; the mean and variance of are and , respectively ) . for the clustered profile, takes its customary exponential form .thus we can see that for small , almost all of the weight of lies in the region where is small ; the exponential tail of , though , results in a larger mean value of and more trajectories in regions where is significant .thus for , the clustered configuration is more efficient . on the other hand when , essentially all the probability weight of lies in the region , where ; however , is actually largest in the region where is small .these trajectories , which correspond to molecules that rapidly diffuse away from the cluster and do not return , generally will not lead to reactions and thus reduce the relative efficiency of the clustered configuration .how can the greater impact of clustering in a spherical geometry be understood intuitively ? on a typical path to the boundary ,a single intermediate particle originated form the center explores only a fraction of the whole sphere .if the same number of enzymes are distributed on a spherical shell further from the center , the effective `` reaction cross - section '' is smaller because the fraction of this shell that an intermediate will typically explore decreases , and with it the fraction of the enzymes in the system to which the intermediate molecule will be exposed .this is in contrast to the one - dimensional case , where the particle passes all enzymes before getting absorbed by the boundary . to derive a benefit from distributing enzymes ,a larger value is therefore required in three dimensions to compensate for this reduction in the effective level of enzymes to which intermediate molecules are exposed .next we investigated the optimal enzyme distribution as a function of the control parameter .as noted above , if the clustering of all enzymes at is permitted then independent of , yielding the maximal possible flux .however , we note again that this configuration with an infinite packing density is not physically realizable .instead we impose a limit to the possible packing density through a minimal radius within which enzymes can not be placed .we adapt the optimization procedure described above by solving eq .[ 3d_dimensionless ] on a radial lattice where each lattice site represents a concentric shell of the system .this fixes a minimal value of at the mid - point of the innermost shell ; larger values of can be prescribed , but not smaller shells .are qualitatively similar in two and three dimensions .( a ) for the three - dimensional system described by eq .[ 3d_dimensionless ] , an extended region of distributed enzymes emerges for .( b ) in two dimensions , the critical value for the emergence of a distributed enzyme fraction is . ][ fig:2d ]  as shown in fig .[ fig:3d_optimal]a , when a finite minimal clustering radius is imposed we again find that a purely clustered configuration is optimal for small while for larger than a critical value an extended region of distributed enzymes emerges , in the same way as in one dimension .the critical value at which this transition occurs is strongly dependent on the minimal allowed radius , .interestingly , for and the transition point is significantly lower than in the one - dimensional system , despite the above arguments that distributing enzymes is less efficient in three dimensions than in one ; only at does the transition point reach the observed in one dimension .the increased penalty , via the reduction in reaction cross - section , of moving enzymes to larger values of is instead reflected in the shape of the extended enzyme `` tail '' : unlike in one dimension , the enzyme density in the extended region of the profile is not constant but decreases roughly as .we have seen that one- and three - dimensional systems have qualitatively similar phenomenology in terms of whether a clustered or uniform profile is preferable and in terms of the optimal enzyme profile , although the quantitative aspects of these transitions vary .the same also holds true for the two - dimensional case : we find that this system also displays a transition from a clustered to uniform configuration being preferable at an value between those at which the transition occurs in one- and three - dimensional systems ( data not shown ) .figure [ fig:2d]b shows that once again an extended optimal profile appears at ( for ; the effect of varying is much weaker in two dimensions than in three ) . in the extended region , the optimal density is found to decrease as .we can therefore see that the underlying physics of these transitions is generic , and is not dependent on the statistics of diffusion in particular dimensions .we now turn to the case of fully non - linear reactions .we begin by considering the case , in which there is no competition with secondary pathways for the intermediate .we furthermore restrict ourselves to a one - dimensional system on the domain  ] , and thereby calculate an effective enzyme exposure for a trajectory as {\rm d}t$ ] taking to be the solution to eq .[ nonlin_dimensionless ] .however , it must be noted that this does not lead to a true decomposition of the reaction flux into diffusion- and reaction - dependent terms because itself depends on the reaction parameter .  .( b ) clustered fraction of enzymes in the optimal profile , as a function of the catalytic activity of .a non - monotonic dependence is seen when . ]  finally , we briefly consider the results of the full model described in section [ sec : model ] , including both a non - linear reaction , , and a competing pathway , , in a radially - symmetric three - dimensional geometry .figure [ fig : full]a shows that the optimal enzyme distributions are qualitatively similar to those in fig .[ fig:3d_optimal]a for a system without competition and with only linear reactions .however , examining the fraction of enzymes that are clustered at ( fig .[ fig : full]b ) , we see that is always larger than in the limiting case .this is consistent with our results above for one - dimensional systems with only competition or only non - linearity in the reaction with enzymes , where we found that increasing the strength of either of these effects will increase the tendency for clustering of enzymes .the non - monotonic dependence of on that was previously found ( fig .[ fig : leakage_reflecting]c ) when only competition is present in the model is also preserved in the case . in a similar way to fig .[ fig : nonlinear ] , fig .[ fig : full]b also suggests that the principle effect of varying is to alter the threshold value of beyond which the purely clustered profile becomes sub - optimal . in summary , these results indicate that the qualitative effects of each of the modifications that we have previously considered individually are representative of the impact of the same elements in the full model .in this work we have demonstrated that the existence of a transition between clustered and distributed optimal arrangements of enzymes is a generic feature of diverse reaction - diffusion systems . while the exact shape of the optimal enzyme profile and the parameter - dependence of this transition varies with the specific system , the fact that a non - trivial optimal profile exists is a general result of the interplay of reaction and diffusion in such systems .the transition ultimately emerges from the stochastic dynamics of individual intermediate molecules , as demonstrated by its dependence on the interplay between the distributions of enzyme exposure and reaction probability . by examining these distributions ,we are led to an intuitive explanation for the transition .when reactions are slow , clustering of enzymes is beneficial because this provides the highest enzyme density in the region in which intermediate molecules are most likely to spend a significant amount of time .when reactions become fast , a limited density of enzymes will already ensure the rapid reaction of these molecules ; in this scenario , it becomes preferable to distribute a fraction of enzymes more widely so as to provide an opportunity to react with those intermediate molecules that rapidly escape from the enzyme cluster .  for specific systems ,our analysis has revealed some further notable features . in systems with competing pathways, the optimal enzyme profile tends to concentrate near the source again as the reaction rate is increased further , which results from the fact that intermediates can be lost from the region near the source as opposed to only at the boundary of the system .we have also seen that the benefit of clustering increases with the effective dimension of the system , as the increased space available to diffusive trajectories means that intermediates typically only have access to a small fraction of distributed enzymes .finally , our results for non - linear reactions suggest that clustering again enhances pathway efficiency if the availability of intermediates ( determined by the activity of the first enzyme in the pathway ) is increased .this observation suggests that it may be desirable to dynamically regulate the localization of enzymes , and specifically the formation of multi - enzyme complexes , in response to the availability of substrate or the flux of upstream reactions .a potential example of such regulation is provided by mammalian hexokinase isoform hkii , which is thought to undergo reversible translocation between the outer membrane of mitochondria and a more diffuse cytoplasmic distribution depending on factors including glucose-6-p and gsk3 , thereby altering the relative flux of glucose through different metabolic pathways .  while there are several well - documented examples of enzyme clustering , including the pyruvate dehydrogenase and cellulosome complexes mentioned above as well as glycolytic enzymes in various cell types , enzyme clustering is not thought to be the default strategy in molecular biology .for example ,while the cellulosome is a conglomeration of enzymes tethered to the outside of bacteria , other enzyme classes such as proteases do not typically form tethered complexes but rather are simply secreted into the extracellular environment .we are not aware of specific enzyme systems that display a combination of a cluster with a more diffuse arrangement near the cluster .observation of such localization patterns will be difficult , due to the relatively low density and dynamic nature of the distributed region in close proximity to the high - density cluster .such enzyme distributions could be generated with the help of pre - existing cellular structures such as the cytoskeleton or membrane sub - domains .simpler arrangements consisting of a localized and a uniform fraction would naturally arise from weak , transient interactions between the enzymes .our principal conclusions could be tested experimentally using , for example , the `` single - molecule cut - and - paste '' technique or dna origami to construct specific arrangements of enzymes .such constructs could also be incorporated into microfluidic chambers featuring localized sources and sinks of substrate .the reaction efficiency could be measured by observing the relative quantities of reacted and un - reacted substrate in the efflux channel , or using fluorogenic substrates .such experiments would allow for quantification of the relative reaction efficiencies of different enzyme arrangements as the parameter is varied by altering , for example , the number of enzymes in the system or the substrate diffusion constant .while the continuous reaction - diffusion models that we have considered here present a useful mesoscopic description of the enzymatic systems , they make a number of approximations that will limit their validity at extremely short length scales .foremost amongst these is that neither enzymes nor intermediate molecules occupy any finite volume .in reality there will be an upper limit to the number of enzymes that can be clustered within a certain region .furthermore , steric hindrance by enzymes will affect the trajectories of intermediate molecules .thus the tight packing of enzymes may strengthen the effect of clustering by physically blocking the escape of intermediates .however , by the same measure , a tight clustering of enzymes may prevent the access of initial substrates into the cluster . at such short length - scales ,it is also not clear to what extent the motion of intermediates can be represented as normal diffusion . additionally , enzymes are not reactive over their entire surfaces but only at specific catalytic binding sites . while rotational orientation can generally be neglected for freely - diffusing enzymes , these effects may become significant if enzymes are attached to rigid scaffolds. a more complete understanding of these issues , together with more complex reaction schemes including cooperativity and allosteric regulation , will be crucial for a complete understanding of the design principles underlying enzyme arrangements in living cells as well as the effective engineering of synthetic biochemical systems .this research was supported by the german excellence initiative via the program ` nanosystems initiative munich ' and the german research foundation via the sfb 1032 ` nanoagents for spatiotemporal control of molecular and cellular reactions ' and via the focus area spp 1617 .ft is supported by a research fellowship from the alexander von humboldt foundation .here we briefly consider a system described by eq . [ reaction_diffuison_leakage ] , but with an absorbing boundary condition rather than the reflecting boundary considered in section [ leakage ] .comparing the known results for the limit  , and for the reflecting boundary condition discussed above , we can predict that there should be a qualitative difference in whether clustered or uniform profiles are preferred as is varied . for small , loss to the competing pathway will be negligible compared to loss through the boundary at .we would therefore expect that as is increased , the system should undergo a transition from a regime in which the clustered configuration is preferable to a regime in which the uniform profile provides a higher efficiency . on the other hand , for large the length scale associated with the loss to the secondary pathwayis short compared to the system size . in this case, the choice of boundary condition of should have little influence on the dynamics , which should resemble that described in section [ leakage ] where the clustered configuration is always preferable .enzymes reveals a qualitatively different behavior in different ranges . for ( a ) a cross - overis observed at which the profile with the higher efficiency changes . for ( b ) clustering is preferable for all values of .( c ) the transition value of dividing the regimes where the clustered or uniform profiles achieve a higher reaction flux , found by solving numerically . above the critical ( dashed line ) the transition disappears and the clustering of enzymes is always more efficient . ]  the reaction fluxes obtained by solving this system for uniform and clusteredenzyme configurations are figure [ fig : leakage_absorbing]a and b confirms that there is a difference in whether the clustered or uniform profile is more efficient in different regimes of , in keeping with our expectations .figure [ fig : leakage_absorbing]c plots the critical value of as a function of , and demonstrates that the transition disappears at a finite value of .when the clustered configuration always achieves a higher reaction flux .34 natexlab#1#1bibnamefont # 1#1bibfnamefont # 1#1citenamefont # 1#1url # 1`#1`urlprefix[2]#2 [ 2][]#2  , * * , ( ) ., , , , * * , ( ) .  , , , ,* * , ( ) .  , , , * * , ( ) ., , , , , * * , ( ) .  ,* * , ( ) .  , * * , ( ) .  , , , * * , ( ) .  , * * , ( ) .  , , , * * , ( ) .  , * * , ( ) .  , , , * * , ( ) .  , , ,, * * , ( ) .  ,* * , ( ) ., , , , * * , ( ) .  , , , * * , ( ) ., , , , , , , , * * , ( ) .  , , , , , * * , ( ) .  , , , , , , * * , ( ) .  , , , * * , ( ) .  , , , , , , * * , ( ) .  , * * , ( ) .  , , , , * * , ( ) .  , , , , , * * , ( ) .  , , , * * , ( ).  , * * , ( ) .  , , , * * , ( ) .  , , , , , , * * , ( ) .  , , , * * , ( ) .  , * * , ( ) .  , * * , ( ) .  , * * , ( ) .  ,* * , ( ) .the reaction - diffusion equation leads us to define the reaction efficiency according to the eq .[ conservation_equation ] of the main text , here we show that the alternative expression ( eq .[ eq : efficiency_traj ] of the main text ) which arises from the examination of individual intermediate trajectories , can be derived from eq .[ eq : efficiency1 ] in the linear regime where .we begin by reformulating the steady - state density in terms of trajectories of diffusing molecules of intermediate .we denote the diffusive trajectory of a single intermediate molecule , in the absence of any enzymes , as .such a trajectory has associated with it a time after which the trajectory is terminated , either by escape across the system boundary or loss to a secondary pathway .the reintroduction of enzymes according to the distribution leads to an instantaneous propensity for conversion to the correct product at each point along the trajectory of .thus the survival probability that an intermediate molecule on the trajectory has not undergone a reaction with before the time follows .this equation can be integrated to yield , \ \t\leq t_{\rm escape}.\ ] ]  at steady state , each trajectory of an intermediate molecule will make a contribution to the total intermediate density at point that depends on the total time that the trajectory spends at , weighted by the probability that the intermediate molecule has not yet undergone a reaction prior to each return to .this later weighting factor is simply the survival probability .therefore , the local enzyme density can be rewritten as ,\ ] ] where the inner integral is the weighted time spent by a single trajectory at , and the outer integral sums over the contributions of all possible trajectories weighted by the probability of a specific trajectory occurring .substituting eqs .[ eq : redensity ] and [ eq : survival ] into eq .[ eq : efficiency1 ] and changing the order of integration , we find  \nonumber \\   & = \int{\rm d}{\lbrace\mathbf{r}(t)\rbrace}\ p({\lbrace\mathbf{r}(t)\rbrace})\int_0^{t_{\rm escape}}{\rm d}t\      \alpha e(\mathbf{r}(t ) ) e^{-\alpha\int_0^t{\rm d}t'\     e(\mathbf{r}(t ' ) ) } \nonumber \\   & = \int{\rm d}{\lbrace\mathbf{r}(t)\rbrace}\ p({\lbrace\mathbf{r}(t)\rbrace})\int_0^{t_{\rm escape}}{\rm d}t\    \frac{\rm d}{{\rm d}t}\left[-e^{-\alpha\int_0^t{\rm d}t'\      e(\mathbf{r}(t'))}\right ] \nonumber \\   & = \int{\rm d}{\lbrace\mathbf{r}(t)\rbrace}\ p({\lbrace\mathbf{r}(t)\rbrace})\left[1-e^{-\alpha\int_0^{t_{\rm escape}}{\rm d}t'\      e(\mathbf{r}(t'))}\right].\end{aligned}\ ] ] finally , defining we can change the variable of integration from to , recovering consider the rescaled reaction diffusion equation as stated in the main text with the source boundary conditions .below we consider the cases of reflecting ( ) and absorbing ( ) boundaries at .the enzyme profile is taken to be clustered at some point , . in the endwe take the limit goes to zero , leading to a clustering of at the origin .we divide the system into two parts , part where and part where . in each parteq . [ appendix_1d_competing_pathway ] reduces to which has the solution with .applying the boundary conditions at and yields in order to determine all constants we impose two additional conditions , firstly the matching condition of the concentration of intermediates at , leading to the second condition , which captures particle conservation in the system , is found by integrating eq .[ appendix_1d_competing_pathway ] from to and taking the limit of small , the last term on the left hand side vanishes in the limit .this leads to the expression -\frac{\alpha}{\sqrt{\beta}}\bigg(a_{i}e^{\sqrt{\beta}x_0}+b_{i}e^{-\sqrt{\beta}x_0 } \bigg)=0.\ ] ] calculate the reaction current yields after some straightforward algebra we arrive at expressions for all four constants .last we plug them into the above equation and take the limit ,    the reaction diffusion equation with a uniform enzyme profile reads the solution is given by applying the boundary conditions at and leads to the conditions similarly to above , the constants and can be obtained straightforwardly , and the efficiency of the pathway is    the approach is very similar to the one the preceding section . the only difference , however , is that the boundary condition leading to a slightly different second condition for the clustered enzymes likewise , we obtain as the second boundary condition for the uniformly distributed enzymes .similarly , to the approach in section [ clu_enzyeme_section ] and [ uni_enzyme_section ] , respectively the corresponding efficiency for the clustered case is given by and for the uniform case shown in the main text , the efficiency of the pathway in terms of the enzyme exposure probability distribution reads to obtain an exact expression of for the respective case it is convenient to calculate the inverse laplace transformation of with respect to .we have seen above that the expression for the efficiency often has the form ; hence the loss term is .the inverse laplace transformation is easily obtained and has the form for the case of a reflecting outer boundary we therefore have for the uniform enzyme profile , and thus ; for the clustered enzyme profile , and .  in the case of an absorbing boundary , the clustered distribution also gives rise to a similar expression for the efficiency , with and thus .however , with a uniform distribution the loss flux does not have the form discussed above , and thus the calculation of the inverse laplace transformation is more involved .we rewrite eq .[ appendix_1d_abs_uni ] and use eq .[ appendix_1d_enzyme_exposure ] to obtain where the second term has the form as we have already discussed above , thus +\beta e^{-\beta e}. \label{partial_inverse}\ ] ] the inverse laplace transformation of the first term is calculated by determining the singularities in terms of and than calculate their residues by using the laurent expansion . finding the poles is here equivalent to determining the roots of the denominator , this is satisfied for for . andthe residues are given by the first term above cancels with the last term in eq .[ partial_inverse ] . combining the remaining terms ,we are left with the overall enzyme exposure distribution three dimensions we impose rotational symmetry and reduce the reaction diffusion equation to depend only on the radial coordinate , we apply the following boundary conditions , and the outer sphere is absorbing .we again begin by considering a clustered distribution of enzymes , , that has been normalized such that . as in section [ clu_enzyeme_section ]we divide the system into two parts , part for and part for . in each part , due to the absence of enzymes , the solution of eq .[ appendix_3d_cluster ] is where .applying the boundary conditions leads to the following two conditions the remaining two conditions come again from the matching of the concentration at , , and the discontinuity of the derivative of the concentration at .hence we get and =0 .\label{3d_deriv}\ ] ] with these four conditions eqs .[ 3d_bcs]-[3d_deriv ] we obtain after some straightforward algebra the expressions for the four constant .since we do not consider a competing pathway in this model the efficiency of the pathway can also be calculated as    for the uniform enzyme profile the reaction - diffusion equation eq .[ appendix_3d_cluster ] reads this is solved by with the boundary conditions we arrive at the following two conditions : at the origin , and at the absorbing outer boundary , in the same way as the case of clustered enzymes the efficiency is given by to the one dimensional case the enzyme exposure probability distribution is obtained by the inverse laplace transformation of the loss current through the boundary .the loss current of the clustered enzyme profile reads which has the general form discussed in section [ enzyme_exp_1d_section ] .hence the enzyme exposure distribution is for the uniform enzyme profile , the calculation again proceeds in the same way as described above .we firstly calculate the singularities of the loss current , the roots of , which are given by this leads then to the following residues we assemble all the individual terms and get for the enzyme exposure probability distribution

Abstract: the spatial organization of enzymes often plays a crucial role in the functionality and efficiency of enzymatic pathways . to fully understand the design and operation of enzymatic pathways , it is therefore crucial to understand how the relative arrangement of enzymes affects pathway function . here we investigate the effect of enzyme localization on the flux of a minimal two - enzyme pathway within a reaction - diffusion model . we consider different reaction kinetics , spatial dimensions , and loss mechanisms for intermediate substrate molecules . our systematic analysis of the different regimes of this model reveals both universal features and distinct characteristics in the phenomenology of these different systems . in particular , the distribution of the second pathway enzyme that maximizes the reaction flux undergoes a generic transition from co - localization with the first enzyme when the catalytic efficiency of the second enzyme is low , to an extended profile when the catalytic efficiency is high . however , the critical transition point and the shape of the extended optimal profile is significantly affected by specific features of the model . we explain the behavior of these different systems in terms of the underlying stochastic reaction and diffusion processes of single substrate molecules .


wireless networks are expected to support a variety of applications with diverse quality - of - service ( qos ) requirements . because of the scarcity of network resources ( i.e. , energy and bandwidth ) , radio resource management is crucial to the performance of wireless networks .the goal is to use the network resources as efficiently as possible while providing the required qos to the users .adaptive modulation has been shown to be an effective method for improving the spectral efficiency in wireless networks ( see for example ) . however , the focus of many of the studies to date has been on maximizing the throughput of the network , and the impact of the modulation order on energy efficiency has not been studied to the same extent .recently , the authors of have studied modulation optimization for an energy - constrained time - division - multiple - access ( tdma ) network .for such a network , they have used a convex - optimization approach to obtain the best modulation strategy that minimizes the total energy consumption under throughput and delay constraints .game - theoretic approaches to power control have recently attracted considerable attention ( see , for example , ) . in ,the authors provide motivations for using game theory to study communication systems , and in particular power control . in , power controlis modeled as a non - cooperative game in which users choose their transmit powers in order to maximize their utilities , where utility is defined as the ratio of throughput to transmit power . a game - theoretic approach to joint power control and receiver designis presented in , and power control for multicarrier systems is studied in .the authors in use pricing to obtain a more efficient solution for the power control game .similar approaches are taken in for different utility functions .game - theoretic approaches to power control in delay - constrained networks are proposed in .  in this work, we use a game - theoretic approach to study the effects of modulation on energy efficiency of code - division - multiple - access ( cdma ) networks in a _ competitive _ multiuser setting . focusing on m - qam modulation, we propose a _non - cooperative _ game in which each user chooses its strategy , which includes the choice of the transmit power , transmit symbol rate and constellation size , in order to maximize its own utility while satisfying its qos constraints .the utility function used here measures the number of reliable bits transmitted per joule of energy consumed , and is particularly suitable for energy - constrained networks .we derive the best - response strategies and nash equilibrium solution for the proposed game .in addition , using our non - cooperative game - theoretic framework , we quantify the tradeoffs among energy efficiency , delay , throughput and modulation order .the effect of coding on energy efficiency is also studied and quantified using the proposed game - theoretic approach .in addition , our framework allows us to illustrate the tradeoff between energy efficiency and spectral efficiency .the remainder of this paper is organized as follows .the system model and definition of the utility function are given in section [ systemmodel ] .we then present a power control game with no delay constraints in section [ pcg ] and derive the corresponding nash equilibrium solution .a delay - constrained power control game is proposed in section [ pcgd ] and the corresponding best - response strategies and nash equilibrium solution are derived .the analysis is extended to coded systems in section [ pcgtcm ] .numerical results and conclusions are given in sections [ numericalresults ] and [ conclusion ] , respectively .we consider a direct - sequence cdma ( ds - cdma ) wireless network in which the users terminals are transmitting to a common concentration point ( e.g. , a cellular base station or an access point ) .the system bandwidth is assumed to be hz .let and be the symbol rate and the transmit power for user , respectively . in this work , we focus on m - qam modulation .hence , each symbol is assumed to be complex to represent the in - phase and quadrature components . for the m - qam modulation ,the number of bits transmitted per symbol is given by since there is a one - to - one mapping between and , we sometimes refer to as the constellation size .we focus on _ square _ m - qam modulation , i.e. , or equivalently , since there are exact expressions for the symbol error probability of square m - qam modulation ( see ) .we can easily generalize our analysis to include odd values of by using an approximate expression for the symbol error probability .we define the utility function of a user as the ratio of its throughput to its transmit power , i.e. , this utility function is similar to the one used in and .throughput in is defined as the net number of information bits that are transmitted without error per unit time ( it is sometimes referred to as _ goodput _ ) , and is expressed as where is the transmission rate , is the output signal - to - interference - plus - noise ratio ( sir ) for user , and is the efficiency function " which represents the packet success rate ( psr ) for the user .we require that to ensure that when . in general, the efficiency function depends on the modulation , coding and packet size .we assume an automatic - repeat - request ( arq ) mechanism in which the user keeps retransmitting a packet until the packet is received at the access point without any errors .based on ( [ eq2 ] ) and ( [ eq3 ] ) , the utility function for user can be written as this utility function , which has units of _ bits / joule _ , measures the number of reliable bits that are transmitted per joule of energy consumed , and is particularly suitable for energy - constrained networks .let us for now focus on a specific user and drop the subscript .assuming a packet size of bits , the packet success rate for square m - qam modulation is given by where and here , represents the symbol sir and is the complementary cumulative distribution function of the standard gaussian random variable . note that at , we have . since we require the efficiency function to be zero at zero transmit power , we define note that when is large ( e.g. , ) .  a non - cooperative power control game , in general , can be expressed as  ] with being the maximum allowed transmit power . throughout this work ,we assume is large .  for all linear receivers, the output sir for user can be written as where is the system bandwidth and is the effective channel gain which depends on the channel gain of user and on the channel gains and transmit powers of other users in the network but is independent of the transmit power and rate of user .for example , for a matched - filter receiver , is given by where is the channel gain for user , and is the noise power .therefore , the utility function in can be written as based on , and by dropping the subscript for convenience , the maximization in can be written as since for a given user , and are fixed , maximizing the user s utility is equivalent to maximizing for that user .it is important to observe that , for a given , specifying the operating sir completely specifies the utility function .let us for now fix the symbol rate and the constellation size .taking the derivative of with respect to and equating it to zero , we conclude that the utility of a user is maximized when its output sir , , is equal to , which is the ( positive ) solution of it is shown in that for an s - shaped ( sigmoidal ) efficiency function , has a unique solution .it can easily be verified that given by is s - shaped .note that is ( uniquely ) determined by physical - layer parameters such as packet size , modulation , and coding .  assuming that is feasible , the maximum utility is hence given by based on , it can be shown that is ( approximately ) given by the solution of we can compute numerically for different values of .table [ table1 ] summarizes the results for a system with bits ( i.e. , 100 bits per packet ) ..summary of the parameters corresponding to the utility - maximizing solutions for different modulation orders for an uncoded system with 100 bits per packet [ cols="^,^,^,^,^,^,^,^,^",options="header " , ]in this section , we quantify the effect of constellation size on energy efficiency of a user with a delay qos constraint .the packet size is assumed to be 100 bits , and the source rate ( in bps ) for the user is assumed to be equal to where is the system bandwidth .we further assume that a user chooses its constellation size , symbol rate , and transmit power according to its best - response strategy corresponding to the pareto - dominant nash equilibrium ( see section [ ne ] ) .for the coded system , we assume an 8-state convolutional encoder with rate 2/3 . the code rate for qpsk is chosen to be 1/2 .[ fig5 ] shows the optimum constellation size , transmit power , throughput , and user s utility as a function of the delay constraint for both uncoded and coded systems .the results for the coded case are obtained by using as an approximation for the coding gain . for all four plots ,the packet delay is normalized by the inverse of the system bandwidth . to keep the spectral efficiency of the two systems the same , we assume that the number of information bits transmitted per symbol is the same for both uncoded and coded systems .the throughput corresponds to the transmission rate for the user which is obtained by multiplying the symbol rate by the number of ( information ) bits per symbol ( i.e. , ) , and is normalized by the system bandwidth .the transmit power and user s utility are also normalized by and , respectively .let us for now focus on the uncoded system .when the delay constraint is large , qpsk ( which is the most energy efficient m - qam modulation ) can accommodate the delay requirement and hence is chosen by the user . as the delay constraint becomes tighter , the user increases its symbol rate and also raises the transmit power to keep the output sir at 9.1 db ( recall that  db when ) . in this case, the user s utility stays constant .once the symbol rate becomes equal to the system bandwidth , the user can not increase it anymore .hence , as the delay constraint becomes more stringent , the user is forced to aim for a higher target sir to meet its delay requirement . in this case, the transmission rate stays constant and the transmit power increases .hence , the user s utility decreases .as the delay requirement becomes tighter , a point is reached where the spectral efficiency of qpsk is not enough to accommodate the delay constraint . in this case, the user jumps to a higher - order modulation ( i.e. , 16-qam ) and the process repeats itself .the trends are similar for the coded system except that , due to coding gain , the required transmit power is smaller for the coded system .this results in an increase in the user s utility .this means that , for the same number of information bits transmitted per symbol , the energy efficiency is higher when tcm is used .in addition , in fig .[ fig6 ] , we have plotted the utility gain achieved due to tcm as a function of normalized packet delay .it is seen that the gain in energy efficiency due to tcm depends on the delay constraint and fluctuates between 1.5 db and 3 db . in general , for the same delay constraint , the uncoded system has to transmit at a slightly higher symbol rate as compared to the coded system .this is because ( see table [ table coded uncoded ] ) . as a result , .the spikes in fig .[ fig6 ] correspond to the cases in which the uncoded system is transmitting at the maximum possible symbol rate and has to increase its target sir to meet the delay constraint .this results in a drop in the utility of the uncoded system .the coded system may still be able to meet the delay constraint without increasing the target sir .[ fig7 ] shows the user s size " , , corresponding to the pareto - dominant nash equilibrium as a function of the packet delay for both uncoded and coded systems . as explained in section [ ne ] , increases as the delay constraint becomes tighter .this makes sense because a user would need to consume more network resources to satisfy a more stringent delay .it is also seen that coding reduces the user s size " and , hence , increases the network capacity .this is because for the same constellation size , the symbol rate and the sir are smaller for the coded system .we have seen throughout this paper that the strategy that maximizes the user s energy efficiency is not spectrally efficient . to illustrate the energy efficiency - spectral efficiency tradeoff ,let us fix the symbol rate to be .for a fixed constellation size , the user s utility ( energy efficiency ) is proportional to .the spectral efficiency is given by . by varying the constellation size, we can quantify the tradeoff between energy efficiency and spectral efficiency .this tradeoff is shown in fig .[ fig8 ] for both uncoded and coded systems . in this figure , we have plotted vs. .different points on the plot correspond to different values of .the energy efficiency - spectral efficiency tradeoff is definitely an interesting and important topic that requires more in - depth analysis .we have studied the effects of modulation order on energy efficiency of wireless networks using a game - theoretic framework . focusing on m - qam modulation ,we have proposed a non - cooperative game in which each user chooses its strategy in order to maximize its utility while satisfying its delay qos constraint .the actions open to the users are the choice of the transmit power , transmit symbol rate and constellation size .the utility function measures the number of reliable bits transmitted per joule of energy consumed and is particularly suitable for energy - constrained networks .the best - response strategies and the nash equilibrium solution for the proposed game have been derived .we have shown that to maximize its utility ( i.e. , energy efficiency ) , the user must choose the lowest modulation level that can accommodate the user s delay constraint . using our non - cooperative game - theoretic framework ,the tradeoffs among energy efficiency , delay , throughput and constellation size have also been studied and quantified . in addition, we have included the effects of tcm and have shown that , as expected , coding increases energy efficiency .the tradeoff between energy efficiency and spectral efficiency has also been illustrated .a. b. mackenzie and s. b. wicker , `` game theory in communications : motivation , explanation , and application to power control , '' _ proceedings to the ieee global telecommunications conference ( globecom ) _ , pp . 821826 , san antonio , tx , usa , 2001 .f. meshkati , h. v. poor , s. c. schwartz , and n. b. mandayam , `` an energy - efficient approach to power control and receiver design in wireless data networks , '' _ ieee transactions on communications _ ,52 , pp . 18851894 ,november 2005 .f. meshkati , m. chiang , h. v. poor , and s. c. schwartz , `` a game - theoretic approach to energy - efficient power control in multicarrier cdma systems , '' _ ieee journal on selected areas in communications ( jsac ) _ , vol .24 , pp . 11151129 , june 2006 .f. meshkati , h. v. poor , and s. c. schwartz , `` a non - cooperative power control game in delay - constrained multiple - access networks , '' _ proceedings of the ieee international symposium on information theory ( isit ) _ , adelaide , australia , september 2005 .f. meshkati , h. v. poor , s. c. schwartz , and r. balan , `` energy - efficient resource allocation in wireless networks with quality - of - service constraints , '' to appear in the _ ieee transactions on communications_.    v. rodriguez , `` an analytical foundation for resource management in wireless communication , '' _ proceedings of the ieee global telecommunications conference ( globecom ) _ , pp . 898902 , san francisco , ca , usa , december 2003 .t. holliday , a. j. goldsmith , and p. glynn , `` wireless link adaptation policies : qos for deadline constrained traffic with imperfect channel estimates , '' _ proceedings of the ieee international conference on communications ( icc ) _ , pp . 33663371 , new york , ny , usa , april / may 2002 .

Abstract: a game - theoretic framework is used to study the effect of constellation size on the energy efficiency of wireless networks for m - qam modulation . a non - cooperative game is proposed in which each user seeks to choose its transmit power ( and possibly transmit symbol rate ) as well as the constellation size in order to maximize its own utility while satisfying its delay quality - of - service ( qos ) constraint . the utility function used here measures the number of reliable bits transmitted per joule of energy consumed , and is particularly suitable for energy - constrained networks . the best - response strategies and nash equilibrium solution for the proposed game are derived . it is shown that in order to maximize its utility ( in bits per joule ) , a user must choose the lowest constellation size that can accommodate the user s delay constraint . this strategy is different from one that would maximize spectral efficiency . using this framework , the tradeoffs among energy efficiency , delay , throughput and constellation size are also studied and quantified . in addition , the effect of trellis - coded modulation on energy efficiency is discussed .  energy efficiency , m - qam modulation , trellis - coded modulation , game theory , utility function , nash equilibrium , delay , quality - of - service ( qos ) , cross - layer design .


neutron interferometry is one of the most precise techniques used to test the postulates of quantum mechanics , and it is also one of the most important and precise techniques used to measure low - energy neutron cross sections .although the fundamentals of neutron interferometry are easily recognizable from common optics , the slow velocity of neutrons ( 1680 for the 2.35 neutrons used in this study ) and low count rates at the detector ( ) demand novel solutions .the most important of these was the development of multiblade , single - crystal interferometery which enables high contrast to be observed with only limited beam alignment .the challenge remains however of making the experiment robust against mechanical vibrations .a typical single - crystal interferometer has path lengths of 10 with a typical 50 travel time of neutrons in the system .small - amplitude low - frequency vibrations may significantly degrade the contrast . herewe propose a solution to robust interferometry based on a four - blade single - crystal geometry that reduces errors introduced by vibrations .the most common three - blade geometry for a perfect crystal neutron interferometer ( ni )  is show in fig .[ fig:4bl_sch ] * a * ) along with the four - blade configuration ( fig .[ fig:4bl_sch ] * b * ) that is less sensitive to low - frequency vibrations .   * a * : a schematic diagram of the three - blade neutron interferometer .a neutron beam ( with neutron velocity ) comes from the left , is split by the first blade , is diffracted on the second blade , and recombines at the third . after passing through the interferometer, the beam is captured by the o- and h - detectors .we model vibrations as oscillations of and around the center of mass of the interferometer , as , where could be ( transverse vibrations ) , ( longitudinal vibrtions ) , and ( rotational vibrations ) . in order to compare oscillations between the three- and four - blades devices the distance between the bladesis set equal to . * b * : a schematic diagram of the proposed interferometer with four blades .instead of one diffracting blade here we have two , which reverses the neutron paths in order to compensate for vibrations .we will compare the performance using the same vibration modes with the same amplitudes .note that for the three - blade interferometer the o - detector has the maximum contrast and in the four - blade interferometer the h - detector has the maximum contrast.,scaledwidth=100.0% ]  in the three - blade case the neutron beam , coming from the left , is coherently split into two paths by the first blade via bragg scattering . after being reflected by the second blade, these two paths are recombined at the third blade .the resultant interference is observed at the o- and h - detectors .note that we align the coordinate parallel to the bragg planes and the coordinate is perpendicular to the bragg planes . in the four - blade casethe situation is nearly identical with the significant difference that the paths are reflected and cross each other without mixing at the center of the interferometer , i.e. without a blade there is no mixing of the states .it is sufficient to take a simple model for noise and to consider vibrations as sinusoidal oscillations around the center of mass of the single crystal , which we write as for we can specify any coordinates , , , or any angles ( such as - rotation around z axis ) . in order to motivate the discussion we adopt a simple model for the neutron - blade interaction that includes all of the necessary physics .the interaction is that of bouncing of a small particle ( neutron ) from a moving heavy wall ( the blade ) where the particle is reflected . when the particle is transmitted there is no interaction ( figure [ fig : inter_sch ] ) .we use conservation of momentum and energy to calculate the neutron s change in velocity after bouncing .we require a small enough amplitude and low enough oscillation frequency that the modified momentum of the neutron still satisfies the bragg condition .vibrations thus modify the neutron velocities and change the path length of the neutron inside the interferometer .   a schematic diagram of the neutron scattering from a blade of the neutron interferometer .due to the crystal movement , the reflected neutron s momentum is modified .for the transmitted case , the momentum remains unaltered.,scaledwidth=70.0% ]  using these approximations , it is clear that vibrations along every axis except the coordinate will reduce the interferometer contrast .the -component of neutron velocity is zero and the paths lengths are independent of crystal vibration along the axis at this level of approximation .we first consider vibrations along the axis .the measure of the quality of the interferometer is it s contrast , so we calculate and plot the contrast versus the frequency of oscillations .the four - blade version returns higher contrast by compensating with the third blade for momentum change introduced by vibration of the second blade .we expect this compensation to be most precise for low - frequency vibrations .assume that the neutron hits the first blade of interferometer at the time .we rewrite eq .[ eq : oscil ] for vibrations along the axis where is a random phase between the arriving neutron and the vibrating blade . velocity of the interferometer at the time is at time , the velocity of the interferometer along the -coordinate is , where is random .conservation of momentum and energy at the moment implies that the velocity of the transmitted neutron does not change , while the reflected neutron s velocities are .the phase difference for the neutron between and is where is the momentum of the neutron and is the path - length vector along which the neutron is moving .  for the neutron to travel between the first two bladestakes a time .the contrast depends on the total phase difference between the paths .notice that under these assumptions the two paths cross the third blade at the same spot and the travel time along these paths remains .so , the loss in contrast seen in the presence of vibration is not due to the finite coherence length of the interferometer but rather is due to the extra phase shifts introduced by the vibrations .  using thesewe find if we assume that is slowly varying on the scale of ( or ) we can approximate the expression as a derivative of   the intensity at the o - detector is and depends on the random phase .we average the intensity over this phase :   the contrast as usual is defined as where we vary the phase to find the max and min .[ fig : vib ] * a * shows the dependence of the contrast on the frequency of vibrations along the axis for the three - blade interferometer . the contrast was calculated for  cm , a neutron velocity of  m / s , and vibration amplitudes of m .here we observe that the contrast starts to decrease near 100 hz .* a * contrast due to vibrations along the axis . *b * contrast due to vibrations along the axis .* c * contrast due to rotational vibrations around axis and the center of mass .note that the scale of the frequency axis in * c * is different from those shown on * a * and * b * , scaledwidth=100.0% ]   as in the three - blade interferometer case , we derive expressions for the phases the neutron acquires while traveling along the two interferometer paths . here , the time for the neutron to travel between the first two blades is .note that the detector corresponding to the neutron paths with equal number of reflections for the three - blade interferometer is o - detector and for the four - blade interferometer it is h - detector .  between the first blade and the second , the phases are identical to the three - blade interferometer case except that changes to .the phase difference between and is again , as for the case of three - blade interferometer , we assume that the function is slowly varying on the scale of ( or ) and we rewrite the phase change in terms of a derivative =\nonumber \\ & & 16\frac{m_n}{\hbar}\tau^3 \left(v_y - u_y(0)\right)\frac{3}{2}\left.\frac{du_y^2(t)}{d^2t}\right|_{t=\frac{5\tau}{4 } } \label{eq : d_phi_4bl_der}.\end{aligned}\ ] ]  notice that the linear term drops out .this is the source of the protection against vibrations .the contrast comparison we make is the o - beam of the three - blades interferometer to the h - beam of the four - blade interferometer .the intensity at the h - detector is and depends on the random phase of vibration .again we average the intensity over this random phase :   we obtain the contrast using eq .[ eq : contr_def ] . in fig .[ fig : vib ] * a * we plot the frequency dependence of the contrast for the four - blade interferometer .notice that and cross the fourth blade at the same spot . from fig .[ fig : vib ] * a * we clearly see that the four - blade interferometer is predicted to be much less sensitive to vibrations .in the case of vibrations along the axis the momentum of the neutron is not modified ( see fig . [fig : inter_sch ] ) . however the path length changes depending on the phase of the oscillations at .  herethe contrast is primarily limited by the neutron coherence length of .the four - blade geometry does not protect against this and indeed the influence of a finite coherence length is slightly worse due to the noise being introduced since the intervals between blades are changed by the vibrations . in the three - blade casethere are two such intervals , while in the four - blade case there are three .so with finite the more blades there are , the worse the noise is .however , since the acceptance of the ni is small , is small and this contribution to the noise is small .   a schematic diagram of the neutron arriving to the third blade . due to the crystal movement , the paths of the neutron will not recombined at the ideal point but at away from the third blade.,scaledwidth=70.0% ]  here we calculate just the contribution to the loss in contrast from the phase shift introduced by vibrations along axis .the vibrations along the axis are in this case the phase shift is due to the paths crossing at a point displaced from ideal as shown in fig . [fig : dx_df_sch ] .once we find for each interferometer , then and the phase difference is   for the three - blade interferometer , we have and where we neglect distance ( m ) in comparison with ( cm ) . for the four - blade interferometer using these we can get , substitute obtained to find intensities , and average intensities over different to obtain the contrasts.  figure [ fig : vib ] * b * shows the contrast dependence on the frequency of vibrations along the axis .this should be viewed as an upper bound on the predicted contrast .the message is that vibrations along axis are not very important and even though the four - blade design is somewhat more sensitive to them than is the three - blade design . in the final analysisthe four - blade design is predicted to have better overall performance .these vibrations are expected to be most limiting since the neutron interferometer has such a small acceptance angle . in the case of rotational vibrations , we rewrite the oscillation in terms of the angle around the axis for small angles , rotational vibrations can be considered as translational vibrations , i.e. , where is the distance from the blade to the center of rotation .   in the three - blade interferometer ,the center of rotation is also the center of mass and the center of the middle blade .for the point [ see fig .[ fig:4bl_sch ] * a * ) ] where the neutron path crosses the blades , the rotational vibrations can be modeled as vibration along the axis for the first and last blade and along the axis for the path crossing the middle blade . in this case , the interaction with the middle blade does not change the velocity of the neutron . at the first blade we have a change in the momentum of the reflected beam and no change for the transmitted neutrons  where velocity of the first blade in the direction .the phase difference between the two paths is \tau = \nonumber \\ 8\frac{m_n}{\hbar } l\theta_0\omega\sin{\varphi } \left[2l\theta_0\omega\sin{\varphi}-v_y\right]4\tau \label{eq : d_phi_3bl_r},\end{aligned}\ ] ] where .  substituting this difference in phase into eq .[ eq : int_3b_y ] for the o - beam intensity and averaging , we find the frequency dependence of the contrast [ eq : contr_def ] . fig .[ fig : vib ] * c * shows this contrast . as an amplitude of vibrations we used .the value for the amplitude comes from the current nist setup , where a vibration isolation table is controlled to this level  .   in the four - blade interferometerthe center of rotation and the center of mass coincide between the blades . for points[ see fig .[ fig:4bl_sch ] * b * ) ] where the neutron path crosses the blades , the rotational vibrations are modeled as vibrations along the axis . as in the three - blade case the component of the neutron velocity does not change .the velocities are modified as follows .  for : \label{eq: v_4b1_r},\end{aligned}\ ] ] where and the sign of the last cosine term is positive because the oscillations of the second and the third blades have a phase - shift difference .  for :   the phases along each path are as follows .  for : for :   the phase difference is   as before , we can find the intensity at the h - detector , average it over the random phase , and extract the contrast .this contrast dependence on the frequency of rotational vibrations is plotted in fig .[ fig : vib ] * c*. we see that for these rotations the four - blade interferometer design is significantly more robust than the three - blade .notice that for rotational vibrations the first moment of the loss of contrast is not equal to zero ; all vibrations contribute to the loss of contrast .our model reconstructs the situation that is normally seen in neutron interferometry .vibrations we use in our simulations ( with amplitude m in translation ) produce changes in the incident angle of the neutron of much less than the acceptance angle of the crystal ( ) and of a similar order for the 50hz frequency range previously measured  . in order to exceed the acceptance anglethe amplitude of vibrations would have to be bigger than 50 m .note that small angle vibrations around the axis will be similar to the translational vibrations along the axis , and small angle vibrations around will be similar to the translational vibrations along . as mentioned before, vibrations along the axis do not influence the contrast .taken together these results bring us to the four - blade experimental geometry for neutron interferometer .although the new four - blade design leads to a loss of half of the neutron intensity we can make up this loss with a more robust , small system that can be moved closer to the beam break .we thus regain and even increase the final neutron intensity at the detectors . in order to achieve high contrastthe proposed four - blade interferometer system still requires excellent temperature stability  .the robust nature of the four - blade interferometer can also be understood as a result of dynamical decoupling or decoherence - free subspace ( dfs ) .the dynamical decouping analogy  is easiest to see ; the third set of blades act to undo the change in momentum introduced by the second blades .provided that the noise ( motion of the ni ) is the same when the neutron encounters both the second and third blades , then the momentum error is completely removed .   a schematic representation of the robust four - blade interferometer imbedded in a five - blade interferometer .the overall interferometer operates over su(4 ) and can be thought of as a tensor product space of two qubits .a simple abstraction of the noise generators is at each internal reflection ( there is no noise at the first and final blades ) .+ when we look over the entire path : two paths have noise ( and ) , and two paths do not have noise ( and ) .the noise generator is proportional to for each reflection internal to ni . ,scaledwidth=80.0% ]  the dfs  picture can be seen if two additional paths are considered .now we have four paths and an interferometer that acts over su(4 ) ; see fig .[ fig : dfs ] .the outer two paths are sensitive to the noise and the inner two are isolated from it .we are presently installing a five - blade interferometer to test these predictions .support provided by nist is gratefully acknowledged .the authors are grateful for discussions with d. l. jacobson , r. laflamme , b. levi , c. ramanathan .discussion with s.a .werner about an earlier idea of a four - blade interferometer for use in gravity and spin - rotation experiments is gratefully appreciated .

Abstract: perfect single - crystal neutron interferometers are adversely sensitive to environmental disturbances , particularly mechanical vibrations . the sensitivity to vibrations results from the slow velocity of thermal neutrons and the long measurement time that are encountered in a typical experiment . consequently , to achieve a good interference solutions for reducing vibration other than those normally used in optical experiments must be explored . here we introduce a geometry for a neutron interferometer that is less sensitive to low - frequency vibrations . this design may be compared with both dynamical decoupling methods and decoherence - free subspaces that are described in quantum information processing . by removing the need for bulky vibration isolation setups , this design will make it easier to adopt neutron interferometry to a wide range of applications and increase its sensitivity .


self - assembly of molecules on surfaces has become a topic of intensive research , both from a fundamental point of view and for developing electronics based on molecular units. for controlling the kinetic growth , the diffusion coefficient is an important parameter .various techniques are used to measure molecular diffusion on surfaces. for fast diffusion , a particularly suited method is an analysis of signal fluctuations caused by molecules entering and leaving the detection area under a probe .corresponding time series can be recorded by a scanning tunneling microscope , or , in principle , also by an atomic force microscope or some other probe with a point like sensor . by fixing the probe locally above the substrate ,the time resolution can be significantly increased , thus allowing to capture high mobilities , as often encountered in the case of molecular motion .different means are available for evaluating the detection events in the time series .originally , the autocorrelation function ( acf ) has been the focus of experimental and theoretical studies . for atomic diffusion , the acf method yields relative changes of diffusion coefficients .recently , this method has been extended to molecules with sizes larger than the step length of translational moves. this way absolute values of become accessible .furthermore , the residence time - distribution ( rtd ) has been discussed and another new method , the interpeak - time distribution ( it d ) , has been introduced .the rtd is the distribution of peak widths and the it d is the distribution of time intervals between successive peaks .both of them are suitable to determine absolute values of .various merits and limits have been discussed for the three variants based on the acf , rtd , and itd. from a general point of view , inter - molecular and molecule - tip interactions can affect the results .the acf is influenced by both of these interactions , while the it d is influenced only by inter - molecular interactions .accordingly , the rtd method can be viewed as the most powerful one , because it is only influenced by the molecule - tip interaction , which can be reduced systematically by adjusting the molecule - probe distance.    so far , the focus lay on molecules , whose shape can be effectively described by a circle .examples of such circular - type molecules , widely used in present studies of self - assembly on surfaces , are shown in fig .[ fig : footprints](a)-(c ) . other molecules in such studies , as shown in fig .[ fig : footprints](d)-(f ) , are better described by rectangles . in this work ,we extend our theoretical treatment to these shapes .rectangular - type molecule render signal fluctuations from rotational diffusion possible . including such rotational moves in the theoretical description increases the complexity significantly .we offer a first approach to this problem , considering thermally activated rotational diffusion uncoupled from translation .our focus in this work will be on the rtd , because of its advantage over the other methods . in the last section [ sec : conclusions ], we will also adress the question how rectangular shapes and rotational diffusion affect the acf and it d . as a further point, we discuss in sec .[ sec : isotropic ] the question how many peaks need to be recorded to obtain faithful results for the diffusion coefficient .the quality of approximations entering our analytical treatments is checked by comparison with results obtained from kinetic monte carlo ( kmc ) simulations .these simulations are carried out on a square lattice with lattice constant ( setting our length unit ) and periodic boundary conditions .the center positions of particles of circular or rectangular shape perform jumps between nearest neighbor lattice sites with rate ( setting our time unit ) , where is the diffusion coefficient .the latter is to be compared to values extracted from the analysis of simulated data , which serve as surrogate for experimental ones .the particles represent freely diffusing molecules in an experiment and are assumed to have a coverage much smaller than a monolayer .we are neglecting any interaction effects between them .when including rotational dynamics , reorientation moves with rate are additionally taken into account , see sec .[ sec : rotation ] .translational and rotational dynamics are implemented by using the reaction time algorithm.  .,width=302 ]  time series are generated by selecting one fixed lattice site , the so - called `` probe - site '' , which represents the point on the surface , from which the signal in an experiment is recorded .the simulated signal is `` on '' if the probe - site is covered by a part of a molecule , while otherwise it is `` off '' .the thus - generated `` on - off '' time series resemble measured ones , if in the latter noise is eliminated and a proper threshold is set , which allows for a reduction to a rectangular signal .the details of this procedure are described in ref . .the example shown in fig .2(b ) of that work may be compared to the simulated signal displayed in fig .[ fig : signal ] .in the following we shortly summarize our previous results from ref .  with respect to the rtd, because we will refer repeatedly to the corresponding equations in the following sections .the methods were developed for molecules , with size large compared to the sensitive area of the probe . this way the probe can be considered as pointlike and the molecule extension gives rise to a detection area , which is defined by the set of molecule center positions around the tip , that will cause a detectable change in the signal . for molecules with approximately circular shapes , as the ones shown in fig .[ fig : footprints](a)-(c ) , a circle with radius can be assigned to the detection area , for example , by taking its gyration radius .the rtd then is given by where is the bessel function of first kind with its -th zero ; is the minimal penetration depth of the center into the detection area under the tip , which is at least necessary to turn the signal `` on '' . for times much larger than the typical time for the molecule to explore the detection area , while a power law characterizes the behavior for . for continuum description breaks down .( ) , 2 ( ) , 3 ( ) , 4 ( ) , 5 ( ) , and 10 ( ) .the dashed lines are least square fits to eq.([eq : psirtdii ] ) .the distributions were vertically shifted for better visibility .( b ) diffusion coefficients from the fitting , divided by the input value in the kmc simulation , as a function of .,width=302 ]  application of eq .( [ eq : psirtd ] ) and its derivates requires to exceed a certain threshold , because for close to or larger than , one would need to consider molecular jumps into and out of the detection area instead of a diffusive motion within this area .the jump motion could be treated as well , but this is not our focus here .corresponding events may be difficult to detect within experimental time resolution . nevertheless , before starting to address our main topics , it is interesting to see , down to which fractions the continuum treatment gives reliable results for .simulated rtds for various fractions are shown in fig .[ fig : extended ] and were fitted with eq .( [ eq : psirtdii ] ) . in these simulations we fixed and , and varied .the fits [ dashed lines in fig .[ fig : extended](a ) ] yield estimates in good agreement with for , see fig .[ fig : extended](b ) .10 ( ) , 10 ( ) , and 10 ( ) .the dashed lines are least square fits to eq.([eq : psirtdii ] ) , and the distributions were vertically shifted for better visibility .( b ) as a function of .,width=302 ]  another essential factor for applications is how many events need to be recorded for obtaining reliable results .the long - time regime of the rtd is most relevant for the fitting , see eq .( [ eq : psirtdii ] ) . from eq .( [ eq : psirtd ] ) it can be calculated that about 4% of rtd events are in this regime .figure [ fig : noevents](a ) shows rtds for different total numbers of events .fits to the exponential decay according to eq .( [ eq : psirtdii ] ) are marked by the dashed lines .already for in total 10 events , i.e. about 40 - 100 events in the long - time regime , is close to , but , as shown in fig .[ fig : noevents](b ) , the error is quite large .taking larger number of events reduces the error . in view of the additional uncertainties in experiments, we suggest to perform measurements capturing at least 10 events in total .so far , a circular detection area with radius was assigned to the molecules .we now extend this ansatz to molecules to which rectangles can be assigned with a longer and shorter edge and , corresponding to an aspect ratio . only for significantly larger than onethis will give rise to notable changes of the rtd compared to a description based on a circular detection area .an analytical expression for the rtd in case of rectangular shapes is readily obtained . considering a molecule center entering the rectangular detection area and its diffusion within the area , we are led to the problem of determining the diffusion propagator in the presence of a closed absorbing boundary formed by the edges of the detection area .a uniform distribution of the molecule center on an inner rectangular contour , displaced by from the absorbing boundary , is used as initial condition .expansion of in terms of the eigenfunctions of the laplacian , \sin[k^{\rm ( s)}_ny] ] ( for ) . using the kmc simulations we again checked that this procedure gives good estimates for both diffusion coefficients and .the model of uncoupled rotation and translation with a fixed jump angle is certainly a strong simplification and further work is necessary to account for non - uniform jump angles , rotation centers displaced from the molecule center , rotation - translation couplings , etc .irrespective of these complications , the possibility to simultaneously determine rotational and translational diffusion coefficients is a promising perspective that calls for an experimental verification .22ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1'''' [ 0]secondoftwosanitize [ 0 ] + 12$12 & 12#1212_12%12[1][0] * * ,  ( ) * * ,  ( ) * * ,  ( ) * * , ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( )  _( , , ) * * ,  ( )

Abstract: analysis of signal fluctuations of a locally fixed probe , caused by molecules diffusing under the probe , can be used to determine diffusion coefficients . theoretical treatments so far have been limited to point - like particles or to molecules with circle - like shapes . here we extend these treatments to molecules with rectangle - like shapes , for which also rotational diffusion needs to be taken into account . focusing on the distribution of peak widths in the signal , we show how translational as well as rotational diffusion coefficients can be determined . we address also the question , how the distribution of interpeak time intervals and autocorrelation function can be employed for determining diffusion coefficients . our approach is validated against kinetic monte carlo simulations .