Theory of Active Intracellular Transport by DNA-relaying
Christian Hanauer, Silke Bergeler, Erwin Frey, Chase P. Broedersz
TTheory of Active Intracellular Transport by DNA-relaying
Christian Hanauer, ∗ Silke Bergeler, ∗ Erwin Frey, and Chase P. Broedersz
1, 2, † Arnold-Sommerfeld-Center for Theoretical Physics and Center for NanoScience,Ludwig-Maximilians-Universit¨at M¨unchen, D-80333 M¨unchen, Germany. Department of Physics and Astronomy, Vrije Universiteit Amsterdam, 1081 HV Amsterdam, The Netherlands (Dated: January 12, 2021)The spatiotemporal organization of bacterial cells is crucial for the active segregation of replicatingchromosomes. In several species, including
Caulobacter crescentus , the ATPase ParA binds to DNAand forms a gradient along the long cell axis. The ParB partitioning complex on the newly replicatedchromosome translocates up this ParA gradient, thereby contributing to chromosome segregation.A DNA-relay mechanism—deriving from the elasticity of the fluctuating chromosome—has beenproposed as the driving force for this cargo translocation, but a mechanistic theoretical descriptionremains elusive. Here, we propose a minimal model to describe force generation by the DNA-relaymechanism over a broad range of operational conditions. Conceptually, we identify four distinctforce-generation regimes characterized by their dependence on chromosome fluctuations. Theserelay force regimes arise from an interplay of the imposed ParA gradient, chromosome fluctuations,and an emergent friction force due chromosome-cargo interactions.
The interior organization of bacterial cells is an essen-tial prerequisite for several vital processes, ranging fromchromosome and plasmid segregation to cell division [1].Dedicated active mechanisms ensure the rapid translo-cation and accurate localization of macromolecular ob-jects, such as low-copy-number plasmids [2], protein clus-ters [3, 4], and carboxysomes [5]. A prominent exampleis the translocation of the partition complex in bacteriasuch as
Caulobacter crescentus . One copy of the partitioncomplex — bound to the newly replicated chromosome —translocates rapidly from the old to the new cell pole, re-sulting in chromosome segregation [6]. The translocationof the chromosome-bound partition complex depends ona protein gradient: the partition complex follows an in-creasing amount of the ATPase ParA in the cell [7–10].However, the physical principles underlying this directedmotion of the partition complex remain unclear.The ATPase ParA belongs to the widely conservedParAB S partitioning system for chromosome and plas-mid segregation [11]. The partitioning complex is a largecentromere-like protein-DNA cluster consisting of inter-acting ParB proteins [11–15]. The ATPase ParA existsin an ADP- and ATP-bound form and its prefered loca-tion in the cell can change dependent on its nucleotidestate [16]: As an ATP-bound dimer, ParA binds non-specifically to DNA and, upon interaction with ParB, itsATPase activity is stimulated leading to detachment ofADP-bound ParA monomers into the cytosol. The inter-actions of ParA ATPases with the partition complex arenecessary for its directed translocation [16].Various mechanisms have been proposed for force gen-eration [7, 17–20], including a class of Brownian-ratchetmodels [21–25]. Specifically, a DNA-relay mechanismwas suggested [21, 26], where DNA-bound ParA pro-teins relay the partition complex up a ParA concentrationgradient by exploiting elastic fluctuations of the chro-mosome [21, 27]. It has been argued using simulations, that this model can explain the experimentally observedtranslocation of the partition complex [21, 27]. However,a theoretical description of the DNA-relay force that re-veals the dependence of the force on key system param-eters is still lacking.Here, we present an analytic theory for force gener-ation by the DNA-relay mechanism. We compute therelay force by evaluating the stochastic binding of DNA-bound ParA-like proteins to a cargo using a Master equa-tion approach. Conceptually, the predicted relay forceoriginates from the interplay of the ParA gradient, chro-mosome fluctuations, and an emergent friction force dueto the interactions of chromosome-bound ParA proteinswith the cargo. These contributions give rise to four dis-tinct force generation regimes, depending on the strengthof chromosomal fluctuations and the cytoplasmic frictionon the cargo. We thus establish a theoretical frameworkto characterize the DNA-relay mechanism over a broadrange of operational conditions, providing conceptual in-sight into active directed transport of ParB-like cargosfor in vivo [21, 28–30] and in vitro [31] settings.To elucidate force generation by the DNA-relay mech-anism [21, 27], we study a minimal model obtained byreducing the full complexity of the partitioning system tokey elements important for DNA-relaying (Fig. 1a). Ourone-dimensional model consists of the cargo and ParA-bound chromosomal elements. To account for the chro-mosomal dynamics in a simplified manner, the chromo-some is modelled as a set of fluctuating elastic springs.In ParAB S -like partitioning systems, the ATPase ParAdetaches from the chromosome at the cargo due to stim-ulation of ATP hydrolysis by ParB, and can only rebindto the chromosome upon ATP binding and dimerization.This dynamics results in a ParA gradient propagatingwith the cargo, as was shown for an in vitro reconsti-tuted partitioning system [31, 32]. Instead of modelingthe ParA dynamics explicitly, we use this observation by a r X i v : . [ phy s i c s . b i o - ph ] J a n FIG. 1.
Minimal model for force generation by DNA-relaying . (a) The relay force F arises from the interactionsof the cargo with ParA ATPases bound to the chromosome,represented by a set of chromosomal elements modelled asa bead-spring system with an associated ParA concentration(indicated by the green tone). We assume that the ParA gra-dient is co-moving with the cargo. Chromosomal elementsfluctuate due to thermal energy, with the magnitude of thefluctuations, σ = p k B T /k (red Gaussian). (b,c) Cargo tra-jectory (b) and the corresponding DNA-relay force (c) ob-tained from the numerical solution of Eq. (1) using Browniandynamics simulation. The horizontal line shows the time-averaged value of F . imposing a co-moving ParA gradient on the cargo.Specifically, the cargo is represented as a line segmentof length 2 r with a reaction radius r , and chromosomalregions are described in a coarse-grained way as a set of N tot beads, equally spaced along a domain of length ‘ (Fig. 1a). Each bead is tethered to a fixed position by aspring with stiffness k , thermally fluctuating with ampli-tude σ = p k B T /k . The ParA concentration associatedwith a chromosomal bead at a distance x i from the cargois set to c ( x i ) = mx i + c . Cargo and chromosomal el-ements interact: beads within the reaction radius of thecargo bind with rate k on c ( x ). Cargo-bound beads unbindwith rate k off . Importantly, due to the elasticity of theDNA, cargo-bound chromosomal elements exert a forceon the cargo. We describe the resulting cargo motion byan overdamped Langevin equation γ c dx c dt = k X i ( x i − y i ) + p γ c k B T η ( t ) , (1)where x c is the cargo position and the index i runs over allcargo-bound chromosomal elements with rest position x i and bead position y i . The white noise term η ( t ) satisfies h η ( t ) i = 0 and h η ( t ) η ( t ) i = δ ( t − t ), and γ c is the frictioncoefficient of the cargo in the cytoplasm.Our goal is to calculate the steady-state DNA-relay force on the cargo for a co-moving ParA gradient. Tocompute the steady-state DNA-relay force using a fi-nite chromosomal domain of size ‘ , we employ periodicboundary conditions, such that there are always N tot chromosomal elements the cargo could interact with [33].For σ (cid:29) ‘ , the limited number of chromosomal elementsbecomes important, allowing us to study finite systemsize effects. In contrast, if σ (cid:28) ‘ , this model is effec-tively identical to one with an infinite system size.To facilitate further theoretical analysis we recast vari-ables and system parameters in a non-dimensional formusing the system size ‘ as a characteristic length, x → x‘ ,and the unbinding time 1 /k off as characteristic time scale, t → t/k off . Using this non-dimensionalized form, weidentify four key parameters that dictate the system’sdynamics: The binding propensity c k on /k off → c char-acterizes the on/off kinetics between the cargo and ParA;the concentration gradient m‘/c → m describes theasymmetry of the ParA gradient on the chromosome; σ/‘ → σ sets the magnitude of chromosomal fluctuationrelative to system size; and the cargo friction coefficient γ c k off ‘ / ( k B T ) → γ c provides a measure for how suscep-tible the cargo is to DNA-relay forces [33].Using Brownian dynamics simulations (Fig. 1b,c) wefind distinct force-generation regimes depending on themagnitude of chromosomal fluctuations σ and the cyto-plasmic friction coefficient γ c of the cargo, each character-ized by a different dependence on σ (Fig. 2). While we ob-serve maximal force under stalling conditions ( γ c → ∞ ),the system’s behavior changes drastically for a movingcargo (finite γ c ). Interestingly, in this parameter rangewe find a maximum in the force at intermediate σ , sug-gesting an optimal operating regime for this transportmechanism.To provide conceptual insight into the DNA-relaymechanism, we develop an analytical theory to calculatethe relay force on the cargo. Specifically, we derive anapproximation for the relay force F = 1 σ X i ( x i − y i ) , (2)which reveals how microscopic system parameters controlthe DNA-relay mechanism. To obtain an explicit analyt-ical expression, we consider the average relay force, anduse a continuum approximation F = 1 σ Z / − / d x Z r − r d y n ( x, y, t )( x − y )= Z / − / d x f ( x, t ) . (3)We moved to the cargo frame of reference, introduced thedensity n ( x, y, t ) of cargo-bound chromosomal elementswith a rest position x and binding position y at the cargo, FIG. 2.
Average relay force F in the weak-bindinglimit ( c (cid:28) ) for different values of the friction coef-ficient γ c of the cargo in the cytoplasm and the mag-nitude of chromosome fluctuations σ . (a) We compareresults from simulations (dots) with theory (lines), obtainedfrom Eq. (8) and Eq. (10) for a static (black) and movingcargo (blue). The dotted vertical line at σ = 1 / and defined the force density f ( x, t ) = 1 σ Z r − r d y n ( x, y, t )( x − y ) . (4)Thus, the relay force can be understood by studying theforce density f , for which we need to calculate n ( x, y, t ).The dynamics of the density n ( x, y, t ) is described by ∂ t n ( x, y, t ) − v ( n, t ) ∂ x n ( x, y, t ) = c ( x ) φ ( y ; x, σ )( N tot − n ( x, t )) − n ( x, y, t ) . (5)For a static cargo ( v = 0), the temporal change in n isdetermined only by a gain and a loss term, correspond-ing to binding to and unbinding from the cargo. For abinding event, a chromosomal bead needs to move withinthe reaction radius of the cargo. We describe the posi-tion y of an unbound bead as a Gaussian random vari-able with mean x and variance σ . The probability thata bead with rest position x is at position y ∈ [ − r, r ] is thus given by the Gaussian probability density func-tion φ ( y ; x, σ ) (Fig. 1a). This is justified under weakchromosome-cargo interactions, i.e. whenever the decor-relation time τ corr = σ γ b N tot is much smaller than thebinding time τ bind = 1 /c . A binding event takes placestochastically with a rate c ( x )( N tot − n ( x, t )), account-ing for the finite density of chromosomal elements avail-able for binding, where c ( x ) = c (1 + mx ) denotes thedimensionless ParA concentration. The total density ofcargo-bound chromosomal beads with rest position x canbe obtained by integrating the density n ( x, y, t ) over allpossible binding positions y on the cargo: n ( x, t ) = Z r − r n ( x, y, t ) d y. (6)Unbinding is described by a constant detachment rate,set by the last term in Eq. (5). Finally, when v = 0 thetemporal evolution of n also includes an advection termto account for cargo motion.We expect the weak-binding limit ( c (cid:28)
1) to be thebiologically relevant parameter regime in this model, be-cause of the high ParA turnover rate caused by ParB-induced ATP hydrolysis of ParA and subsequent detach-ment of ParA from the cargo [34]. Henceforth, we thusconsider only this limit, for which saturation effects ofthe cargo by bound chromosomal elements are negligible.For completeness, we provide our results for the strong-binding limit [33] and find that the conceptual insightsgained from the weak-binding limit largely apply.Having established a theoretical framework to studyforce generation by DNA-relaying, we first consider thecase of a static cargo ( v = 0). Put simply, we computethe stalling force of the cargo. This static case allows usto study basic features of the force generation mechanismand provides insights that will also be relevant for themoving cargo scenario. We first calculate the steady-state solution of Eq. (5), and with this an expression forthe steady-state force density [33]: f ( x ) = c ( x )( φ ( x ; r, σ ) − φ ( x ; − r, σ )) . (7)This expression for the force density constitutes one ofour key findings and allows us to understand how theDNA-relay force is generated and how it depends on sys-tem parameters.The force density encodes the contribution of a chro-mosomal element with rest position x to force generation.Intuitively, this force density is determined by the inter-play between how likely it is for a chromosomal elementto bind to the cargo and how much force is exerted onthe cargo in this configuration. In the limit σ (cid:29)
1, chro-mosomal beads exhibit strong fluctuations, and withouta ParA gradient ( m = 0) every bead thus has approx-imately the same binding probability. Here, only thedistance of a chromosomal element from the cargo mat-ters for force generation and therefore the force density FIG. 3.
The influence of the ParA concentration gra-dient m and the cargo velocity v on the force density f ( x ). (a,b) f ( x ) for a static cargo given by Eq. (7) without( m = 0) and with ( m = 2) a ParA gradient. (c,d) f ( x ) fora static ( v = 0) and a moving ( v = 0 .
05) cargo both with m = 2. The force density for a moving cargo is obtained nu-merically. We compare results from simulations (dots) andour theoretical results (lines). Note that the dark green andthe light blue curves in (a,c) and (b,d) show the same data. increases linearly with the distance of the bead from thecargo (Fig. 3a, light green). Because of the symmetry of f ( x ), forces exerted on the cargo from chromosomal ele-ments positioned behind and in front of the cargo cancel,such that no net force is generated. By contrast, if theParA concentration on the beads increases towards theright ( m > σ (cid:28) f ( x ) becomes asymmetric,resulting in a net force on the cargo. In all cases, ouranalytical predictions for the force density are in accordwith Brownian dynamics simulations.Having analyzed the steady-state force density f ( x ),we next evaluate the cargo stalling force F sc in the weak-binding limit using Eq. (3): F sc = mc Z / − / d x x ( φ ( x ; r, σ ) − φ ( x ; − r, σ )) (8)Upon performing this integral, we obtain the dependenceof the cargo stalling force on σ (Fig. 2). Remarkably,for σ (cid:28) F sc is independent of σ . Uponincreasing σ , more chromosomal elements are recruitedto contribute to force generation. However, this increase in participation is precisely compensated by the softeningof the springs resulting in a stiffness independent DNA-relay force F sc = const. For σ (cid:29)
1, we obtain F sc ∝ /σ .Here, the finite size of the system affects force generation.Due to the limited number of beads, the softening of thesprings can not be compensated anymore by an increasedamount of beads interacting with the cargo. Therefore,the force on the cargo decreases.To understand force generation for a dynamic cargo,we first consider the case of a cargo that moves withan imposed velocity v . To this end, we study the steady-state force density, which determines the relay force F ( v ).We calculate the steady-state solution of Eq. (5) for afixed velocity v and obtain the corresponding force den-sity f ( x ) using Eq. (4). We observe that, for v > /k off ), result in an increasedamount of chromosomal beads pulling the cargo back-wards.Interestingly, we find that a moving cargo experiencesthe force F ( v ) = F sc − v σ N sc , (9)which has two contributions: the static relay force andan additional force term linear in v . This term can beinterpreted as an emergent friction force with the frictioncoefficient γ e = σ N sc = σ rc , where N sc denotes aver-age number of cargo bound beads for a static cargo [33].Next, we use this result for imposed motion to obtainthe DNA-relay force exerted on a cargo that moves au-tonomously due to diffusion and the interactions withParA-bound beads. First, we self-consistently determinethe velocity v of a self-propelled cargo using force bal-ance γ c v = F ( v ). From this analysis, we obtain an ex-plicit expression for the generated force associated to thistranslocation velocity F = F sc γ e γ c . (10)Interestingly, the force on an autonomously moving cargocan be entirely calculated from quantities obtained for astatic cargo.The interplay of self-propulsion and emergent frictionforce gives rise to four distinct force generation regimes,as depicted in the phase diagram in Fig. 2b. As in thestatic limit, we can distinguish force generation for smalland large chromosomal fluctuations. Importantly how-ever, the qualitative dependencies on the strength of thechromosome fluctuations can differ because of the emer-gent friction force. In the limit where the cytoplasmicfriction dominates the emergent friction, γ c (cid:29) γ e , the dy-namic relay force is well approximated by the static relayforce (Fig. 2a, black line). Upon lowering the cytoplas-mic friction slightly, the emergent friction only reducesforce generation for small σ . Here, the σ -dependenceof the emergent friction, γ e ∝ /σ , combines with theconstant static cargo force to F ∝ σ (Fig. 2a, dark blueline). Upon lowering γ c further the emergent friction alsoinfluences the regime σ (cid:29)
1. For this parameter regime,the decrease in driving and friction force with increasing σ combine to F ∝ /σ (Fig. 2a, light blue line). In thelimit σ → ∞ , we find that the relay force vanishes, asfor a static cargo. In all cases, we find that our analyt-ical predictions agree well with the Brownian dynamicssimulations.Our work complements previous studies on numeri-cally and phenomenologically modeling cargo motion inParAB S -like systems [21–24, 27, 35, 36] by providingan analytical microscopic theory for force generation byDNA-relaying. It is still debated whether the main con-tribution to force generation in ParAB S systems derivesfrom chromosome elasticity (DNA-relay force) [21–24, 26]or chemophoresis [20, 25, 37]. We contribute to this openquestion by developing a quantitative mechanistic the-ory. Our analytical predictions for the dependence ofthe DNA-relay force on microscopic parameters couldbe tested in in vitro experiments with a stiffness con-trolled DNA-carpet [31]. 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1, 2, † Arnold-Sommerfeld-Center for Theoretical Physics and Center for NanoScience,Ludwig-Maximilians-Universit¨at M¨unchen, D-80333 M¨unchen, Germany. Department of Physics and Astronomy, Vrije Universiteit Amsterdam, 1081 HV Amsterdam, The Netherlands (Dated: January 12, 2021)
I. BROWNIAN DYNAMICS SIMULATION
FIG. S1.
Schematics to illustrate the implementation of our model to investigate the steady-state DNA-relayforce exerted on the cargo. (a) To obtain steady-state cargo forces without simulating large chromosomal regions, weconsider a finite number of beads N tot around the cargo, which correspond to a chromosomal region of size ‘ . When thecargo moves to the right, beads further away from the cargo move outside of the region [ x c − ‘/ , x c + ‘/ We perform one-dimensional Brownian dynamics simulations of cargo motion. The cargo position, x c is described bythe overdamped Langevin equation γ c dx c dt = F + p γ c k B T η ( t ) , (S1)where γ c refers to the cytosolic friction coefficient of the cargo and the white noise term η ( t ) satisfies h η ( t ) i = 0 and h η ( t ) η ( t ) i = δ ( t − t ). The DNA-relay force term reads F = − k X i ( y i − x i ) , (S2)where the sum runs over all cargo-bound chromosomal elements, x i refers to the rest position and y i to the positionof bead i . We use the Euler-Maruyama scheme [1] to find a numerical solution to Eq. (S1) at time steps t k = k ∆ t with k = 0 , . . . , N steps via the relation x k +1 = x k + F ( x k ) γ c ∆ t + s k B Tγ c ∆ W k , (S3)where the independent Wiener increments ∆ W k are Gaussian random variables with ∆ W k = N (0 , ∆ t ). To reducecomputational complexity we do not explicitly use a Brownian dynamics scheme to simulate the positions of thebeads. Instead, we draw the current bead position y i from the Gaussian distribution N ( x i , σ ).Given that a bead is in the cargo reaction radius ( | x c − y i | ≤ r ), we simulate the reactions between cargo andbead using a simple stochastic algorithm. First, we generate a uniform random number ν in the interval (0 , ν < k on c ( x i )∆ t , the bead attaches to the cargo. If ν ≥ k on c ( x i )∆ t , no reaction takes place in this time step. Once a r X i v : . [ phy s i c s . b i o - ph ] J a n a bead is cargo-bound, an unbinding event can occur at any time step and is simulated with the same algorithmexplained above, but with the rate k off . We choose ∆ t sufficiently small, such that it is very unlikely that more thanone stochastic event takes place during ∆ t .Our goal is to obtain the steady-state DNA-relay force and its dependence on the model parameters withoutconsidering boundary effects of the chromosome. Since the simulation of an infinitely large system is not possible,we approximate an infinitely large system by using a finite chromosomal region around the cargo. When the cargomoves to the right, beads to the left of the cargo can move outside of the region [ x c − ‘/ , x c + ‘/
2] around the cargo(Fig. S1). These beads are then reintroduced in front of the cargo, such that the cargo is always surrounded by thesame N tot beads. Furthermore, we assume that the ParA concentration gradient on the beads is co-moving with thecargo position, such that the cargo experiences the same gradient when it is moving through the system: c ( x ) = m ( x − x c ) + c (S4) TABLE I. Summary of model parameters. If not otherwise stated, we use the parameter values listed here for our simulation.The notation [ · , · ] indicates a range of parameters.Parameter Symbol Unit Fig. 1 Fig. 2a Fig. 2b Fig. 3 Fig. S2 Fig. S3Cargo friction coefficient γ c k B T s µ m − − , , [10 − , ] 10 k k B T µ m −
100 [10 − , ] [10 − , ] 0 . ,
100 0 . , [10 − , ]Binding rate k on µ m s − .
01 0 .
01 0 .
01 0 .
01 [10 , , ] [10 − , , ]Unbinding rate k off s − . c µ m − m µ m − ,
2] 2 2Number of beads N tot r µ m 0 .
05 0 .
05 0 .
05 0 .
05 0 .
05 0 . ‘ µ m 1 1 1 1 1 1 II. NON-DIMENSIONALIZATION
Here we describe how we choose the time and length scales as well as the characteristic scale of the density n ( x, y, t )to arrive at a non-dimensional differential equation for n ( x, y, t ) (Eq. (5) in the main text). In dimensional units, thetime evolution of the density n is given by ∂ t n ( x, y, t ) − v ( n, t ) ∂ x n ( x, y, t ) = k on c ( x ) φ ( y ; x, σ ) (cid:18) N tot ‘ − n ( x, t ) (cid:19) − k off n ( x, y, t ) , (S5)together with γ c v = F and F = k Z ‘/ − ‘/ d x Z r − r d y n ( x, y, t )( x − y ) . (S6)The ParA concentration gradient is denoted by c ( x ) = mx + c and the position x of a chromosomal element withrest position y is described by the Gaussian probability density φ ( y ; x, σ ).Next, we express lengths in terms of the system size ‘ , x = ˜ x‘ , and time in terms of the average time until acargo-bound bead detaches from the cargo, t = ˜ t/k off : k off ∂ ˜ t n ( x, y, t ) − v ( n, ˜ t ) ‘ ∂ ˜ x n ( x, y, t ) = k on ( m‘ ˜ x + c ) φ (˜ y ; ˜ x, ˜ σ ) ‘ (cid:18) N tot ‘ − n ( x, t ) (cid:19) − k off n ( x, y, t ) . (S7)The density n ( x, y, t ) becomes dimensionless by using n ( x, y, t ) = ˜ n (˜ x, ˜ y, ˜ t ) /‘ and therefore n ( x, t ) = ˜ n (˜ x, ˜ t ) /‘ .Multiplying the above equation by ‘ /k off yields ∂ ˜ t ˜ n (˜ x, ˜ y, ˜ t ) − ˜ v (˜ n, ˜ t ) ∂ ˜ x ˜ n (˜ x, ˜ y, ˜ t ) = k on k off ( m‘ ˜ x + c ) φ (˜ y ; ˜ x, ˜ σ ) (cid:0) N tot − ˜ n (˜ x, ˜ t ) (cid:1) − ˜ n (˜ x, ˜ y, ˜ t ) . (S8)Upon defining c = ˜ c k off /k on and m = ˜ mc /‘ , we obtain: ∂ ˜ t ˜ n (˜ x, ˜ y, ˜ t ) − ˜ v (˜ n, ˜ t ) ∂ ˜ x ˜ n (˜ x, ˜ y, ˜ t ) = ˜ c (˜ x ) φ (˜ y ; ˜ x, ˜ σ ) (cid:0) N tot − ˜ n (˜ x, ˜ t ) (cid:1) − ˜ n (˜ x, ˜ y, ˜ t ) , (S9)with ˜ c (˜ x ) = ˜ c (1 + ˜ m ˜ x ).The expression for the relay force, in terms of non-dimensional quantities, reads F = k B T‘ σ Z / − / d˜ x Z ˜ r − ˜ r d˜ y ˜ n (˜ x, ˜ y, ˜ t )(˜ x − ˜ y ) (S10)= k B T‘ ˜ F . (S11)Finally, we find by γ c v = F (S12) k off ‘ k B T γ c ˜ v = ˜ F (S13)that the characteristic scale of the friction coefficient is k B T /k off ‘ . In the following, as well as in the main text, weomit the tilde to simplify the notation. III. THEORETICAL APPROXIMATION FOR DNA-RELAY FORCE
The time evolution of the density of chromosomal elements n ( x, y, t ) that are bound to the cargo with rest position x and bead position y is given by: ∂ t n ( x, y, t ) − v ( n, t ) ∂ x n ( x, y, t ) = c ( x ) φ ( y ; x, σ )( N tot − n ( x, t )) − n ( x, y, t ) , (S14)with the cargo velocity v , the rescaled expression for the ParA density, c , and the Gaussian probability density function φ . The density n ( x, t ) is the density of all cargo-bound beads with rest position x : n ( x, t ) = Z r − r n ( x, y, t ) d y. (S15)Integrating Eq. (S14) over all possible bead-binding positions at the cargo, yields an equation for the density of boundbeads with rest position x : ∂ t n ( x, t ) − v ( n, t ) ∂ x n ( x, t ) = c ( x ) (cid:18)Z r − r φ ( y ; x, σ ) d y (cid:19) ( N tot − n ( x, t )) − n ( x, t )= c ( x ) p f ( x )( N tot − n ( x, t )) − n ( x, t ) , (S16)with the finding probability p f ( x ). For a rest position x , we calculate the integral over all deflections weighted by n ( x, y, t ): f ( x, t ) = 1 σ Z r − r n ( x, y, t )( x − y ) d y. (S17)To get the total relay force, we need to integrate over all deflections of cargo-bound beads: F = Z / − / f ( x, t ) d x. (S18)With Eq. (S14), the temporal evolution of the relay force density is given by: ∂ t f ( x, t ) = v ( n, t ) (cid:18) ∂ x f ( x, t ) − σ n ( x, t ) (cid:19) + c ( x )( N tot − n ( x, t )) 1 σ Z r − r φ ( y ; x, σ )( x − y ) d y − f ( x, t ) . (S19)Eq. (S16) and Eq. (S19) constitute a system of partial differential equations. Next, we obtain steady-state solutionsfor n and f and use them to find expressions for the amount of cargo-bound beads and the force on a cargo. A. Stationary cargo case
For a stationary cargo, i.e. v = 0, the steady-state distribution n ( x ) is given by n ( x ) = N tot c ( x ) p f ( x )1 + c ( x ) p f ( x ) . (S20)With this expression, we calculate the steady-state expression for f in the stationary cargo case: f ( x ) = 1 σ c ( x )( N tot − n ( x )) Z r − r φ ( y ; x, σ )( x − y ) d y (S21)= 1 σ N tot c ( x )1 + c ( x ) p f ( x ) Z r − r φ ( y ; x, σ )( x − y ) d y. (S22)Using these results, we compute the average number of bound beads for a stationary cargo N sc = Z / − / n ( x ) d x = Z / − / N tot c ( x ) p f ( x )1 + c ( x ) p f ( x ) d x (S23)and the stalling force: F sc = Z / − / f ( x ) d x = 1 σ Z / − / d x N tot c ( x )1 + c ( x ) p f ( x ) Z r − r d y φ ( y ; x, σ )( x − y ) (S24) B. Dynamic cargo case
To obtain an expression for the force density in the dynamic cargo case, we first consider a cargo with an exter-nally imposed velocity v . Hence, we need to solve the following differential equations to determine the steady-stateexpressions of the density of cargo-bound beads n and the relay force density f : − v∂ x n ( x, v ) = c ( x ) p f ( x )( N tot − n ( x, v )) − n ( x, v ) , (S25) − v (cid:18) ∂ x f ( x, v ) − σ n ( x, v ) (cid:19) = 1 σ c ( x )( N tot − n ( x, v )) Z r − r φ ( y ; x, σ )( x − y ) d y − f ( x, v ) . (S26)A numerical solution of these equations is shown in Fig. S2. To determine approximate analytical solutions, we write n ( x, v ) and f ( x, v ) as Taylor expansions in the velocity v : n ( x, v ) = n ( x ) + v n ( x ) + O ( v ) , (S27) f ( x, v ) = f ( x ) + v f ( x ) + O ( v ) . (S28)We insert these expressions into Eq. (S25) and (S26). To zeroth order in v we get the following two equations0 = c ( x ) p f ( x )( N tot − n ( x )) − n ( x ) , (S29)0 = 1 σ c ( x )( N tot − n ( x )) Z r − r φ ( y ; x, σ )( x − y ) d y − f ( x ) , (S30)which are solved by the expressions for the static cargo case. The terms that are first order in v , lead to expressionsfor n ( x ): ∂ x n ( x ) = n ( x )(1 + c ( x ) p f ( x )) , (S31) ⇒ n ( x ) = ∂ x n ( x )1 + c ( x ) p f ( x ) , (S32)and f ( x ): − ∂ x f ( x ) + 1 σ n ( x ) = − σ c ( x ) n ( x ) Z r − r φ ( y ; x, σ )( x − y ) d y − f ( x ) , (S33) ⇒ f ( x ) = ∂ x f ( x ) − σ n ( x ) − σ c ( x ) n ( x ) Z r − r φ ( y ; x, σ )( x − y )d y. (S34)We insert this result in the general relation for the relay force (Eq. (S18)) and find that the force on a cargo movingwith imposed velocity v is F ( v ) = 1 σ Z / − / d x Z r − r n ( x, y, v )( x − y ) d y = Z / − / d x f ( x, v )= F sc + v Z / − / d x f ( x ) + O ( v )= F sc + v Z / − / d x (cid:18) ∂ x f ( x ) − σ n ( x ) − σ c ( x ) ∂ x n ( x )1 + c ( x ) p f ( x ) Z r − r φ ( y ; x, σ )( x − y )d y (cid:19) + O ( v )= F sc − vσ N sc − vσ Z / − / d x c ( x ) ∂ x n ( x )1 + c ( x ) p f ( x ) Z r − r φ ( y ; x, σ )( x − y )d y + O ( v ) (S35)We use this result in the force balance equation γ c v = F ( v ) to self-consistently determine the velocity and thereforethe force on a self-propelled cargo, as shown in the main text for the weak-binding limit. C. Weak-binding limit
In the weak-binding limit ( c (cid:28) c , the expression for F sc (Eq. S24) can be solved analytically: F sc = Z / − / f ( x ) d x = N tot c σ Z / − / d x (1 + mx ) Z r − r d y φ ( y ; x, σ )( x − y ) (S36)= N tot mc Z / − / x ( φ ( x ; r, σ ) − φ ( x ; − r, σ )) d x (S37)= N tot c m r π σ (cid:18) e − (1+2 r )28 σ − e − (1 − r )28 σ (cid:19) + r (cid:18) erf (cid:18) − r √ σ (cid:19) + erf (cid:18) r √ σ (cid:19)(cid:19)! (S38)For σ (cid:28)
1, the width σ of the Gaussian function φ ( x ; ± r, σ ) is much smaller than the system size and we find F sc ≈ N tot rc m using Eq. (S37). For σ (cid:29) F sc ≈ N tot mc σ Z / − / xσ √ πσ (cid:18) − ( x − r ) σ − x − r ) σ (cid:19) d x (S39)= N tot r mc √ πσ σ Z / − / x d x (S40)= N tot r mc √ π σ
112 (S41)Therefore, we find that F sc = const ( σ (cid:28)
1) and F sc ∝ /σ ( σ (cid:29) N sc ≈ Z / − / c ( x ) p f ( x ) d x = Z / − / N tot c (1 + mx ) p f ( x ) d x = N tot c Z / − / p f ( x ) d x. (S42)In the last step we used that p f ( x ) is even in x , such that the integral over xp f ( x ) is zero. For σ a lot smaller thanthe system size, σ (cid:28)
1, we find N sc ≈ rc N tot .In the case of the dynamic cargo, the equation for the force Eq. (S35) can be approximated to linear order in c by: F ( v ) = F sc − vσ N sc . (S43) FIG. S2.
Distributions of the density of cargo-bound chromosomal elements n ( x ) with rest position x and thecorresponding force density f ( x ) in the strong-binding limit. (a,b,c,d) n ( x ) and f ( x ) for a static ( v = 0) and a moving( v = 0 .
05) cargo in the regime of large chromosomal fluctuations for different values of the binding propensity c . (e,f,g,h) n ( x )and f ( x ) for a static ( v = 0) and a moving ( v = 0 .
05) cargo in the regime of small chromosomal fluctuations for different valuesof the binding propensity c . The density of cargo-bound beads n ( x ) and the force density f ( x ) for a static cargo are obtainedfrom Eq. (S20) and Eq. (S22), while the profiles for a moving cargo are obtained from a numerical solution of Eq. (S16) andEq. (S19). We compare results from simulations (dots) and theory (lines). D. Strong-binding limit
While in the weak-binding limit saturation effects can be neglected, we now explain how the relay force is alteredin the strong-binding limit. In analogy to our discussion in the main text, we first consider the static cargo case.To understand the effect of an increase of c on force generation, we consider the density profiles of cargo-boundchromosomal elements n ( x, t ) with rest position x (Fig. S2a,e). In the case of large chromosome fluctuations ( σ (cid:29) c leads to a binding profile n ( x, t ) that quickly approaches a uniform distribution (Fig. S2a). When allbeads bind to the cargo with the same probability, no net force is generated. This observation explains the reductionof the force for large bead fluctuations for very strong binding (Fig. S3a). On the other hand, for small chromosomefluctuations ( σ (cid:28) n ( x, t ) is not saturated(0 < n < c further, enlarges the region around the cargo where n ( x, t ) is saturated,but as long as this region is smaller than the system size, there is an imbalance of the cargo-bound beads and hencea net force is generated (Fig. S3a).In the case of a dynamic cargo, the generated force is the result of the driving force and additional friction due tocargo-bound beads. For very large binding propensities, not only the driving force is reduced (for large fluctuations)as discussed above, but also the number of cargo-bound beads is increased (Fig. S2c,g), which leads to larger effectivefriction coefficients for the cargo (Fig. S2d,h). ∗ These authors contributed equally to this work. † [email protected][1] P. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Stochastic Modelling and Applied Prob-ability (Springer Berlin Heidelberg, 2011). FIG. S3.
Average relay force F in the strong-binding limit ( c (cid:29) ) for different values of the binding propensity c and different values of the bead fluctuations σ ..