Diffusion of tagged particles in a crowded medium: supplementary material. Paper submitted to Europhys. Lett.

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1 Diffusion of tagged particles in a crowded medium: supplementar material. Paper submitted to Europhs. Lett. Marta Galanti 1,2,3, Duccio Fanelli 2, Amos Maritan 4, Francesco Piazza 3 1 Dipartimento di Sistemi e Informatica,Università di Firenze and IF, Via S. Marta 3, IT Florence, Ital 2 Dipartimento di Fisica e Astronomia, Università di Firenze and IF, Via Sansone 1, IT Sesto Fiorentino, Ital 3 Université d Orléans and Centre de Biophsique Moléculaire CBM, CRS-UPR 4301, Rue Charles Sadron, Orléans, France 4 Dipartimento di Fisica e Astronomia G. Galilei Università di Padova, CISM and IF, via Marzolo 8, 35131, Padova, Ital PACS numbers: I. DERIVATIO OF THE MEA FIELD EQUATIOS I TWO DIMESIO As stated in the main tet, the equations governing the evolution of the particles, which we obtained for a 1D setting, can be etended to higher dimensions. In the following we will provide a detailed derivation for the mean field equations in 2D. We will first generalized the procedure described in the main bod of the paper and then turn to consider and alternative approach inspired to the work of Landman and collaborators see e.g. Simpson, Landman and Hughes Phs Rev E To progress in the analsis we assume each site of the the two-dimensional lattice to be labelled with two indices i, j. In the 2D case, the nearest neighbour sites the selected particle can jump to are four. This is at variance with the 1D geometr where each site has just two adjacent neighbors. The binar variables at time k are hence indicated as m i,j k and n i,j k. The stochastic process reads: m i,j k + 1 m i,j k = z + i 1 m i 1,jk[1 m i,j k][1 n i,j k] + z i+1 m i+1,jk[1 m i,j k][1 n i,j k] +z + j 1 m i,j 1k[1 m i,j k][1 n i,j k] + z j+1 m i,j+1k[1 m i,j k][1 n i,j k] z + i m i,jk[1 m i+1,j k][1 n i+1,j k] z i m i,jk[1 m i 1,j k][1 n i 1,j k] z + i m i,jk[1 m i,j+1 k][1 n i,j+1 k] z i m i,jk[1 m i,j 1 k][1 n i,j 1 k] 1 where the stochastic variables z ± are defined in analog to the one dimensional case. The equation governing the evolution of n i,j can be equivalentl modified. After introducing the one-bod occupanc probabilities and assuming a mean-field factorization for the two-bod and three-bod correlations, one eventuall ends up with ρ i,j k = m i,j k 2 φ i,j k = n i,j k 3 ρ i,j k + 1 ρ i,j k = q ρ i 1,j k + ρ i+1,j k + ρ i,j 1 k + ρ i,j+1 k [1 ρ i,j k] [1 φ i,j k] q ρ i,j k[4 ρ i 1,j k + ρ i+1,j k ρ i,j 1 k + ρ i,j+1 k φ i 1,j k + φ i+1,j k 4 φ i,j 1 k + φ i,j+1 k φ i+1,j kρ i+1,j k + φ i 1,j kρ i 1,j k + φ i,j+1 kρ i,j+1 k + φ i,j 1 kρ i,j 1 k] φ i,j k + 1 φ i,j k = w φ i 1,j k + φ i+1,j k + φ i,j 1 k + φ i,j+1 k [1 φ i,j k] [1 ρ i,j k] w φ i,j k[4 φ i 1,j k + φ i+1,j k φ i,j 1 k + φ i,j+1 k ρ i 1,j k + ρ i+1,j k ρ i,j 1 k + ρ i,j+1 k + φ i+1,j kρ i+1,j k + φ i 1,j kρ i 1,j k + φ i,j+1 kρ i,j+1 k + φ i,j 1 kρ i,j 1 k] 5

2 2 Assume the concentration of the tagged particles to be small, namel ρ i,j 1. Then the following approimated relations are found: ρ i,j k + 1 ρ i,j k = qρ i 1,j k + ρ i+1,j k + ρ i,j 1 k + ρ i,j+1 k [1 φ i,j k] qρ i,j k[4 φ i 1,j k + φ i+1,j k + φ i,j 1 k + φ i,j+1 k] φ i,j k + 1 φ i,j k = wφ i 1,j k + φ i+1,j k 2φ i,j k + φ i,j 1 k + φ i,j+1 k 2φ i,j k After introducing the continuous variables ρ,, t = lim ρ i,jk, φ,, t = lim φ i,jk a, t 0 a, t 0 6 where a and t respectivel stand for the linear size of the lattice site and the characteristic time step of the microscopic dnamics. B defining the diffusion coefficients according to the standard practice see main bod of the paper, we get the sought generalized model: ρ = 2 [D ρ 1 φρ] + 2D ρ ρ φ φ = D φ 2 φ 7 As anticipated, we shall now turn to discussing an alternative derivation of the above equation. To this end, we define the variable 1, if the site i, j is occupied b an agent of the species m at time step k γ i,j k = 2, if the site i, j is occupied b an agent of the species n at time step k 0 if the site i, j is empt at the time step k 8 We write then the master equation for the motion of the tagged particles as: P 1 γ i,j k + 1 = 1 P 1 γ i,j k = 1 = α [P 2 γ i,j k = 1, γ i+1,j k = 0 + P 2 γ i,j k = 1, γ i 1,j k = 0 ] + P γ 2 i,j k = 1, γ i,j+1 k = 0 + P 2 γ i,j k = 1, γ i,j 1 k = 0 + α [P 2 γ i,j k = 0, γ i+1,j k = 1 ] + P γ 2 i,j k = 0, γ i 1,j k = 1 + P 2 γ i,j k = 0, γ i,j+1 k = 1 + P 2 γ i,j k = 0, γ i,j 1 k = 1 9 where P 2 stands for a joint probabilit, while P 1 is the probabilit of a single event and α is the rate of success of the selected jump. The master equation for the population n is similar, the variable γ assuming the value 2 instead of 1 and β labeling the associated jump rate in principle the two population can have different jump probabilities. In the mean field limit, we factorize the joint probabilities P 2 in the master equations as in P 2 γ i,j k = 1, γ i,j+1 k = 0 = P 1 γ i,j k = 1 P 1 γ i,j+1 k = 0. To perform the continuum limit we emplo a Talor epansion for the three different probabilit functions P 1 as:

3 3 Pγ i±1,j k = 1 = ρ ± a ρ a2 2 ρ 2 + oa2 Pγ i,j±1 k = 1 = ρ ± a ρ a2 2 ρ 2 + oa2 Pγ i±1,j k = 2 = φ ± a φ a2 2 φ 2 + oa2 Pγ i,j±1 k = 2 = φ ± a φ a2 2 φ 2 + oa2 Pγ i±1,j k = 0 = µ ± a µ a2 2 µ 2 + oa2 Pγ i,j±1 k = 0 = µ ± a µ a2 2 µ 2 + oa2. 10 where again a represents the linear size of each site. Making use of the relation µ = 1 ρ φ and defining as usual we obtain αa 2 lim a, t 0 t = D ρ βa 2 lim a, t 0 t = D φ, ρ = D ρ1 φ 2 ρ + ρ 2 φ for the tagged species. Under the hpotesis of low concentration of the tagged agents, we finall get: φ = D φ 2 φ for the evolution of the bulk densit. II. O THE VALIDITY OF THE MEA FIELD APPROXIMATIO: COMPARIG STOCHASTIC AD MEA FIELD SIMULATIOS I 2D To test the adequac of the proposed mean field model we have performed a campaign of numerical simulations, with reference to the 2D setting. More specificall we have implemented a montecarlo scheme to solve the stochastic process under scrutin and so trace the evolution of the tagged particle in time. At each time iteration, all crowders and the tagged particle can update their position, moving at random, and with equal probabilit, in one of the four allowed directions, provided the selected target site is unoccupied. If the site at destination is occupied, the move is rejected and the particles keep their original positions. The order of selection of the particles is, at each iteration, randomized. B averaging over man independent realizations, one can reconstruct the normalized histogram of the position visited b the crowders at a given time t and compare it with the densit profile ρ obtained upon integration of the mean field sstem 7. To carr out the numerical integration of the above partial differential equations we assumed a forward difference approimation in time and replaced the spatial derivatives b centered approimations. The result of the comparison is reported in figures 1a and 1b, for two choices of the initial conditions. In figure 1a, the tagged particle is initiall positioned in the middle of a two dimensional waterbag, filled with crowders, with average densit equal to φ 0. In figure 1a, the tagged particle is instead positioned, at time t = 0, in the center of an empt region, a square of assigned size. The crowders are instead assumed to occup an adjacent domain with average uniform densit φ 0. In both cases, the agreement between stochastic and mean field simulations is satisfing. III. ALTERATIVE DERIVATIO OF THE MODEL USIG A COARSE GRAIED PICTURE In this final section we discuss an alternative derivation of model 10, which assumes a coarse-grained decription of the scrutinized problem. The derivation follows a different philosph: it is here carried out in one dimension, but readil generalizes to the relevant d = 3 setting. We consider the phsical space to be partitioned in Ω patches, also called urns. Each patch has a maimum carring capacit - it can be filled with particles at most. Labelling m i the number of tagged particles contained in urn i, and with n i the corresponding number of crowders, one can write: n i + m i + v i = where v i stands for the number of vacancies, the empt cases in patch i that can be eventuall filled b incoming particles. The ecluded-volume prescription is here implemented b requiring that particles can move onl into the nearest-neighbor patches that ehibit vacancies, i

4 ρ 0,2 tagged particles, of tpe the crowders or a vacanc belonging to the i-patch. This is stochastic process governed, under the Markov hpothesis, b a Master equation for the probabilit P n, m, t of finding the sstem in a given state specified b the 2Ω dimensional vector n, m = n 1,..., n Ω, m 1..., m Ω at time t. The Master equation reads: , P n,m,t = n n [T n, m n, mp n, m + 12 T n, m n, m P n, m T n, m n, mp n, m T n, m n, mp n, m] where T a b is the rate of transition from a state a to a compatible configuration b. The allowed transitions are those that take place between neighboring patches as dictated b the chemical reactions 11. For eample, the transition probabilit associated with the second of equations 11 reads ρ FIG. 1: Comparison between stochastic thin solid lines and mean field simulations thick solid line. The stocastic simulations are averaged over realizations. Here φ 0 = 0.5. Two dimensional snapshots of the stochastic simulations are also displaed. The crowders are plotted as small black online circles. The tagged particle is represented b the filled red online square. Upper panel: initial condition see first snapshot, top left panel originating a super-diffusive transient. Lower panel initial condition see first snapshot, top left panel, causing a sub-diffusive transient. as eemplified b the following chemical reactions zω M i + V j M j + V i 11 zω i + V j j + V i Here z is the number of nearest-neighbor patches and M i, i, V i are respectivel a particle of tpe M the T n i 1, n j +1 n i, n j = n i v j zω = n i zω 1 n j m j. 13 The transition rates bring into the equation an eplicit dependence on the amount of molecules per patch, the so-called sstem size. To proceed in the analsis, we make use of van Kampen sstem size epansion [? ], which enables one to separate the site-dependent mean concentration φ i t from the corresponding fluctuations ξ i in the epression of the discrete number densit of species. The fluctuations become less influent as the number of the agents is increased, an observation which translates in the following van Kampen ansatz: n i t = φ it + ξ i. 14 In the following we will also assume just one tagged particle, the analsis etending straightforwardl to the case where a bunch of diluted particles is assumed to be dispersed in the background of crowders. Since the tagged particle belongs to one of the patches, it is convenient to look at the evolution of P k n, t = P n, 0, 0,..., 0, 1, 0,..., 0, 0, t }{{}}{{} k 1 Ω k in the master equation 12. P k n, t is the probabilit that the target particle be in the k-patch, for a particular configuration n of species. The Master equation can be hence written in the following compact form

5 5 P k n, t = + j i 1,i+1 j i 1,i+1 ɛ j ɛ+ i 1T n i 1, n j + 1 n i, n j P k n i, n j, t 1 zω 1 n j P k + j i 1,i+1 1 zω 1 n k P j 15 where use has been made of the shift operators: equation 13 takes the form ɛ ± i f..., n i,... = f..., n i ± 1,... Under the van Kampen prescription [? ], one can epand the transition rates in power of 1/. For eample, T n i 1, n j + 1 n i, n j = { φ i 1 φ j + 1 [ ξi 1 φ j ξ j φ i ] + 1 zω [ ξ iξ j m j φ i ] + 1 [ ] } mj ξ 3 i 2 and also epress the shift operators in terms of differential operators: ɛ j ɛ+ i 1 = O 1 ξ j 2 ξ j [ ɛ j ɛ+ i 1T n i 1, n j + 1 n i, n j P n i, n j, t = 1 ξ j zω ξ i1 φ j + ξ j φ i Π k ξ, t [ ξ j ] zω φ i1 φ j Π k ξ, t 2 ] ξ j zω φ i1 φ j Π k ξ, t + O otice that m i / cannot be approimated as a continuum-like densit, the continuum limit being not appropriate for the case of a single tracer. We then define a new probabilit distribution Π k ξ, τ, function of the vector ξ and the scaled time τ = t Ω. In terms of the new probabilit distribution Π k ξ, τ the left hand side of 15 becomes The leading order contribution in 1 gives: 1 Ω Π k φ i = which ields Π k φ i = z and finall: zω j {i 1,i+1} φ i 1 φ j Π k Π k ξ j 16 Π k 2φ i φ i 1 φ i+1 17 P k = 1 Ω Ω Π k φi + 1 Π k Ω. φ i = 2 φ i 18 where is the discrete Laplacian operator defined as φ i = 2 z j i φ j φ i, where j i means a summation

6 6 over the sites, j, which are nearest-neighbors of site i. B taking the size of the patches to zero, one recovers the standard diffusion equation for species φ, in agreement with the result reported in the main bod of the paper. Consider now the following identities: z j {i 1,i+1} = z = z z = 2 ξ j ξ i 1 φ j ξ j φ i Π k ξ i 1 φ i 1 ξ i 1 φ i Π k + 1 z ξ i 1 φ i+1 ξ i+1 φ i Π k +1 ξ i 1 φ i 1 ξ i 1 φ i Π k + ξ i 1 φ i+1 ξ i+1 φ i Π k ξ i+1 1 φ i ξ i φ i+1 Π k z ξ i Π k ξ i 1 1 φ i ξ i φ i 1 Π k and z j {i 1,i+1} ξ j 2 φ i 1 φ j Π k = 2 φ z ξ 2 i 1 φ i 1 Π k φ i ξi 2 i 1 φ i+1 Π k φ ξi+1 2 i 1 φ i+1 Π k + φ 2 ξi 1 2 i 1 φ i 1 Π k φ 22 i 1 φ i 1 Π k φ 22 i 1 φ i+1 Π k 1 +1 = 2 2 z ξ 2 2φ i + φ i 1 + φ i+1 2φ i φ i+1 + φ i 1 Π k + 2φ i 1 φ i 1 Π k i φ i 1 φ i+1 Π k +1 Making use of the above relations, at the net to netto-leading corrections one eventuall gets Π k = ξ i Π k + i+1 B i,j Π k 2 ξ i 2z ξ i=i 1 i ξ j 1 φ k Π k 1 2 φ k+1 φ k 1 Π k + 1 φ k Π k+1 + z Here B represents the diffusion matri, whose entries are B i,i = 2φ i + φ i 1 + φ i+1 2φ i φ i+1 + φ i 1 B i,i 1 = 2φ i 1 φ i 1 B i,i+1 = 2φ i 1 φ i To provide a mean-field description of the inspected problem, we consider the probabilit function of the

7 7 tagged agent integrated over the fluctuations of the - particles. In formulae: ρ k t = Π k dξ whose evolution is governed b ρ k = z 1 φk ρ k 1 2 φ k+1 φ k 1 ρ k + 1 φ k ρ k+1 = 2 ρ k φ k ρ k + ρ k φ k The the last epression involves the discrete laplacian defined above. In the continuum limit, and considering a straightforward generalization to higher dimensions, one gets ρr, t = D ρ 1 φr, t 2 ρr, t + D ρ ρr, t 2 φr, t where D ρ is the diffusion coefficient of the tagged particle. One can finall write the non-linear equation for ρ as a Fokker Plank equation: ρr, t = D1 2 φr, tρr, t +2D ρr, t φr, t Hence, b neglecting the role of fluctuations, which amounts to operating in the mean-field limit, a nonlinear partial differential equation is found for the densit of the tagged species, coupled to a standard diffusion equation for the background densit: φr,t = D 2 φr, t ρr,t = 2 D1 φr, tρr, t + 2D ρr, t φr, t 20 This sstem constitutes the generalization of model 10 to higher dimensions. It is worth emphasising that the second of eqs. 20 can be also cast in the alternative form: ρr, t = D 1 φr, t ρr, t + Dρr, t φr, t