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Effective temperature of active complex matter

We use molecular dynamics simulations to study the dynamics of an ensemble of interacting self-propelled semi-flexible polymers in contact with a thermal bath. Our intention is to model complex systems of biological interest. We find that an effective temperature allows one to rationalize the out of equilibrium dynamics of the system. This parameter is measured in several independent ways -- from fluctuation-dissipation relations and by using tracer particles -- and they all yield equivalent results. The effective temperature takes a higher value than the temperature of the bath when the effect of the motors is not correlated with the structural rearrangements they induce. We show how to use this concept to interpret experimental results and suggest possible innovative research directions.

study. These are systems with competing interactions between their constituents and no external energy input that in some conditions (such as sufficiently low temperature or high density) cannot equilibrate with their environment. Glassy systems are the typical example. An important outcome of the theoretical modeling of these systems, later confirmed with numerical simulations, is the generation of an effective temperature, T ef f [13]. T ef f should be understood as a time-scales dependent parameter which takes a thermodynamic meaning in systems with slow dynamics only [13,14]. Experimental measurements of T ef f in different glassy systems [15][16][17][18][19][20] have been performed.
In [21] we took a first step in the direction of analyzing T ef f in active matter. We studied the dynamics of an ensemble of self-propelled particles, meant to be a reasonable model for real but simple active matter such as bacterial colonies, with molecular dynamics simulations. Here we get closer to more complex cases and we study a model of filamentous semi-flexible polymers [22]. We analyze its structure and dynamics and we study T ef f by using a variety of independent measurements. In particular, we demonstrate that tracer particle techniques, that have already been exploited in the study of the mechanical properties of real biological systems [23,24], can also be useful to obtain a direct characterization of the out-of-equilibrium state. Details of the calculations will be given in a forthcoming publication [25].
The model.-We consider the model for a passive semiflexible linear polymers (filaments) of Ref. [26]. The optimal values of the parameters have been fixed by preliminary calculations [25,26]. Polymer chains are coarsegrained and each segment is formed by identical beads, with mass m and diameter r 0 . Each bead is in contact with a thermal environment that is described by a random noise and a viscous drag. The dynamical evolution at a bath temperature T is therefore controlled by a Langevin equation. The deterministic mechanical conservative force among beads is given by intramolecular and intermolecular contributions. Connectivity of the chains is assured by harmonic springs acting between nearestneighbor monomers. Chain rigidity is controlled by a purely repulsive interaction between next nearestneighbor monomers. The actual choice of parameters allows one to control the flexibility of the chains, ranging from semi-rigid to completely flexible [25,26].
Having in mind that no simple technique can take into account chemical activity in a molecular dynamics calculation, physical insight has been used to choose a reasonable model for motor action. Only a fixed fraction of polymers are provided with motors that have a gentle but still detectable effect on the behavior of the system. We localize the non-conservative motor activity on the monomer at the center of the polymer [27]. The motor effect is given by a time series of random isotropic kicks generated by a suitable stochastic process [21]. The action of the motors is independent of the structural rearrangements they induce (adamant [28] motors). The force of all motors, f M , is the same and a fraction of the mean conservative mechanical force acting on the equivalent passive system, F . We thus quantify the motor activity with the control parameter f = f M /F . More details on the model are given in Refs. [21,25]. We express all quantities in dimensionless reduced units [29] that we associate to reasonable values of the experimentally measured counterparts [30]. The typical unit of energy is 2k B T ; typical lengths are measured in units of 0.4 nm implying forces of order 20 pN at ambient temperature, T = 300 K.
Structure.-We first investigate the structure of the semi-flexible active polymer melt by keeping all external macroscopic conditions fixed and varying the motor activity f . The analysis of the static structure factors and pair distribution functions (not shown) is consistent with a picture in which the average nearest neighbor distance between beads pertaining to different polymers decreases upon increasing f . Overall, the motors have the mixed effect of making the melt more compact and more disordered simultaneously. This picture is corroborated by the extremely complex probability distributions of the gyration radius, R g , which are highly non-Gaussian and very asymmetric (Fig. 1). Motor activity has the effect of pushing the filaments closer and these, at the same time, fold substantially. We interpret this behavior in terms of a competition between two effects: crowding, which is re- lated to excluded volume effects and tends to slow down the dynamics, and folding of the chains, which dynamically lifts the topological constraints, therefore decreasing entanglement. These have important consequences on the long-time evolution of the system, as it is verified below.
Dynamics.-We next quantify the effect of motor activity on the long time dynamics. Figure 2 a) shows the self-intermediate scattering function, F s (Q, t), at the wave-vector corresponding to the main peak in the static structure factor and as a function of time for the investigated motor forces. The mean-squared displacement data (not shown) confirm the picture whereby the collective dynamics of the filaments gets faster under stronger motors. This is consistent with the gyration radius data and supports our interpretation: the augmented folding of the filaments seems to solve local topological constraints and diminishes the effect of entanglement. A quantitative analysis is presented in Fig. 2 b), where we plot the dependence of the diffusion coefficient and the relaxation time upon the strength of the motors. Quantities are shown renormalized to the value of the passive system. It is very interesting to note that the two Strikingly, the f -dependence of the diffusion coefficient is reminiscent of what was found in [31,32]. For the active colloidal particles used in these studies, the effective diffusion coefficient can be written as D/D o = (1 + 1/9P 2 e ), where D o is the diffusion coefficient of the passive system and P e is the Peclet number, which characterizes the particle activity [31,33]. In our case the exponent (2.3 ± 0.1) is close to 2, the differences being possibly ascribed to the uncertainties on the determination of D or to the presence of corrections to the proposed scaling. We speculate that here f plays the role of the Peclet number. A similar rationale should be valid for the striking linear dependence of the inverse of the relaxation time.
Effective temperatures.-The analysis above proves that both structure and dynamics of an active system are influenced by motor activity in a very complex fashion. We ask now whether it is possible to embed all this complexity in a single parameter, also prone to be directly determined in experiments. The chosen quantity is the effective temperature T ef f . One way of measuring the effective temperature in a non-equilibrium steady state consists in using the generalization of the fluctuationdissipation theorem to out-of-equilibrium conditions [13]. The most visual way of analyzing the data is to construct a parametric relation between the integrated linear response, χ AB (t), and the correlation C AB (t) of a wisely chosen pair of observables, A and B, and associating minus the inverse of its slope with a possibly time-dependent parameter T ef f (t) [13,34]. Such measurements were performed in different out-of-equilibrium conditions [35][36][37][38] demonstrating that T ef f (t) is (asymptotically) piecewise constant taking, typically, two values: the one of the environmental bath at short times and an additional one characterizing the structural relaxation at long times, in cases with slow dynamics.
We performed these measurements for different values of the forcing f and the results are shown in Fig. 3. (The needed observables are chosen such that the considered correlation functions are the F s (Q, t) of Figure 2 a). Details can be found in Ref. [36].) The curves clearly show two regimes. The first one, corresponding to high frequencies, represents the fast vibrations of the monomers, before any structural relaxation takes place. The second regime, at long times, describes the actual structural relaxation in the sample. In all cases, the parametric plot is rather well described by a straight line, implying that T ef f is a constant over this time-scale. These data are also shown in Fig. 4 with open (red) squares. The construction detailed above is very powerful, direct, and easily exploitable in computer simulations. In experiments, however, its implementation is far from obvious. This difficulty has been circumvented by considering the dynamics of tracers (or intruders), normally of spherical shape. These objects of micrometric size are nowadays quite simple to control and manipulate by active microrheology techniques. When immersed in an extended complex environment, the latter acts as an external reservoir and the intruders' dynamics provide a wealth of information on the underlying matrix state. In our simulations we have implemented these ideas in two ways.
A possible realization of an adapted thermometer that couples to the structural relaxation of the sample [13] is a free tracer particle with mass much larger than the one of the polymer beads. Its kinetic energy can be related to the system's effective temperature via an equipartition theorem [36,38]. Our results are shown in Fig. 4

as (blue) open circles.
Another measurement of T ef f is based on the comparison between (free) diffusion and (driven) mobility of tracer particles immersed in the active sample. Conform-ing to the Einstein relation, T ef f should be the parameter linking the tracers' mean-square displacement and their driven displacement under a weak external applied force, in the long time limit in which the dynamics is diffusive. Data for T ef f obtained in this way are included in Fig. 4 as open (green) triangles. Figure 4 collects all our results in the form T ef f /T against f . We also include (diamonds) average values with error bars calculated from the entire set of data. The figure demonstrates that independent determinations of T ef f yield consistent results and suggests that, in our system, there is a single frequency-independent T ef f parameter, with a precise thermodynamic meaning. Data tend to T ef f = T in the limit of vanishing activation, as they should, and the deviation from the environmental temperature monotonically increases with increasing forcing. Most importantly, data are represented by the empirical law T ef f /T = 1 + γf 2 , with γ = 15.41 (dashed line). Remarkably, this finding supports our conjecture that the parameter f plays here a role analogous to the Peclet number for colloidal active particles considered in a few very recent experiments [31,33].
In conclusion, we studied the out of equilibrium behavior of a system of active motorized filaments, a model for complex biological structures, using a genuine microscopic approach that is still quite innovative in the field. We described the influence of motor activity on both structure and dynamics, finding a rationale for very similar recent experimental observations. We confirmed the relevance of the effective temperature, a concept well established in statistical mechanics of glassy systems, for complex active matter. Most important, we demonstrated that this parameter, relatively simple to measure by microrehology techniques, can be directly related to chemical activity, a process of extraordinary complexity. It would be interesting to explore the possibilities offered by the molecular dynamics approach used in this paper to clarify the extremely complex phenomenology of cell mechanical stability [39,40]. In particular, one could try to give an answer to the question as to how mechanical properties (such as elastic moduli) change with motor activity and whether these changes can be rationalized in terms of an effective temperature.