Active-Brownian Dynamics

Consider a particle undergoing random motion over a potential \(V\). For our development, we focus on 1-dimensional transport, but the final results are extended to 3 dimensions. The particle is subjected to thermal (emph{i.e.} Brownian) force \(f_{B}\) and active forces \(f_{A}\) that represent transient perturbations from the surrounding enzyme activity. The temporal evolution of the position \(x(t)\) is governed by a modified Langevin equation

(1)\[\xi \frac{d x}{d t} = f_{V} + f_{B} + f_{A},\]

where \(f_{V} = - \frac{\partial V}{\partial x}\) is the potential force on the particle at position \(x\).

The Brownian force \(f_{B}\) is governed by the fluctuation dissipation theorem, which states that \(f_{B}\) is a Gaussian-distributed random force that satisfies

\[\begin{split}\langle f_{B}(t) \rangle & = & 0, \\ \langle f_{B}(t) f_{B}(t') \rangle & = & \kappa_B ( |t - t'|) = 2 k_{B}T \xi \delta (t - t').\end{split}\]

This assumes that the environment is purely Newtonian, leading to the instantaneous decorrelation of Brownian forces (i.e. Gaussian white noise). Diffusive transport in a viscoelastic fluid leads to temporal memory in the Brownian force, reflecting temporal correlation in the frictional stress.

In our work, we assume the active force \(f_{A}\) are also Gaussian-distributed with an arbitrary temporal correlation, such that

\[\begin{split}\langle f_{A}(t) \rangle & = & 0, \\ \langle f_{A}(t) f_{A}(t') \rangle & = & \kappa_{A}(|t-t'|),\end{split}\]

where \(\kappa_{A}(t)\) represents the temporal correlation between active forces.

We develop a path integral formulation of the Active-Brownian particle that results in an expression for the joint probability \(\mathcal{P}[x(t)|x_0;t;f_A^{(0)}]\). This function governs the probability that if a particle begins at \(x_{0}\) experiencing an active force \(f_{A}^{(0)}\) at time \(t = 0\), the particle will be located at position \(x\) at time \(t\). Carrying out the integral over the active forces is performed by noting a Gaussian form of active forces

\[\mathcal{P}[f_A(t)]\propto \exp \left\{ -\frac{1}{2}\int_0^t dt_{1}\int_0^t dt_2 f_A(t_1)\kappa_A^{-1}(|t_1-t_2|)f_A(t_2) \right\}\]

This is used in the path integral formulation, and after functional integration over Brownian and Active forces, we arrive at the expression

\[\begin{split}& & \mathcal{P}[x|x_0;t,f_A^{(0)}, t_{0}] \nonumber \\ & & =\int_{x_0}^{x} \mathcal{D}[x(t)]\int \mathcal{D}[w(t)] \int d\eta \exp \left\{ -k_{B}T\xi\int_0^t [w(t_{1})]^2dt_{1}+i\xi\int_0^t \dot{w}(t_{1})x(t_{1})dt_{1} \right. \nonumber \\ & & \left. -i\xi w(t)x(t)+i\xi w(0)x(0)+i\int_0^t w(t_{1})f_V[x(t_{1})]dt_{1} -\frac{1}{2} \int_0^t \! \! dt_1 \int_{0}^{t} \! \! dt_2 w(t_1)\kappa_A(|t_1-t_2|) w(t_2) \right. \nonumber \\ & & \left. -i\eta f_A^{(0)} -\eta\int_0^t dt_1 \kappa_A(|t_1-t_0|)w(t_1)-\frac{1}{2}\eta^2\kappa_A(0) \right\}\end{split}\]

The theoretical development so far is general to any form of the spatially varying potential \(V(x)\) and does not assume any special form for the conservative forces between particles.

Dynamic behavior of an Active-Brownian polymer

We apply our results of an active-Brownian particle in a harmonic potential to the problem of an active-Brownian polymer chain. Since the normal-mode expansion of the Rouse polymer adopts the form of a particle in a harmonic potential, we can frame our results in the previous section to find the behavior of the pth normal mode.

We defined the normal-mode correlation function for the pth mode to be

\[\begin{split}C_{p}(\tau) & = & \langle \vec{X}_{p}(\tau) \cdot \vec{X}_{p'} (0) \rangle \\ & = & \frac{3 k_{B}T}{k_{p}} \left\{ \exp \left(-\frac{ p^{2} \tau}{N^2} \right) + \frac{\Gamma}{1- p^4/(K_{A}^{2} N^{4})} \left[ \exp \left(-\frac{p^{2} \tau}{N^2} \right) - \frac{ p^{2}}{K_{A} N^{2}} \exp \left( - K_{A} \tau \right) \right] \right\} \delta_{pp'}\end{split}\]

The dimensionless time \(\tau = t/t_{b}\) is scaled by the diffusive time \(t_{b} = b^{2} \xi/(k_{B}T)\), and we define the dimensionless Active force \(F_{A}^{2} = \Gamma K_{A} = f_{A}^{2} b^{2}/(k_{B}T)^{2}\) and dimensionless active rate constant \(K_{A} = t_{b} k_{A}\).

We find the center-of-mass mean-square displacement to be

(2)\[\mathrm{MSD}_{\mathrm{com}} = \langle \left( \vec{X}_{0}(t) - \vec{X}_{0}(0) \right)^{2} =\frac{6Nk_BT}{k_b}\Big[ \Big(\frac{1+\Gamma}{N^2}\Big)\tau+\frac{\Gamma}{N^2K_A}\Big(e^{-K_A \tau}-1\Big) \Big]\]

We note that Eq. (2) has the short-time asymptotic behavior \(\mathrm{MSD}_{\mathrm{com}} \rightarrow 6 [k_{B}T/(\xi N)] t\) as \(t \rightarrow 0\), which coincides with the mean-square displacement for a Brownian-only polymer. The long-time asymptotic behavior \(\mathrm{MSD}_{\mathrm{com}} \rightarrow 6 [k_{B}T/(\xi N)] (1 + F_{A}^{2}/2) t \) reveals the long-time effective temperature due to Brownian and active fluctuations, given by \(T_{AB} = T (1 + F_{A}^{2}/2)\).

The mean-square displacement of a segment of the polymer chain is define as

(3)\[\text{MSD}(\tau) = \langle ( \vec{r}(n,t) - \vec{r}(n,0))^{2} \rangle\]

for the chain position n. In this work, we focus on the midpoint motion at \(n=N/2\), but this is easily extended to other polymer positions. We insert our normal-mode representation into Eq. (3), resulting in the expression

\[\text{MSD}(\tau) = \text{MSD}_{\text{com}}(\tau) + 4 \sum_{p=1}^{\infty} \triangle C_{2p}(\tau)\]

where \(\triangle C_{p}(\tau) = C_{p}(0) - C_{p}(\tau)\). In addition, we define the mean-square change in distance (MSCD) for a polymer chain. This quantity is defined as

\[\mathrm{MSCD} = \langle \left( \Delta \vec{R}(t) - \Delta \vec{R}(0) \right)^{2} \rangle\]

where \(\Delta \vec{R}(t) = \vec{r}(N/2 + \Delta, t) - \vec{r}(N/2 - \Delta,t)\). Thus, MSCD is a measure of the change in separation distance between two points on the polymer that are \(2 \Delta\) segments from each other. This results in the expression

\[\text{MSCD}(\tau) = 16 \sum_{p=0}^{\infty} \triangle C_{2p+1}(\tau) \sin^{2} \! \left[ \frac{\pi(2p+1)\Delta}{2N} \right],\]

where \(\triangle C_{p}(\tau) = C_{p}(0) - C_{p}(\tau)\)

Functions contained with the ‘active_brown’ module

Active-Brownian dynamics

Notes

Detailed derivations for MSD and MSCD are found in “Interplay of active and thermal fluctuations in polymer dynamics”

wlcstat.active_brown.flow_spec_active(k, t, length_kuhn, ka, gamma, b=1, num_nint=1000, num_modes=20000)[source]
Parameters

  • k –

  • t –

  • length_kuhn –

  • ka –

  • gamma –

  • b –

  • num_nint –

  • num_modes –

wlcstat.active_brown.gen_conf_rouse_active(length_kuhn, num_beads, ka=1, gamma=0, b=1, force_calc=False, num_modes=10000)[source]

Generate a discrete chain based on the active-Brownian Rouse model

Parameters

  • length_kuhn (float) – Length of the chain (in Kuhn segments)

  • num_beads (int) – Number of beads in the discrete chain

  • ka (float) – Active force rate constant

  • gamma (float) – Magnitude of the active forces

  • b (float) – Kuhn length

  • num_modes (int) – Number of Rouse modes in calculation

Returns

r_poly – Conformation of the chain subjected to active-Brownian forces

Return type

(num_beads, 3) float

wlcstat.active_brown.gen_pymol_file(r_poly, filename='r_poly.pdb', ring=False)[source]
Parameters

  • r_poly ((num_beads, 3) float) – Conformation of the chain subjected to active-Brownian forces

  • filename (str) – File name to write the pdb file

  • ring (bool) – Boolean to close the polymer into a ring

wlcstat.active_brown.mscd_active(t, length_kuhn, delta, ka, gamma, b=1, num_modes=20000)[source]

Compute mscd for two points on an active-Brownian polymer.

Parameters

  • t ((M,) float, array_like) – Times at which to evaluate the MSCD

  • length_kuhn (float) – Length of the chain (in Kuhn segments)

  • delta (float) – Length of the chain between loci (in Kuhn segments)

  • ka (float) – Active force rate constant

  • gamma (float) – Magnitude of the active forces

  • b (float) – The Kuhn length (in desired length units).

  • num_modes (int) – how many Rouse modes to include in the sum

Returns

mscd – result

Return type

(M,) np.array<float>

wlcstat.active_brown.mscd_plateau_active(length_kuhn, delta, ka, gamma, b=1, num_modes=20000)[source]

Compute mscd plateau for two points on an active-Brownian polymer.

Parameters

  • length_kuhn (float) – Length of the chain (in Kuhn segments)

  • delta (float, array_like) – Length of the chain between loci (in Kuhn segments)

  • ka (float) – Active force rate constant

  • gamma (float) – Magnitude of the active forces

  • b (float) – The Kuhn length (in desired length units).

  • num_modes (int) – how many Rouse modes to include in the sum

Returns

mscd_plateau – result

Return type

(M,) np.array<float>

wlcstat.active_brown.msd_active(t, length_kuhn, ka, gamma, b=1, num_modes=20000)[source]

Compute msd for the midpoint points on an active-Brownian polymer.

Parameters

  • t ((M,) float, array_like) – Times at which to evaluate the MSCD

  • length_kuhn (float) – Length of the chain (in Kuhn segments)

  • ka (float) – Active force rate constant

  • gamma (float) – Magnitude of the active forces

  • b (float) – The Kuhn length (in desired length units).

  • num_modes (int) – how many Rouse modes to include in the sum

Returns

msd – result

Return type

(M,) np.array<float>

wlcstat.active_brown.phi_active(t, length_kuhn, ka, gamma, b=1, num_modes=20000)[source]
Parameters

  • t –

  • length_kuhn –

  • ka –

  • gamma –

  • b –

  • num_modes –

Returns

Return type

phia

wlcstat.active_brown.structure_factor_active(k, t, length_kuhn, ka, gamma, b=1, num_nint=1000, num_modes=20000)[source]
Parameters

  • k –

  • t –

  • length_kuhn –

  • ka –

  • gamma –

  • b –

  • num_nint –

  • num_modes –