Active regeneration unites high- and low-temperature features in cooperative self-assembly

Robert Marsland, III and Jeremy L. England

Phys. Rev. E 98, 022411 – Published 31 August 2018

Abstract

Cytoskeletal filaments are capable of self-assembly in the absence of externally supplied chemical energy, but the rapid turnover rates essential for their biological function require a constant flux of adenosine triphosphate (ATP) or guanosine triphosphate (GTP) hydrolysis. The same is true for two-dimensional protein assemblies employed in the formation of vesicles from cellular membranes, which rely on ATP-hydrolyzing enzymes to rapidly disassemble upon completion of the process. Recent observations suggest that the nucleolus, p granules, and other three-dimensional membraneless organelles may also demand dissipation of chemical energy to maintain their fluidity. Cooperative binding plays a crucial role in the dynamics of these higher-dimensional structures, but is absent from classic models of one-dimensional cytoskeletal assembly. In this paper, we present a thermodynamically consistent model of active regeneration with cooperative assembly, and compute the maximum turnover rate and minimum disassembly time as a function of the chemical driving force and the binding energy. We find that these driven structures resemble different equilibrium states above and below the nucleation barrier. In particular, we show that the maximal acceleration under large binding energies unites infinite-temperature local fluctuations with low-temperature nucleation kinetics.

Received 1 November 2017
Revised 13 July 2018

DOI:https://doi.org/10.1103/PhysRevE.98.022411

Physics Subject Headings (PhySH)

Biomolecular self-assembly Nonequilibrium fluctuations

Biomolecules Cytoskeleton

Condensed Matter, Materials & Applied PhysicsStatistical Physics & ThermodynamicsPhysics of Living Systems

Authors & Affiliations

Robert Marsland, III^* and Jeremy L. England

Physics of Living Systems Group, Department of Physics, Massachusetts Institute of Technology, 400 Technology Square, Cambridge, Massachusetts 02139, USA

^*Email: marsland@bu.edu; current address: Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, MA 02215.

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Vol. 98, Iss. 2 — August 2018

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Images

Figure 1
Thermodynamically consistent model of actively regenerating cooperative self-assembly process. (a) Blue circles and red squares represent two distinct internal states of the assembling monomers with different binding energies ( $Δ G > 0$ and 0, respectively). Arrows represent binding or unbinding and state-changing reactions. A steady-state probability current is maintained around this cycle by coupling the state change to ATP hydrolysis. The circles are stabilized by binding to ATP (small circles), and the squares by binding to ADP (diamonds). The free energy of hydrolysis biases the state-change reaction of a bound circle towards the noninteracting square state. The square rapidly dissociates into the solution, where nucleotide exchange (double-headed arrows) is much faster than hydrolysis, and transforms the particle back to the circle state. The particle can now bind to the structure again, completing the cycle. (b) Shaded curves are steady-state probability distributions $p_{ss} (m)$ over the occupancy fraction $m$ , for increasing values of the system size $N$ , with $Δ μ = 0, β J = 8$ , and $c_{A} = 0.02$ . The black line is the effective free-energy density, defined as $f (m) = - {lim}_{N \to \infty} p_{ss} (m) / N$ .
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Figure 2
Active regeneration accelerates turnover and disassembly. (a) Local turnover rate $k^{off}$ in assembled phase for four values of $β J$ . $c_{A}$ and $Δ μ_{0}$ are tuned to the coexistence point, with $c_{I} = 0.0001$ . (b) Mean time $τ_{diss}$ for switching from high to low $m$ states under the same conditions, for a structure of size $N = 100$ . (c) Effective free-energy landscapes $f (m)$ on the $β J = 12$ curve at the four values of $Δ μ$ indicated by the dots with corresponding colors in panel (a). (d) Stochastic trajectories for $N = 100$ at the three nonzero $Δ μ$ values, plotted in the same colors.
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Figure 3
Driven steady state combines features of two equilibrium landscapes. (a) Effective free-energy landscape with $Δ μ = 6.5 k_{B} T$ , below the nonequilibrium phase transition (black), compared with the equilibrium landscape at the same temperature/coupling $β J = 12$ and monomer concentration $c_{A} = 8.1 \times 10^{- 3}$ (blue, left) as well as a modified equilibrium landscape with the same temperature but a smaller monomer concentration $c_{A} = 8.8 \times 10^{- 4}$ (red, right). (b) Same plots for $Δ μ = 15.2 k_{B} T$ , except that the modified equilibrium landscape (red, right) is obtained by making the temperature infinite, and setting the (constant) off-rate equal to $k_{I} = 0.01$ .
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Figure 4
Equilibrium approximations are generalizable. (a) Effective free energy of the driven system and the equilibrium free energy under a reduced monomer concentration, for $Δ μ$ up to 10 $k_{B} T$ and $β J = 12$ . (b) Same comparison, but with $β J = 14$ . (c) Effective free energy of the driven system and the equilibrium free energy at infinite temperature, for $Δ μ$ above 11 $k_{B} T$ and $β J = 12$ . (d) Same comparison, but with $β J = 14$ .
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