Emergent systems energy laws for predicting myosin ensemble processivity

Abstract

In complex systems with stochastic components, systems laws often emerge that describe higher level behavior regardless of lower level component configurations. In this paper, emergent laws for describing mechanochemical systems are investigated for processive myosin-actin motility systems. On the basis of prior experimental evidence that longer processive lifetimes are enabled by larger myosin ensembles, it is hypothesized that emergent scaling laws could coincide with myosin-actin contact probability or system energy consumption. Because processivity is difficult to predict analytically and measure experimentally, agent-based computational techniques are developed to simulate processive myosin ensembles and produce novel processive lifetime measurements. It is demonstrated that only systems energy relationships hold regardless of isoform configurations or ensemble size, and a unified expression for predicting processive lifetime is revealed. The finding of such laws provides insight for how patterns emerge in stochastic mechanochemical systems, while also informing understanding and engineering of complex biological systems.

Fig 1. Schematic of processive myosin system with dissociation.

Schematic of a myosin ensemble propelling actin at unloaded velocity v _u. Myosin states are stochastic, with myosins being detached, attached and power-stroking (light yellow point of contact), or attached and drag-stroking (dark red point of contact). Initially three myosins are attached (top); later the filament has translated and one myosin is attached (middle); at processivity termination, all myosins are detached (bottom).

Fig 2. Agent-based simulation of myosin systems.

(a) Logic rules that each individual myosin agent autonomously follows each step of the simulation. (b) Three rendered frames of myosin ensembles interacting with a single long actin filament. Myosins only generate force when attached to actin and based on their state generate positive force promoting filament motility (left pointing arrows) or negative force retarding motility (right pointing arrows).

Fig 3. Comparison of agent-based molecular simulation and analytical methods to experimental data.

(a) A datum isoform (squares) has k _on = 900s ⁻¹, k _off = 1600s ⁻¹, and δ ₊ = 5nm that correspond to empirical measurements [27], whereas extrapolated isoforms have one perturbed parameter each as labeled on the chart (e.g. the “k _on = 1500s ⁻¹” isoform has values k _on = 1500s ⁻¹, k _off = 1600s ⁻¹, and δ ₊ = 5nm). Each line corresponds to analytical outputs while symbols refer to simulation data, with the exception of solid circles that represent experimental data. (b) v _u as myosin isoforms vary for analysis and simulations, with each isoform normalized to one perturbed parameter as other parameters remain constant. The k _off perturbation (blue diamonds) has k _on = 900s ⁻¹, k _off = 3500s ⁻¹, and δ ₊ = 10nm, normalization; the k _on perturbation (orange rectangles) has k _on = 3500s ⁻¹, k _off = 1000s ⁻¹, and δ ₊ = 10nm normalization; the δ ₊ perturbation (pink triangles) has k _on = 900s ⁻¹, k _off = 800s ⁻¹, and δ ₊ = 13nm normalization. Experimental data corresponds to the δ ₊ [14] and k _off [33] values.

Fig 4. Stochasticity affects velocity and attachment among myosins and filaments for varied ensemble sizes.

Histograms for simulated ensembles of (a) N = 25 myosins and (b) N = 100 myosins when δ ₊ = 10nm, k _on = 900s ⁻¹, and k _off = 1600s ⁻¹ as the number of attached myosins are counted for 1000 random samplings (c) The normalized unloaded filament velocity v _u for the isoform from “a” when considering ensemble size for analytical predictions (N = 60 myosins), simulation (N = 60 myosins), and empirical data (normalized to 100μg/mL concentration of chicken skeletal myosins added to a flowcell). (d) Analytical curve of contact probability P _C and average number of attached myosins N _att for a median isoform of δ ₊ = 10nm, k _on = 2000s ⁻¹, and k _off = 2500s ^-1 as ensemble size varies. All simulated isoforms are identical to the median except for one perturbed parameter as indicated in the key, with all outputs collapsing on a single curve. Therefore, systems have nearly identical contact probabilities for a given number of attached myosins, independent of ensemble size or isoform configuration.

Fig 5. Processive myosin simulation rendering.

The rendering illustrates six periods of time during the agent-based simulation of a single processive run-length event. In the first (top) frame, no myosins are attached, then myosins begin attaching and propelling the filament until a period of time greater than 1ms when no myosins are attached, which leads to systems dissociation (bottom frame). The duration of time recorded for the run length event is measured from the initial point of myosin contact with a filament until system dissociation.

Fig 6. Trends among isoform variations for simulated processive lifetimes and ensemble energy usage.

(a) Simulation measurements of processive lifetime $\mathcal{P}$ and contact probability P _C when ensemble size N varies. Isoforms include a median (red with δ ₊ = 10nm, k _on = 2000s ⁻¹, and k _off = 2500s ⁻¹), with other isoforms having one perturbed parameter as indicated, therefore higher contact probabilities lead to longer $\mathcal{P}$, and lower detachment rates k _off lead to higher processive lifetimes for a given contact probability. (b) The minimum average system energy consumption E required for $\mathcal{P}\ge 500ms$. Isoforms are all perturbed from a configuration where parameters are half of their normalized value; isoforms of higher k _off require more energy to reach the same $\mathcal{P}$ for a given P _C.

Fig 7. Master curve that predicts processive lifetime of ensembles composed of many different myosin isoforms.

Processive lifetimes for isoforms when adjusted system energy consumption E* varies. Isoforms are identical to Fig 6A, except for additional low (δ ₊ = 5nm, k _on = 1000s ⁻¹, and k _off = 1500s ⁻¹) and high (δ ₊ = 15nm, k _on = 3000s ⁻¹, and k _off = 3500s ⁻¹) isoforms, which demonstrate the master curve holds as multiple myosin parameters are altered. The master curve analytically captures the overall response predicted through the unified expression $\mathcal{P}\mathcal{ }=\mathcal{ }A{e}^{B\bullet E^{*}}$, with A ≈ 14.5 and B ≈ 4.5(10⁻⁴). Here, processive lifetime is predictable regardless of individual myosin configuration and the master curve asymptotes are indicative of energy thresholds for perpetual processivity.