# A Numerical Integrator for Forward Dynamics Simulations of Folding Process for Protein Molecules Modeled as Hyper-Redundant Robots

Amal Kacem, Khalil Zbiss, Alireza Mohammadi

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Amal Kacem, Khalil Zbiss, Alireza Mohammadi. A Numerical Integrator for Forward Dynamics Simulations of Folding Process for Protein Molecules Modeled as Hyper-Redundant Robots. 2023 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2023), IEEE Robotics and Automation Society (RAS), Oct 2023, Detroit, United States. hal-04216078

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Submitted on 23 Sep 2023

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# A Numerical Integrator for Forward Dynamics Simulations of Folding Process for Protein Molecules Modeled as Hyper-Redundant Robots

Amal Kacem, Khalil Zbiss, and Alireza Mohammadi†

**Abstract**— This paper investigates development of an efficient numerical integrator for forward dynamics simulation of the protein folding process, where protein molecules are modeled as robotic mechanisms consisting of rigid nano-linkages with many degrees-of-freedom. To address the computational burden associated with fixed step-size explicit Euler methods, we develop a fast numerical scheme with an adaptive step-size strategy for computing the folding pathway of protein molecules.

## I. INTRODUCTION

Developing efficient numerical integrators plays a crucial role in forward dynamics simulations of many emerging robot models such as continuum and soft robots (see, e.g., [1]). Protein molecules, according to the kinetostatic compliance method (KCM), can also be modeled as a large number of rigid nano-linkages folding under the effect of interatomic forces [2]–[4]. The KCM framework, which is a computationally promising approach to numerical simulation of folding dynamics, relies on the explicit Euler integration scheme with fixed step-size to integrate the protein dynamics towards a steady-state associated with a folded molecule conformation. However, each integration step requires heavy computations of interatomic force fields. Accordingly, the explicit Euler method with fixed step-size, which requires a larger than needed number of iterations for convergence to steady-state, imposes an unnecessary computational burden.

In this paper, we address the aforementioned shortcoming by utilizing the pseudo-transient continuation (ΨTC) framework [5] and developing a fast numerical integrator with an adaptive step-size control strategy for computing the folding pathway of protein molecules evolving according to the KCM-based dynamics.

## II. THE EXPLICIT ΨTC NUMERICAL INTEGRATOR

The flow chart of the explicit ΨTC numerical scheme with step-size adaptation tailored to the underlying KCM-based folding dynamics is depicted in Figure. We first present the explicit ΨTC solution to PFPCP under the assumption of a fixed step-size and discuss its convergence and numerical stability properties 1. The step-size adaptation rule of the numerical integrator is based on the so-called *switched evolution relaxation* (SER) technique that is widely used in the ΨTC literature. Figure 2 depicts the free energy $G(\theta)$ of the protein molecule during the folding process associated with a protein backbone chain with a configuration vector of

The authors are with the University of Michigan, Dearborn. This work is supported by the National Science Foundation (NSF) under Award CMMI-2153744. † Corresponding author: A. Mohammadi, Email: amohmmad@umich.edu

dimension 32, as well as its transient conformations, and the associated step-size evolution, namely, $\delta_k$.

<figure><img src="image_2.png" /><figcaption>Fig. 1: The flow chart of the proposed explicit ΨTC scheme for integrating the KCM-based forward dynamics of protein molecules modeled as hyper-redundant serial robotic mechanisms.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 2: The free energy of the backbone chain of a protein molecule with a 32-dimensional dihedral angle vector (blue curve; $G(\theta)$) on the right axis), its transient conformations, and the step-size of the explicit ΨTC scheme (black curve; $\delta_k$ on the left axis). The five plotted backbone chain conformations from left to right correspond to iterations 100, 300, 500, 1000, and 1400, respectively.</figcaption></figure>

## REFERENCES

[1] J. Till, V. Aloi, and C. Rucker, “Real-time dynamics of soft and continuum robots based on Cosserat rod models,” *Int. J. Robot. Res.*, vol. 38, no. 6, pp. 723–746, 2019.

[2] P. Tavousi, M. Behandish, H. T. Ilieş, and K. Kazerounian, “PROTO-FOLD II: Enhanced model and implementation for kinetostatic protein folding,” *J. Nanotechnol. Eng. Med.*, vol. 6, no. 3, 2015.

[3] A. Mohammadi and M. W. Spong, “Quadratic optimization-based nonlinear control for protein conformation prediction,” *IEEE Control Syst. Lett.*, vol. 6, pp. 373–378, 2022.

[4] —, “Chetaev instability framework for kinetostatic compliance-based protein unfolding,” *IEEE Control Syst. Lett.*, vol. 6, pp. 2755–2760, 2022.

[5] T. S. Coffey, C. T. Kelley, and D. E. Keyes, “Pseudotransient continuation and differential-algebraic equations,” *SIAM J. Sci. Comput.*, vol. 25, no. 2, pp. 553–569, 2003.