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1. A Numerical Integrator for Kinetostatic Folding of Protein Molecules Modeled as Robots with Hyper Degrees of Freedom

   https://doi.org/10.3390/robotics13100150  

   Kacem, Amal; Zbiss, Khalil; Mohammadi, Alireza (October 2024, Robotics (Special Issue Bioinspired Robotics: Toward Softer, Smarter and Safer))

   The kinetostatic compliance method (KCM) models protein molecules as nanomechanisms consisting of numerous rigid peptide plane linkages. These linkages articulate with respect to each other through changes in the molecule dihedral angles, resulting in a kinematic mechanism with hyper degrees of freedom. Within the KCM framework, nonlinear interatomic forces drive protein folding by guiding the molecule’s dihedral angle vector towards its lowest energy state in a kinetostatic manner. This paper proposes a numerical integrator that is well suited to KCM-based protein folding and overcomes the limitations of traditional explicit Euler methods with fixed step size. Our proposed integration scheme is based on pseudo-transient continuation with an adaptive step size updating rule that can efficiently compute protein folding pathways, namely, the transient three-dimensional configurations of protein molecules during folding. Numerical simulations utilizing the KCM approach on protein backbones confirm the effectiveness of the proposed integrator. 

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2. Predicting Protein Folding Pathways with Quadratic Constraints on Rates of Entropy Change: A Nonlinear Optimization-Based Control Approach

   https://doi.org/10.1115/1.4068504  

   Mohammadi, Alireza; Spong, Mark W (April 2025, Journal of Dynamic Systems, Measurement, and Control)

   Abstract This paper addresses the problem of algorithmic prediction of protein folding pathways, namely, the transient three-dimensional conformations of protein molecules during folding, under constrained rates of entropy change. We formulate the physics-based prediction of folding pathways as a control synthesis problem, where the control inputs guide the protein folding simulations. These folding control inputs are obtained from largescale trust-region subproblems (TRS) utilizing a computationally efficient algorithm with no need for outer iterations. The proposed control synthesis approach, which leverages the solutions obtained from a special generalized eigenvalue problem, avoids potentially cumbersome and unpredictable iterative computations at each protein conformation. Moreover, the TRS-based control inputs align the closed-loop dynamics closely with the kinetostatic compliance method (KCM) reference vector field while satisfying ellipsoidal constraints on the folding control inputs. Finally, we provide conditions for existence and uniqueness of the resulting closed-loop solutions, which are the protein folding pathways under constraints on the rate of entropy change. Numerical simulations utilizing the KCM approach on protein backbones confirm the effectiveness of the proposed framework. 

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3. Sign Gradient Descent Algorithms for Kinetostatic Protein Folding

   https://doi.org/10.1109/MARSS58567.2023.10294128  

   Mohammadi, A; Al_Janaideh, M (October 2023, IEEE)

   This paper proposes a sign gradient descent (SGD) algorithm for predicting the three-dimensional folded protein molecule structures under the kinetostatic compliance method (KCM). In the KCM framework, which can be used to simulate the range of motion of peptide-based nanorobots/nanomachines, protein molecules are modeled as a large number of rigid nano-linkages that form a kinematic mechanism under motion constraints imposed by chemical bonds while folding under the kinetostatic effect of nonlinear interatomic force fields. In a departure from the conventional successive kinetostatic fold compliance framework, the proposed SGD-based iterative algorithm in this paper results in convergence to the local minima of the free energy of protein molecules corresponding to their final folded conformations in a faster and more robust manner. KCM-based folding dynamics simulations of the backbone chains of protein molecules demonstrate the effectiveness of the proposed algorithm. 

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4. Protamine loops DNA in multiple steps

   https://doi.org/10.1093/nar/gkaa365  

   Ukogu, Obinna A; Smith, Adam D; Devenica, Luka M; Bediako, Hilary; McMillan, Ryan B; Ma, Yuxing; Balaji, Ashwin; Schwab, Robert D; Anwar, Shahzad; Dasgupta, Moumita; et al (May 2020, Nucleic Acids Research)

   null (Ed.)

   Abstract Protamine proteins dramatically condense DNA in sperm to almost crystalline packing levels. Here, we measure the first step in the in vitro pathway, the folding of DNA into a single loop. Current models for DNA loop formation are one-step, all-or-nothing models with a looped state and an unlooped state. However, when we use a Tethered Particle Motion (TPM) assay to measure the dynamic, real-time looping of DNA by protamine, we observe the presence of multiple folded states that are long-lived (∼100 s) and reversible. In addition, we measure folding on DNA molecules that are too short to form loops. This suggests that protamine is using a multi-step process to loop the DNA rather than a one-step process. To visualize the DNA structures, we used an Atomic Force Microscopy (AFM) assay. We see that some folded DNA molecules are loops with a ∼10-nm radius and some of the folded molecules are partial loops—c-shapes or s-shapes—that have a radius of curvature of ∼10 nm. Further analysis of these structures suggest that protamine is bending the DNA to achieve this curvature rather than increasing the flexibility of the DNA. We therefore conclude that protamine loops DNA in multiple steps, bending it into a loop. 

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5. Error estimates for a linear folding model

   https://doi.org/10.1093/imanum/drad004  

   Bartels, Sören; Bonito, Andrea; Tscherner, Philipp (February 2023, IMA Journal of Numerical Analysis)

   Abstract An interior penalty discontinuous Galerkin method is devised to approximate minimizers of a linear folding model by discontinuous isoparametric finite element functions that account for an approximation of a folding arc. The numerical analysis of the discrete model includes an a priori error estimate in case of an accurate representation of the folding curve by the isoparametric mesh. Additional estimates show that geometric consistency errors may be controlled separately if the folding arc is approximated by piecewise polynomial curves. Various numerical experiments are carried out to validate the a priori error estimate for the folding model. 

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