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A quantum compatible classical continuum model explains mechanical responses of cell membranes and membrane crosslinkers

Jichul Kim

Published Dec 21, 2024• 0 citations
Publisher

Abstract

Abstract Both classical mechanics and quantum mechanics explain the Brownian motion. However, it is unclear whether they are compatible with each other as the physical and mathematical identity of the wavefunction in quantum mechanics has been elusive. Here, a continuum theory using grammars in classical mechanics modeling but compatible with the quantum wavefunction is introduced. The theory explains the confined Brownian motion of cell membrane inclusions interacting with extracellular matrices or cytoskeletons via elastic molecular crosslinkers. This crosslinker theory is combined into the Canham-Helfrich-Evans model for fluid membranes. Calculations through the provision of a finite element method for the combined theory reproduced measured data from adhesion molecular machineries and cell membranes. Overall, by providing physical and mathematical interpretations of the quantum wavefunction, the presented theoretical model provides improved capabilities for the realistic simulation of classical and quantum biomechanical aspects of cell membranes and membrane linker proteins.

DOI
10.1101/2024.12.20.628332

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Abstract Both classical mechanics and quantum mechanics explain the Brownian motion. However, it is unclear whether they are compatible with each other as the physical and mathematical identity of the wavefunction in quantum mechanics has been elusive. Here, a continuum theory using grammars in classical mechanics modeling but compatible with the quantum wavefunction is introduced. The theory explains the confined Brownian motion of cell membrane inclusions interacting with extracellular matrices or cytoskeletons via elastic molecular crosslinkers. This crosslinker theory is combined into the Canham-Helfrich-Evans model for fluid membranes. Calculations through the provision of a finite element method for the combined theory reproduced measured data from adhesion molecular machineries and cell membranes. Overall, by providing physical and mathematical interpretations of the quantum wavefunction, the presented theoretical model provides improved capabilities for the realistic simulation of classical and quantum biomechanical aspects of cell membranes and membrane linker proteins.
The connection between the Schrödinger equation and Einstein's diffusion theory based on the Brownian motion of independent particles is well known. However, in contrast to diffusion theory, quantum mechanics theory has suffered controversial interpretations due to the counterintuitive concept of wavefunction. Here, while we confirm there is no difference in the mathematical form of these two equations, we derive the imaginary version of displacement. Using the diffusion theory of particles in a medium, as simple as it is, we describe that quantum mechanics is just an elegant and subtle equation to describe the probability of all the trajectories that a particle can take to propagate in time by a predictive wavefunction. Therefore, information on the position of particles through time in quantum theory is embedded in the wavefunction, which predicts the evolution of an ensemble of individual Brownian particles.
In two prior papers of this series, it was proposed that a wavefunction model of a heavy particle and a collection of light particles might generate ``Brownian-Motion-Like" trajectories as well as diffusive motion (displacement proportional to the square-root of time) of the heavy particle, but did not exhibit a concrete instance. Here we introduce a one-space-dimensional model which, granted a finite perturbation series, fulfills the criteria for BML trajectories and diffusion. We note that Planck's constant and the molecular mass of light particles make an appearance in the diffusion coefficient, which further differentiates the present theory from the work of Poincar{é} and Einstein in the previous century.
The additivity of classical probabilities is only the first in a hierarchy of possible sum rules, each of which implies its successor. The first and most restrictive sum rule of the hierarchy yields measure theory in the Kolmogorov sense, which is appropriate physically for the description of stochastic processes such as Brownian motion. The next weaker sum rule defines a generalized measure theory which includes quantum mechanics as a special case. The fact that quantum probabilities can be expressed "as the squares of quantum amplitudes" is thus derived in a natural manner, and a series of natural generalizations of the quantum formalism is delineated. Conversely, the mathematical sense in which classical physics is a special case of quantum physics is clarified. The present paper presents these relationships in the context of a "realistic" interpretation of quantum mechanics.
Predicting chemical reactions in biomolecules is challenging because reactive centers are embedded within complex environments that strongly influence their energetics and mechanisms. Here, we present a hybrid quantum-classical workflow that combines molecular dynamics, density functional theory, and correlated wavefunction methods, tightly integrating CPUs, GPUs, and quantum processing units (QPUs) to enable simulations across multiple length and accuracy scales. We apply this approach to ADGRG1 autoproteolysis, a maturation reaction in an adhesion G protein-coupled receptor that has not previously been studied using correlated electronic-structure methods. Our simulations treat a biologically realistic system containing more than one million atoms, including the full-length uncleaved receptor embedded in an endoplasmic-reticulum membrane, while incorporating a 54-qubit IQM quantum processor and up to 1,200 NVIDIA H100 GPUs. These results demonstrate how quantum computers can serve as accelerators within scalable scientific workflows, extending correlated electronic-structure methods to biologically relevant systems.
Classical mechanics describes the laws of motion in the macroscopic material world, while quantum mechanics describes the laws of motion in the microscopic material world that classical mechanics cannot explain, and achieves a highly accurate mathematical representation of the laws of microscopic physical motion. But even with such a successful theory, there is confusion about the probability wave. Therefore, this paper attempts to improve the physical definition of the conceptual basis of quantum mechanics, thus solving the confusion of quantum mechanical probability waves, and finally proposes a feasible interpretation of the wave-particle duality origin of microscopic matter particles.
The explanation of quantum mechanics has never been satisfactory. The successful application of quantum mechanics is generally attributed to the mathematical system of quantum mechanics. The application research of quantum encryption communication and quantum computers using quantum entanglement is very difficult. The binary opposition between quantum and classical has hindered human scientific thinking. The widespread quest to quantize gravity mirrors humanity's enduring drive to unify quantum and macroscopic physical frameworks. This research initiates its inquiry by integrating quantum mechanics with classical mechanics, yielding novel analytical perspectives, and introduces the Schrödinger-Tu equation—a wave function that incorporates gravitational potential energy and is capable of describing macroscopic objects. Validating the practical viability of merging quantum and classical approaches, several computational cases involving atoms and molecules yield successful outcomes. Additionally, this study derives a wavemechanical formulation of classical mechanical laws, furnishing direct empirical evidence that quantum mechanics and classical mechanics are not mutually exclusive but rather compatible and complementary. Further substantiation of their compatibility encompasses the following key points: The mass term m in the Schrödinger equation (SE) can be scaled to sufficiently large values to characterize macroscopic entities; The steady-state SE (Tψ+Vψ=Eψ) inherently integrates the wave function ψ with the classical energy conservation relation T+V=E; Potential energy functions within the SE can be derived from macroscopic force fields; Newton's second law F=ma and the SE are mutually derivable, establishing a fundamental theoretical linkage; Electron diffraction experiments concurrently demonstrate wave-particle duality, a phenomenon that bridges both mechanical frameworks. This paradigm shift—recasting the relationship between quantum mechanics and classical mechanics from one of incompatibility to compatibility—represents a pivotal theoretical advancement, with the potential to catalyze the development of a generalized wave mechanics.