Dynamic Model For The Collective Behavior Of Molecular Motors

Molecular motors are nanomachines that convert chemical energy into mechanical energy in cells.They are involved in almost all aspects of life activities,including intracellular cargo transport,signal transduction,and cell movement.The collective behavior of molecular motors is an important way to achieve their functions,and has become a hot topic in related research in recent years.Muscle contraction and beating of flagella and cilia,relying on the collective work of a large number of myosin and dynein,are two of the most typical collective behaviors of the molecular motors.Although many extensive and extensive researches have been done on this area,the underlining work mechanism is still uncertain.Spontaneous oscillatory contraction?SPOC?is considered to be the third state of the muscle,and its activation level is between the contraction and relaxation of the muscle.The experimental data show that the SPOC of the muscle is sensitive to the concentration of the chemical substances in solution.This means that the SPOC contains profound chemical mechanisms.Similar to the spontaneous oscillatory contraction of muscles,in vitro experiments about flagellar beat confirmed that the concentration of substrate?ATP?and products?ADP,Pi?and Ca²⁺,Mg²⁺plasma in solution had a significant effect on the waveform and period of flagellar beating.In order to explain the phenomenon of SPOC,many theoretical models have been proposed.These theoretical models take full account of the mechanical properties of myosin single molecules,as well as the structural characteristics of muscles,but rarely involve biochemical processes of muscle contraction.The theoretical research related to flagellar beatIing mostly focuses on the structural characteristics of the axial filament and the mechanical properties of the flagellar bending,and it is difficult to explain the influence of chemical factors on the flagellar bending.SPOC and flagellar beating are driven by myosin ? and dynein,respectively,which require the conversion of the chemical energy of ATP hydrolysis into the mechanical energy.Modeling the mechanochemical cycle of myosin ? and axonal dynein is an effective theoretical way to study the mechanical and chemical coupling mechanism of SPOC and flagellar beating.Based on this idea,the following works were done in this paper:?1?We first?to our knowledge?taken into account the effect of cooperativity between the two heads on the rates of the mechanochemical cycle,and proposed a mechanochemical model for myosin ?.This model gives all possible states of myosin ? dimer and the transition between these states.According to mass-action law,the proportions of enzyme species?the myosin ? dimmers in different states?were calculated.To our surprising,if we assume that attached myosin heads will help the partner heads rapidly attaching to the actin filament,the proportions of myosin ? dimers in different states will periodically change with time,which leads to the sustained oscillations of contractive tension and serves as the major contribution to SPOC.As a check on the model,the equation of force balance for SPOC under isotonic conditions was built,by which the time-dependent change in the sarcomere length was calculated.The simulation results were consistent with the experimental observations.The results show that the cooperative behavior between the two heads of myosin may be an intrinsic mechanism for the muscle contraction system,which makes the muscle contraction exhibit more abundant nonlinear dynamics.?2?Based on the fact that the myofilament lattice can significantly reduce the diffusion of adenine nucleotides,this paper established the reaction-diffusion equation about ATP.It was unexpected that the concentration of ATP inside myofilament periodically changes with time.The earlier theoretical works considered that the concentration of ATP in the whole muscle fiber was fixed,but recent experimental and theoretical studies confirmed that the complex lattice spacing inside the muscle fibers has a significant effect on the diffusion of ATP.The reaction diffusion equation established in this paper provides a theoretical analysis tool for related researches,which needs to be verified and corrected by experiments.?3?Quantitative analysis of the binding probability of myosin to actin filament and its related chemical reaction rate constants is very important to understand the intrinsic mechanism of muscle contraction.In this paper,based on the experiment results of spontaneous oscillatory contraction of muscle?SPOC?,the quantitative relationship of the binding probability to the concentration of solution,sarcomere length and the sliding speed of actin filaments are deduced from the kinetic equation describing SPOC.Mathematical law reflected the change of the chemical reaction rate with the mucle contraction velocity is established also.From the results we obtain the follow three conclusions:one is that the baseline value of the binding probability is determined by the concentration of the main chemical components in solution,one is that the binding probability is directly proportional to the sliding speed and inversely proportional to the sarcomere length,and another is that the chemical reaction rate is exponentially related to the contraction speed.?4?Cosidering the influence of the force generated by the deformation of flagellar bending on the chemical reaction rate of axonal dynein,a three-state mechanochemical cycle model of single dynein head was established.Based on the cooperativity between double headed,the six-state mechanochemical cycle model was proposed.Numerical simulations show that the period of flagellar beat decreases wiht the increasing concentration of Ca²⁺,which is consistent with the results observed in the experiment.The simulation results also showed that the average value of the ATP concentration inside the flagella changed with time,similar to the change in ATP concentration in the muscle fibers during SPOC.It is suggested that the diffusion of substrates and products in cell may be a common phenomenon.