Physical mechanisms of exit dynamics in microchannels of nonequilibrium transport systems

1. Introduction

Simulations of stochastic biochemical transport processes mainly focus on multiple exit kinetic system constructions and exit kinetic property discussions.^1,2,3,4,5 As for such systems, the impacts of tremendous initial conditions are vital.⁶ Substrates can flow along different product directions.⁶ Moreover, the working process of the whole system yields to fruitful nonlinear physical models.^7,8,9,10,11 For instance, chemical substrate nucleotide small molecules enter from the same entrance, and can eventually get a variety of products like DNA and RNA.¹² Stimulated by different antigen proteins, T cells of the immune system can be activated in different degrees and directions.¹³ During these processes, the determination of exit kinetic properties of the system directly affects the known and potential molecular biochemical processes due to the system complexity.¹⁴

Based on the fact that studies on dynamical mechanisms of complex systems are always hot issues in nonlinear science,^15,16,17 the kinetics mechanisms of the above-mentioned systems should be emphasized. Mechanisms of these kinetics are mainly studied through a single time scale.⁵ However, the single time scale used in classical kinetic systems usually ignores molecule interactions and the interaction between exit properties and properties of the main part.¹⁸ Thus, developing more effective time scales and emphasizing multiple exit kinetics of biochemical reaction kinetic systems become a key scientific problem that needs to be solved urgently.^19,20,21,22

As for the study of multiple exit dynamics models, average exit time and flow in the multiple exit dynamics model were initially studied and served as different time scales.²³ Exit dynamics properties in the multiple exit dynamics model were found not to be independent, but affected by the conversion rate of other adjacent microchannels.²³ Related physical mechanisms were explained from classical mechanics.²³ Then, the molecular transition path was studied, which indicated the short and rare shift of molecule positions in the transition process.²⁴ Transition time under equilibrium conditions was obtained, which showed reversal symmetry and was independent of reaction direction.²⁴ At the same time, the forward or backward symmetry failure was found to occur only when two conditions were met at the same time.²⁴ On the basis and premise of the above two predecessors’ researches,^23,24 the transition of particles at the diffusion points at both ends of two-dimensional conical diffusion was studied.²⁵ Such trajectory was found to be divided into cycle and transition path segments.²⁵ Brownian dynamics simulations were performed to reveal that transition path time was not sensitive to transition direction.²⁵

Beyond previous work emphasizing dynamic models and properties of non-Markov processes,^23,24,25 a continuous-time Markov chain with homogeneous exit distribution and approximate value time of occupancy measure related to the time of exiting the domain was solved by applying the exit time finite state projection method.²⁶ The first passage time related to gene expression and the fixed time of competitive species affected by demographic noise were reported.²⁶

Recently, multiple exit dynamic system models were applied to stochastic biochemical processes.^{27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49} Biochemical signal networks dealing with the inevitable quantitative fluctuations in molecular components,²⁷ the influence of the interaction in the channel between particles on transmission characteristics of various species,²⁸ random channel transmission of two substances,²⁹ relevant characteristics of antimicrobial peptide (AMP) on bacterial clearance kinetics,³⁰ a set of general biochemical network theory with arbitrary internal waiting time distribution,³¹ ligand selection,³² ribosome and T cell receptor,³³ protein multi-scale dynamic analyses and the boundary average first passage time and reachability probability of group continuous-time Markov chains³⁴ were measured via applying discrete-state stochastic method,²⁷ state space method,²⁸ complementary master equation methods,²⁹ active and passive CTRW,³¹ macro-data characterizing mechanical reaction networks and reaction fluxes³³ and martingale theory and stochastic process technology.³⁴ Discrete-state stochastic framework with first channel analyses studying trade-off among typical enzyme attributes,²⁸ particle interaction determining the random transition,²⁹ the heterogeneity of AMP action adjusting the entrance rate,³⁰ non-Markov biochemical reaction processes on network,³¹ RIG-I ligand selection,³² functional importance of pathway heterogeneity and dynamic selection and constraints on the statistical moment passing through the time distribution for the first time³⁴ were found.

More recently, kinetic networks and related dynamical properties were addressed.^{8,9,10,11,45,46,47,48,49} Dynamical models about flagellar length,⁸ AMPs and bacteria in single and population views,⁹ error corrections,¹⁰ tRNA aminoacylation,¹¹ SARS-CoV RdRp complex,⁴⁵ proofreading scheme,⁴⁶ dynamic proofreading,⁴⁷ dynamic proofreading network⁴⁸ and nonequilibrium proofreading system⁴⁹ were reported. Fruitful and interesting results like level-crossing statistics,⁸ theoretical framework,⁹ MFPT,^10,11 cost and benefit effects,^45,46 binding energy effects,^47,48 catalytic effects⁴⁹ etc.

However, although previous studies have made some contributions in studying the time scale of molecular dynamics system and particle transport orbital system,^{27,28,29,30,31,32,33,34,35,36,37} it is still urgent to solve the following key scientific problems, that is, overcoming previous researches^23,28,30,37 that usually consider only one or two orbits for the multi-orbital particle transport system, only the influence of adjacent lattices (i.e., assuming that dynamic properties of a particle in lattice k are only related to the dynamic properties of lattices $k+1$ and $k-1$) on the premise that particles start from the initial lattice and reach the final lattice without passing through any other lattices and only a fixed parameter in the evolution of the exit dynamic properties of the system while other parameters are set as unit one. Therefore, in order to beyond the above-mentioned limitations and expand the system universality, establishing new systems mapped into multiple microchannels, proposing typical dynamic parameters (i.e., the average exit time, etc.) and exploring exit dynamic properties of multi-exit dynamic systems affected by various dynamical parameters and evolution laws of exit dynamic properties are presented in this work. Innovations of this work are mainly reflected in the following aspects:

(1)	Beyond previous studies, newly established systems expand from three parallel microchannels to countable parallel ones. Besides, countable parallel orbit particle transport systems with the external particle source are in the state of unique connection and all connections are also established. Exit dynamics corresponding to these two subsystems are studied independently. At the same time, in order to further reflect the generality of the established new systems and their consistency with general biochemical processes such as enzymatic reaction and particle transport process, intermediate lattices are set up. Countable parallel orbits are taken as the subsystem of each initial lattice, which fully considers the dynamic properties of external particle sources to better reflect the nature and evolution of the general biochemical process and particle transport process.
(2)	Different from previous studies, by abandoning the traditional method that mainly discusses the dynamic properties of adjacent lattices, the influence of all lattices on the particle transport path on dynamic properties of lattices is fully considered via combining practical physical processes with classical mechanical views so as to reflect the universality and potential practical applications of established systems.
(3)	Different from previous work, three groups of parameters are chosen as the basis to analyze the dynamic properties of systems through the change of average motion time. High-priority parameters are introduced to compare their influences on the average motion time, including the comparison of the right conversion rate and left conversion rate of the same track in the transverse direction and the comparison of left and right conversion rates of different tracks in the longitudinal direction. Dependence mechanisms of average motion time on these critical parameters are found.
(4)	Different from previous studies, random values of critical parameters dominating system dynamics are emphasized to better illustrate stochastic dynamics of general biochemical processes and particle transport processes. New dynamic properties are found to support the rationality and effectiveness of the established systems, which are also explained by stochastic mechanics.

Finally, the main contents of this work are organized in the following order. Dynamics in microchannel transport affected by three microchannels, countable parallel ones, unique and complete connections with an external source and universal topology with universal reaction paths are presented in Secs. 2–6, respectively. Conclusions are given in Sec. 7. In Appendices A–E, detailed derivations of the unique time scale of the above-mentioned new mesoscopic transport systems are given. Probability density function describing the properties of particles at the initial lattice is first established. Laplace transform is carried out. The time probability is obtained by setting $s=0$. Moreover, exit time of particles at different initial lattices is obtained by solving derivative of probability density function. Finally, the probability that the particle is located in the initial lattice of different initial microchannels and the average exit time of the whole particle moving to the right are determined via the inter-microchannel transport rate.

2. New Transport System Based on Three Parallel Microchannels

2.1. System constructions

For clarity, arrows in all newly established systems indicate reaction directions. All rates are set as random numbers from 0 to 1. Core physical parameters and related physical meanings are reflected in Table 1. First, an initial system model is established in Fig. 1. The initial system is set as a whole in a static state. Particles are set to be located in lattices 1, 2 and 3 in the middle position and move in a directional direction along three microchannels, respectively. Particles can reach the final lattice without passing through any middle lattice. This system can be used to realize the basic application and expansion of dynamic correction mechanisms in biological systems.^38,39,40,41 Moreover, the overall movement direction of the particle is taken as the right. The last lattice is set as lattice 1${}^{\prime }$. Then, for instance, the initial particle located at lattice 1 can reach the last lattice 1${}^{\prime }$ at the rate of $u_1$ in microchannel $L_1$. The corresponding inverse reaction movement rate is $w_1$. The initial particle 1 moves to lattice 2 on microchannel $L_2$ at the rate of $b_1$. Similarly, the particles of lattice 2 move to lattice 1 of microchannel $L_1$ at the rate of $a_1$ and then react to the last lattice. At the same time, the particles at lattice 2 also tend to move to lattice 3${}^{\prime }$ of microchannel $L_3$ at the rate of $b_2$. Furthermore, a particle at lattice 3 can also finally move to the final lattice 1${}^{\prime }$ through lattices 2 and 1.

2.2. Dynamics of system outlet

Then, the average movement time of particles from lattices 1–3 moving to the right to lattice 1${}^{\prime }$ and the corresponding probability are defined as $T_{1,1^{\prime }},T_{2,1^{\prime }}$, $T_{3,1^{\prime }},{\prod }_{1,1^{\prime }},{\prod }_{2,1^{\prime }}$ and ${\prod }_{3,1^{\prime }}$, respectively. These dynamic characteristics can be detailly evaluated according to each conversion rate, which are shown in Appendix A. Transport time can be obtained as follows :

Similarly, under the set initial conditions, the different time and corresponding probabilities of the particle moving to the right to lattices 2${}^{\prime }$ and 3${}^{\prime }$ can be obtained as follows :

and

3. New Transport System of Particles in Countable Parallel Channels

3.1. System establishment

First, in order to further generalize the above model, establish a new n-orbital particle transport system is established in Fig. 2. The initial particle is at the middle lattice of the microchannel and at rest. All particles are assumed to move to the last lattice without passing through any intermediate nodes. Moreover, the average motion time of particle motion is extracted and analyzed. Take the overall right direction of particle motion as an example. In the n-orbit model, the transformation rate of particles at lattice k from the starting lattice to the end lattice of the same orbit corresponds to $u_k$, respectively. Besides, the transformation rate of particles at the ending lattice $k^{\prime }$ to the starting lattice of the same orbit is $w_k$. Similar to the transport and transformation of substances in organisms,⁴² there are different conversion rates between the established n orbits.

3.2. System dynamics

Then, by taking the particle moving to lattice 1${}^{\prime }$ to the right as an example, the steady-state expression in the corresponding state is obtained as follows :

where $1<k<n$ is satisfied for the second equation.

Afterward, the probability ${\prod }_{{k,1}^{\prime }}$ corresponding to the calculated average motion time $T_{k,1^{\prime }}$ can be obtained as follows :

Thus, time scale can be obtained as follows :

Similarly, the time scale and probability corresponding to the particle moving to the right and reaching other end lattices are detailly presented in Appendix B. After obtaining the time scale equation (8) of the corresponding particles reaching different ending lattices, the average motion time of the particles moving to the right as a whole is obtained as follows :

where $P_i$ is the probability that the particle is in microchannel i in the initial state.

3.3. Evolution of system dynamics

In this part, the dynamic properties of the particle transport system under countable parallel microchannels are studied by performing numerical simulations. First, 100 parallel microchannels are set. As for the initial conditions, particle at the microchannel i can move to the right and the left from the starting lattice at the rate $u_i$ and $w_i$, respectively. Besides, the particle at the microchannel $i+1$ converts to the microchannel i at the rate $a_i$. While, the particle at the microchannel i converts to the microchannel $i+1$ at the rate $b_i$.

Then, the average exit time $T^{+}$ from the initial lattice of all parallel orbits to all final lattices is extracted and used as the core parameter to characterize exit dynamics. Afterward, in order to give consideration to representativeness and convenience, Figs. 3–5 are calculated by studying the effects of motion rates $u_1$, $w_1$, $u_{50}$, $w_{50}$, $u_{100}$, $w_{100}$ and conversion rates $a_1$ and $b_1$ on outlet dynamics of the system by numerically solving Eq. (9).

Then, by comparing Figs. 3–5, it can be seen that the influence of the orbital transport rate of the above three parallel orbits on the exit dynamic properties of the system remains the same as a whole. When inter-orbital conversion rates $a_1$ and $b_1$ approach 0, the overall right-hand average exit time tends to increase. It can be explained by the following physical sense. When the particle transport rate in the new system becomes smaller, there must be more movement time to support particles to reach the last lattice on the same path. At the same time, due to the randomness of orbit selection, the conversion rates between other orbits have a similar effect on exit dynamics. Moreover, transport rates of three special tracks (i.e., the 1st, 100th and 50th tracks) are chosen as an independent variable. The influence of two dominating dynamic parameters $u_i$ and $w_i$ on the average right exit time is found to be basically the same. However, under the condition that the right direction is the overall movement direction and particles finally reach the ending lattice on the right side, exit dynamics are found to be more sensitive to the forward conversion rate $u_i$, which is consistent with the basic set motion direction.

Afterward, by setting parameters as random values and analyzing numerical calculation diagrams of the different dependence of the particle transport system on transport rates $u_i$ and $w_i$ on different orbits and conversion rates $a_1$ and $b_1$ between orbits, respectively, the variation range of the right average motion time of particles with the transport rate on orbit is found to be small. However, the exchange of particles between different orbits has a much greater impact on the exit dynamics of the system. By taking the transformation and transportation of particles between different orbits as the inflow and outflow of particles in a single orbit system, exit dynamics accord with views of classical mechanics and energy conservation. By comparing Figs. 3–5, it can be found that the new system not only expands the number of orbits and conversion rate in a wide range, making countable parallel orbits universal, but also suitable for particle transport and conversion in the biochemical system via analyzing outlet dynamic properties of the model by setting typical dominating parameters as random numbers to prove the calculation generality.

To be summarized, a countable parallel orbital particle transport system is set up here and the evolvement of its exit dynamics is obtained. It can be seen that with the increase of conversion rates $a_i$ and $b_i$ between orbits and conversion rates $u_i$ and $w_i$ of initial and final lattices in the orbit, the right-hand average exit time of the system shows an obvious downward trend. The probability of particles in the initial lattices in different orbits decreases significantly with the conversion rate between initial lattices in different orbits. When the mutual conversion rate between the initial lattice and the final lattice in the orbit increases, the exit time of particles from the initial lattice along the same path to the final lattice decreases. Thus, the average right exit time of the system decreases, that is, the exit dynamic properties of the system are affected.

At the same time, as for three-dimensional evolution diagrams, it can be seen that even with the increase of $a_i$, $b_i$, $u_i$ and $w_i$, the average right exit time shows a downward trend as a whole. However, this property is affected by relevant parameters $u_{50}$ and $w_{50}$ of the initial lattice in the middle of the system. The degree of decline is significantly greater than that of $u_1$, $w_1$, $u_{100}$ and $w_{100}$ of the initial lattice at the boundary. This is because under the condition of only considering the influence of the dynamic properties of adjacent lattices, relevant parameters of the middle initial lattices are more than boundary initial lattices. Thus, the influence of middle initial lattices by adjacent lattices is greater than boundary initial lattices. Therefore, when parameters show the same degree of change, the change range of the average right exit time is much greater than that of boundary lattices. For increasing system universality and reducing different influence degrees of different initial lattice-related parameters (i.e., $u_i$ and $w_i)$, single and all connections between the external particle source and the system are discussed to further analyze outlet dynamics.

4. New Countable Parallel Transportation System with a Unique Particle Exchange Path

4.1. System establishment

As for the above-mentioned newly established systems, exit dynamics of particle transport under multiple parallel orbits are analyzed by discussing the dependences of the exit average motion time on different conversion rates. However, an n-orbital particle transport model with particle exchange with external particle sources is built in Fig. 6 to further enhance the calculation universality and further fit the transformation of particle transport in biochemical reactions.⁴³ Particle exchange quantified as f between particles and external particle source through lattice 1 is assumed. Besides, $f>0$ indicates that the particle source transports particles to the system. However, $f<0$ means that the system particles are output to the outside reservoir. Moreover, f = 0 indicates that the system has just reached the exchange state of dynamic balance with the reservoir.

4.2. Dynamics analyses

Hereafter, by taking the particle moving to the right to lattice 1${}^{\prime }$ as an example, the steady-state expression in the corresponding state is obtained as follows :

where $1<k<n$ is satisfied for the second equation.

Afterward, probability ${\prod }_{{k,1}^{\prime }}$ related to the calculated average motion time $T_{k,1^{\prime }}$ is obtained as follows :

Thus, time scale is derived as follows :

4.3. Evolutions of system outlet dynamic properties

Then, the right average motion time $T^{+}$ of the particle transport system with exchange with external particle sources is calculated. Numerical calculations of the dynamic properties of the system outlet are shown in Figs. 7–9. From the above numerical results, it can be seen that outlet dynamic properties of this system are similar to those of the system in Sec. 3. According to Figs. 7–9, the dependence of the average right motion time of particles on $u_i$, $w_i$, $a_i$ and $b_i$ in the variation trend and value range is highly similar to that of the system without particle exchange. As the established system has considered particle input, the average right motion time of particles is much less dependent on $u_i$, $w_i$, $a_i$ and $b_i$ than that in the nonparticle exchange system, which can be revealed from three-dimensional diagrams. The change trend of average motion time with these parameters slows down obviously. This is because the newly established system is essentially the process simulation of substrate products. When the amount of substrate decreases, the process rate will slow down to a certain extent. Thus, the average right movement time increases. Moreover, the above change trend of the system has been alleviated to a certain extent via a supplemented external source, thus, showing the change trend of the three-dimensional diagrams in Figs. 7–9. Furthermore, detailed derivations about exit dynamic properties of the system are shown in Appendix C to further prove the innovation and authenticity.

From four-dimensional evolution results, it can be seen that exit dynamic properties of this system are related to mutual conversion rates $a_i$ and $b_i$ between parallel orbits and transport rates $u_i$ and $w_i$ on orbit. The average right exit time decreases with $a_i$, $b_i$, $u_i$ and $w_i$. The effect can be accumulated. Besides, when the four groups of parameters increase, the average right exit time decreases more. Exit time of particles at the initial lattice to different ending lattices will decrease without passing through any intermediate lattice, when transport rates on the orbit increase under the same transport path. The reduction of mutual conversion rates reduces the probability $P_i$ of particles in each orbit of the initial lattice. Therefore, the average right exit time decreases, and the exit dynamic properties of the system are affected.

At the same time, it can be seen from three-dimensional evolutions that the influence of $u_{50}$ and $w_{50}$ on outlet dynamic properties of the system is greater than $u_1$, $w_1$, $u_{100}$ and $w_{100}$. Thus, the variation trend of the right average outlet time with $u_{50}$ and $w_{50}$ is greater than $u_1$, $w_1$, $u_{100}$ and $w_{100}$. This is because when exit dynamics are investigated, dynamic properties of the initial lattices of each orbit are limited. Besides, only the influence of particle dynamic properties of adjacent initial lattices is considered. However, as for particles in initial lattices 1 and 100 on both sides, they are only affected by the dynamic properties of one adjacent lattice, which are unlike the middle initial lattice as particle properties are related to two adjacent lattices. Therefore, as for the middle initial lattice, the number of parameters affecting its dynamic properties is more, and the change range of particle dynamic properties is greater. As for a certain transportation route, the change trend of average exit time is more obvious. Additionally, as the only channel connected with the external particle source is set in the new system, there is particle exchange between lattice 1 and external reservoir. Thus, particle dynamics at the initial lattice 1 are more vulnerable to influence, and its average motion time has a larger variation range than that of the initial lattice of the right boundary. Thus, exit dynamics are more variable than that of countable parallel orbits in Sec. 3.

5. New Countable Parallel Microchannel Transport System Connected with External Reservoir

5.1. Necessity of connecting the external source with the transport system

In the previous part, there is particle exchange between the particle transport system and the external particle source, which is quantitatively expressed by is conversion rate f between starting lattice 1 and the external particle source. In order to further enable the established system to characterize the outlet kinetics in the reaction process more consistent with actual biochemical processes,⁴⁴ it is assumed that there is particle exchange between all initial lattices and the external source. Moreover, the rate of particle exchange between lattice k and the source is $\frac{f}{n}$, where $0<k<(n+1)$ and k is a positive integer. Moreover, $f>0$ reflects particles flow into the system. However, $f<0$ reflects particles flow into the external reservoir. However, f = 0 indicates that the system and the particle source are in the dynamic balance of particle exchange.

5.2. Dynamics analyses

Afterward, by taking the particle moving to the right to lattice 1${}^{\prime }$ as an example, the steady-state expression in the corresponding state is obtained as follows :

where $1<k<n$ is satisfied for the second equation.

By using mean-field method, the time scale T and corresponding probability of particles at different lattices reaching lattice 1${}^{\prime }$ are obtained as follows :

and

5.3. Dynamic property evolutions

Hereafter, the corresponding time scale and probability of particles from the initial lattice to different final lattice are calculated, which are detailly expanded in Appendix D. Therefore, the right average motion time $T^{+}$ in which the initial lattice has particle exchange with the external particle source is obtained by numerically solving Eq. (15). The dependence of the time scale on other parameters is depicted in Figs. 11–13. Outlet dynamic properties of this system are found to be similar to those in Secs. 3 and 4, when effects of $a_1$, $b_1$, $u_i$ and $w_i$ are calculated. However, three-dimensional evolution diagrams of the right average motion time based on $a_1$ (or $b_1)$, $u_i$ and $w_i$ have different results.

First, as there is a certain particle inflow at the initial lattice of the new system, outlet dynamics are more sensitive to the change of $u_i$. At the same time, the average motion time based on the change of $w_i$ is larger than that based on $u_i$. From the perspective of dynamics, as the set overall motion direction of particles is right, particles flowing into the system also tend to move to the right, resulting in a large average movement time based on the reverse movement rate $w_i$. Second, it can be found that as for the first orbit and the 50th one, conversion rates $a_1$ and $b_1$ have a much greater impact on the time scale than the transport rate on the orbit. Finally, when the last orbit is discussed, the influence of $a_1$ and $b_1$ on the time scale is greatly reduced compared with the previous two groups. Thus, the change trend is not completely consistent on three-dimensional graphs.

Based on the connected system in Sec. 4, a countable parallel orbital particle transport system connected with the external particle source is set up. The right-hand average outlet time serving as the visual representation of outlet dynamic properties of the system gradually decreases with the increase of inter-orbital mutual conversion rates $a_i$ and $b_i$, and mutual conversion rates $u_i$ and $w_i$ of initial and final lattices on the orbit. Besides, this effect has a cumulative effect. Therefore, when $a_i$, $b_i$, $u_i$ and $w_i$ increase at the same time, the average exit time decreases significantly. It can be explained by following physical facts. Under the same transportation and transformation path, the larger the conversion rate between initial lattices, the smaller the exit time of particles reaching each final lattice, and the smaller the average exit time, which are revealed by evolution processes in Figs. 11–13.

At the same time, the influence of dynamic properties of initial lattices at boundaries on the average exit time is similar to that of the middle initial lattice. With changes of $u_1$, $w_1$, $u_{50}$, $w_{50}$, $u_{100}$ and $w_{100}$, the change trend and amplitude of the average right exit time are similar. Compared with the only connected countable parallel orbital particle transport system, the system connects all initial lattices with the external particle source, which weakens the influence of dynamic properties of intermediate initial lattices and the boundary initial lattices due to adjacent lattices to a certain extent. Thus, exit time depicting exit dynamic properties of all initial lattices reaching the final lattice is relatively similar. Therefore, the dependence of the system average exit time on different parameters $u_1$, $w_1$, $u_{50}$, $w_{50}$, $u_{100}$ and $w_{100}$ is similar. To be summarized, by comparing new systems in Secs. 3–5, the parallel orbital particle transport system is gradually connected with the external particle source. The dependence of exit dynamic properties of the system on $u_1$, $w_1$, $u_{50}$, $w_{50}$, $u_{100}$ and $w_{100}$ and other parameters also shows that the middle initial lattice is completely different from the boundary initial lattice. All initial lattice kinetic properties have similar effects on outlet dynamics.

6. New Transport System with Intermediate Lattices and Multiple Subsystems

6.1. System modeling

Based on multiple systems in Secs. 2–5, exit dynamics of the multi-orbital particle transport model with particle exchange with external sources are analyzed via the developed mean-field method. Exit dynamics are reflected by the average motion time of particles from the starting lattice to the final lattice for the first time without any intermediate lattice. At the same time, the dependence of $T^{+}$ on the conversion rate between different orbits and lattices on the same orbit is simulated. More specific and practical connections between the inner system and the external reservoir are given.

First, particles of initial lattices of n orbits are set to come from one lattice redefined as the initial lattice, which are shown in Fig. 14. Besides, the lattice of particle input from the initial lattice is defined as the intermediate lattice. Moreover, the final lattice is defined as the lattice at which the particle finally reaches, which is also the target of analyses on exit dynamic properties. Each subsystem is assumed to contain $n^{\prime }$ microchannels. $n^{\prime }$ initial lattices are arranged on each orbit. Furthermore, the conversion rate from the particle at lattice k to the particle at lattice $k+1$ is defined as $x_k$. However, the inverse conversion rate is defined as $y_k$.

Second, each initial lattice is connected with $n^{\prime }$ intermediate lattices. The conversion rate from the initial lattice to each intermediate one is $\frac{f}{n}$. The conversion relationship between the intermediate lattice and the final lattice is consistent with Sec. 2. At the same time, as for subsystems corresponding to different starting lattices, conversion rates between particles of the middle lattice and between particles of the middle lattice and the last lattice are different. Multiple parameters are set to consider the universality of the new system to the largest extent.

Third, as for the interaction of dynamic properties of starting lattices, the influence of all starting lattices on the particle transport path for different starting lattices is considered instead of just considering the interaction of adjacent starting lattices,^23,28,30,37 which further widens the universality of the new system and shows the innovation of particle transport system in parallel orbit. Finally, n microchannels where the initial lattices are located are set as an external source. The interaction among lattices is considered. When particles are input into the subsystem, particles tend to output from or input into the system via various conversion rates.

6.2. Dynamic properties of system outlet

Take the following case as an example. The particle moves to the right from the external source and passes through the middle lattice to lattice 1”. The following governing equation in the corresponding state is obtained :

where $2\le k\le n-1$ is satisfied for the second equation.

By using the mean-field method, the time scale and probability corresponding to the particle moving from the starting lattice 1 to the ending lattice 1${}^{\prime \prime }$ are obtained as follows :

and where $2\le k\le n-1$ is satisfied for the second equations of Eqs. (17) and (18).

Afterward, the corresponding motion time and probability from the particle motion of the starting lattice to different ending lattices of the same subsystem are obtained. The corresponding time scale from the particle motion of its lattice to the ending lattice of other subsystems is also obtained. Finally, the average motion time corresponding to the particle moving from the starting lattice to the right to the last lattice is obtained as follows :

Detailed derivations are presented in Appendix E.

6.3. System outlet dynamic properties

Hereafter, through numerical calculations of three groups of parameters, the evolution trend of the average right motion time is shown in Fig. 15. By overcoming the previous method^23,28,30,37 just considering the influence of particles located in adjacent lattices on system dynamic properties, the influence of all lattices in the trajectory is analyzed. At the same time, the countable parallel orbital particle transport system is studied as a subsystem of each initial lattice. Moreover, a set of parameters is selected. The dependence of the average motion time on these representative parameters is obtained to further verify the system rationality, which is shown in Fig. 15.

Then, according to Fig. 15, the right-hand average exit time will gradually decrease with the increase of $x_i$ and $y_i$. These two groups of parameters have a cumulative effect on exit dynamics. The downward trend of the right-hand average motion time is more obvious, when $x_i$ and $y_i$ increase at the same time. The downward range also becomes larger. Thus, the larger the particle conversion rate between initial lattices, the smaller the overall movement time of particles reaching final lattices under the same path, and the smaller the right average exit time.

Furthermore, the influence of mutual conversion rates (i.e., $x_{50}$ and $y_{50})$ of the middle two lattices of initial lattices of the global system on the average exit time is far less than that of conversion rates between initial lattices 1 and 2 at the boundary and that of conversion rates between initial lattices 99 and 100, which is reflected by the decline range of the average exit time of Fig. 15(b) is far less than that of Figs.15(a) and 15(c). Additionally, mutual conversion rates among boundary lattices are found to have the greatest influence on the right average exit time. Besides, the influence decreases as the lattice gets closer to the middle initial lattice. The reason for this phenomenon can be explained by following physical facts.

By overcoming the tradition method of only considering the influence of dynamic properties of adjacent initial lattices, influences of all lattices in the path from the initial lattice to the final lattice are taken into account. Thus, when the studied initial lattice becomes closer to the middle, two paths from other initial lattices to the concerned lattice tend to be consistent. Therefore, the influence of the conversion rate between lattices on outlet dynamic properties of the system becomes smaller, which indicates the smaller the variation range of the average outlet time in Fig. 15.

7. Conclusion

In this work, beyond previous studies, new and fruitful topological structures of new systems composed of multiple channels are established. Reasonable theoretical bases for the simulation of stochastic biochemical processes are provided. Three parallel microchannels as subsystems, countable parallel microchannels as subsystems, unique and multiple connections with the external source, etc., are extensively discussed. Afterward, more reasonable simulations of relevant biochemical processes in the organism are given. Hereafter, by performing Laplace transformation and applying mean-field approaches, fruitful time scales are obtained. The dependences of the right average motion time on typical dynamic parameters are discovered. Dynamic properties of complex dynamic systems with multiple exits are explored. Moreover, the physical mechanisms of all numerical results are explained by energy conservation and the viewpoint of classical mechanics to prove self-consistency.

Beyond previous studies^23,28,30,37 focusing on exit kinetic properties of one-orbit and two-orbit transport systems depicting single- and double-orbit enzymatic reactions, three-orbit and countable orbit particle transport systems are first built to broaden previous studies^23,28,30,37 and achieve more universal and consistent results with more general biochemical processes and particle transport simulations. Fruitful dynamic properties are found.

Then, by connecting newly established systems with the external particle source alone or all, evolutions of the right average motion time are explored, which overcomes previous studies^23,28,30,37 focusing on closed systems and not connecting with the external source. Numerical calculations prove that newly established systems can be more consistent with specific biochemical processes and transportation processes and better balance differences of inlet kinetic properties between boundary initial lattices and intermediate initial lattices.

Afterward, beyond previous studies^23,28,30,37 focusing on particle interactions of adjacent lattices of adjacent microchannels, the interaction between all orbital particles passing through the transport trajectory is fully considered. By setting the universal countable parallel orbital particle system as the subsystem of each initial lattice and setting up intermediate lattices, a new system is proposed to simulate random biochemical processes. Moreover, by extending the single outlet system to a multi-outlet dynamic system with different properties of multiple lattices, the dynamic properties of the outlet are analyzed. The average exit time is found to be positive. The larger the mutual conversion rates between initial lattices, the smaller the average exit time under the same transportation path.

In addition, beyond previous studies^23,28,30,37 focusing on only studying the influence of a certain parameter on the average motion time in a certain range and setting all other parameters as 1, random numbers ranging from 0 to 1 are set for all parameters to further show the generality of new systems proposed here. All relevant conversion parameters of boundary initial lattices and middle initial lattices are extensively studied for investigating representative evolution laws. Evolution mechanisms show that the smaller the average exit time under the same transportation path with increasing mutual conversion rates of initial lattices. Related numerical calculations are explained by classical mechanics. For convenience, multiple exit dynamic properties are studied by calculating influences of specific dominating dynamical parameters.

Finally, detailed derivations of time scale and average exit time of five new systems established are displayed. By establishing the probability density function describing properties of particles at the initial lattice, and performing Laplace transformation, the time probability is obtained by taking zero for the complex variable and probability density derivation to obtain exit time of particles at different initial lattices. Then, the probability that particles are located at initial lattices of different initial orbits and the average exit time of the overall right movement are determined via inter-orbital transport rates.

To be summarized, in the field of nonequilibrium transport, the study of dynamic properties of multiple exit dynamic systems is an effective way to solve issues of general biochemical processes and particle transport. By studying general multiple exit dynamic systems via establishing countable parallel orbits, connecting the transport system with the external particle source, setting up intermediate lattices and considering the influence of dynamics of all initial lattices on transport paths, new multi-inlet, multi-intermediate and multi-outlet dynamic systems are established to keep in line with outlet dynamic properties of more general biochemical processes. Multi-directional biochemical processes like multiple orbital transports and products of a single substrate are described. Physical mechanisms of nonequilibrium particle transport are effectively studied to provide a theoretical basis for biochemical issues. In the future, further outlet dynamics of multiple outlet systems will be investigated by considering the structural states of molecular proteins and structural changes along microtubules.

Acknowledgment

This research is supported by projects below: National Natural Science Foundation of China (Grant No. 11705042), Project of the Ministry of Education on the Cooperation of Production and Education (Grant No. 202101073009), Anhui Provincial Quality Engineering Project (Grant No. 2020kfkc400), Hefei University of Technology Curriculum Ideological and Political Research Project (Grant No. 11020-03392021003), Hefei University of Technology Publishing Fund Project (Grant No. HGDCBJJ2020039), China Postdoctoral Science Foundation (Grant Nos. 2018T110040, 2016M590041) and the Fundamental Research Funds for the Central Universities (JZ2018HGTB0238). Prof. Yu-Qing Wang and Dr. Da-Sen Wei equally contributed.