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[MATH] to [MATH] In our models below, [MATH] for a arrow from protein [MATH] to protein [MATH] , and [MATH] for a horizontal bar instead of arrowhead from protein [MATH] to protein [MATH] (Fig. |
). And it is indispensable to point out that self-connections haven’t been taken into consideration in our models for simplicity. |
Similar to Proposition II.6 in supporting information, with the same initial distribution, when [MATH] , and [MATH] [MATH] , then the stochastic Boolean network model converges to the corresponding deterministic Boolean network model with parameter [MATH] ; and when [MATH] , and |
[MATH] [MATH] , then the stochastic Boolean network model converges to the corresponding deterministic Boolean network model with parameter [MATH] |
III Steady states and synchronized dynamics of the p53 pathways From biochemical perspective, the microscopic variables for a cellular regulatory network are the concentrations, or numbers, of various mRNAs, regulatory proteins, and cofactors. If all the biochemistry were known, then the dynamics of such a network woul... |
The present study will build simple deterministic and stochastic Boolean network models for several negative feedback loops of the p53 pathways, and more important, provides a sound mathematical explanation of the synchronized dynamics and stochastic limit cycles in the stochastic Boolean network model after being pert... |
III.1 The core regulation Negative feedback loops, composed of one transcription arm and one protein-interaction arm, are a common network motif across organisms. p53, the tumor suppressor, transcriptionally activates Mdm2, which in turn targets p53 for degradation. Although it is believed that the existence of negativ... |
A bit out of expectation is, if we use the deterministic Boolean network even to the simplest case(Fig. ), the existence of negative feedback already gives rise to a limit cycle corresponding to oscillations in a biological system, when there exists the outside signal (i.e. [MATH] ). |
Model: From Section II(For Fig. ). The first node [MATH] : p53; and the second node [MATH] : Mdm2. [MATH] is the number of nodes in the model. For simplicity, we set thresholds as [MATH] . And the interacting matrix |
[EQUATION] The deterministic model with [MATH] has a global attractor [MATH] , which corresponds to the fact that the main function of the negative feedback between [MATH] and [MATH] is to keep [MATH] at a low steady state level in normal cells. |
On the other hand when [MATH] , the deterministic model has a unique limit cycle consist of [MATH] state, which is described by Table |
. It corresponds to the fact that the stress signal (e.g. DNA damage) will activate the protein p53(i.e. time-2 point in Table ) and induce a transition to oscillations of p53 level after being perturbed from the outside environment. |
Moreover, as long as the stress signal strength [MATH] is sufficiently low [MATH] or sufficiently high [MATH] , the stochastic Boolean network model would preserve the dynamics of the corresponding deterministic model at a certain low level of noise. As it illustrates the same phenomenon as the Cyclin G/Mdm-2 loop give... |
Similar model can also be built and analyzed exactly by the same reasoning, to other ubiquitin ligases that promote p53 ubiquitination and subsequent proteasomal degradation (Fig. 10 in Harris et al. 2005). We only use the simplest example as a start, and pass directly to a really delicate interesting case. |
III.2 Cyclin G/Mdm2 loop Model: From Section II. The first node [MATH] : p53; The second node [MATH] : cyclin G; The third node [MATH] : PP2A cyclin G; The last node [MATH] : Mdm-2. |
N=4 is the number of nodes in the model. For simplicity, we set thresholds as [MATH] . And from Fig. , the interacting matrix [EQUATION] |
The deterministic model with [MATH] has a global attractor [MATH] , which corresponds to the fact that [MATH] is kept at a low steady state level in normal cells. |
On the other hand, the deterministic model when [MATH] has a unique limit cycle consist of [MATH] state, which is described by Table |
. It corresponds to the fact that the stress signal (e.g. DNA damage) will activate the protein p53(i.e. time-2 point in Table ) and induce a transition to oscillations of p53 level after being perturbed from the outside environment, which roughly corresponds to the p53 pathway or biological trajectory described in (Ha... |
Numerical Simulation of the stochastic Boolean network model The numerical results about the cyclic motion of this stochastic model are quite similar to the stochastic Boolean network model of the cell-cycle, which we have recently discussed (Ge et al. 2007). The main conclusion is that, given the structure of the nega... |
Numerical computations for the current model(Fig. ), are carried out with the famous Gillespie’s method (Gillespie 1977) of the stochastic Boolean network model using MATLAB, and the results are given in the following figures. The network with [MATH] binary nodes has a total of [MATH] number of states. Here, we can pre... |
[MATH] on a line. This 1-d system is reversible if and only if the 4-d system is reversible. Fig. are the basic behavior of a random trajectory. The upper panel shows that there arises the phenomenon of local rapid synchronization like that observed in (Hopfield 1995) during a very short time period after the value of ... |
to [MATH] at time [MATH] , when [MATH] and [MATH] are sufficiently large. The lower panel is a random trajectory over a longer time. Little deviation is shown from the deterministic trajectory in Table |
after being perturbed at time [MATH] , which implies that the stochastic model still leads to well pronounced oscillations when the perturbation from the environment is sufficiently high. |
Fig. shows the stationary distribution of the state [MATH] in the stochastic model which increasingly approaches [MATH] when [MATH] tends to infinity. This excellently corresponds well to the dynamics of the deterministic model when |
[MATH] . At large [MATH] (low temperature or small noise level), the low level state [MATH] is the most probable state of the system. So analogous to the concept of the deterministic model, this state [MATH] can be regarded as the global attractor of the stochastic model. Moreover, one observes a phase-transition like ... |
(similar behavior has been seen in (Qu et al. 2002)). Then we turn to investigate the synchronized dynamics when the signal parameter [MATH] is sufficiently high(i.e. |
[MATH] ). The keys to understand synchronization behavior in stochastic Boolean network models are ( [MATH] ) establishing a correspondence between a stochastic dynamics and its deterministic counterpart; and ( [MATH] ) identifying the cyclic motion in the stochastic models. |
As there is a growing awareness and interest in studying the effects of noise in biological networks, it becomes more and more important to quantitatively characterize the synchronized dynamics mathematically in stochastic models, because the concepts of limit cycle and fixed phase difference no longer holds in this ca... |
In case of the stochastic models of biological networks, there does exist a rather complete mathematical theory for the cyclic motion of the corresponding Markov chains, which has been developed for more than twenty years (Jiang et al. 2004, Kalpazidou 1995). One of the most important concepts in this mathematical theo... |
To further characterize the synchronized dynamics, we give Fig. which shows the Fourier power spectrum of the stochastic trajectory with different values of [MATH] respectively. Using MATLAB, the discrete Fourier transform for time series |
[MATH] is defined as [EQUATION] Therefore, by the Herglotz theorem (Qian et al. 1997 p. 331), the power spectrum of discrete trajectory has a symmetry |
[MATH] . For different sets of parameters, we found all the calculations give the same outstanding main peak in the Fig. . It is important to mention here that by ergodicity, different trajectories give the same power spectrum for any [MATH] |
that is sufficiently large. The single dominant peak in Fig. implies there exists a global synchronization and a globally attractive phenomenon. Note that by representating our maps, one-to-one, from the [MATH] binary nodes to the integers [MATH] , the synchronized behavior is preserved. It is possible that the map wil... |
To further illustrate the synchronized behavior, Fig. shows the power spectra of all the 4 individual nodes in the network. While subtle details are different, all exhibit the dominant peak, similar to that of the overall dynamics. This demonstrates further that synchronized dynamics is presented in the network. |
Fig. plots the magnitude and the period of the dominant peak of the power spectrum in Fig. respectively, as functions of the noise strength, i.e., the parameter [MATH] . It shows, as we have predicted, that the period converges to 8 which corresponds perfectly to the number of states in the main cycle of Table when [MA... |
Finally, Fig. shows how the net circulation of the dominant, main cycle varies with [MATH] , applying the determinant presentation of circulations according to Theorem II.4 in the supporting information. Note that circulation is just the time-averaged number of appearance of certain cycle along the stochastic trajector... |
[MATH] that is just the reciprocal of the number of states in the main cycle, which implies the appearing of more and more distinct synchronization and global attractive behavior with increasing [MATH] . The direction of the net circulation does not change. It is also found that the circulation of negative direction al... |
The net circulations of all the cycles are very small when [MATH] and [MATH] are near zero, since the system is close to equilibrium(reversible) state when [MATH] according to lemma II.9 in supporting information. |
For large [MATH] , the net circulations of all the cycles except the main cycle are also very small by numerical simulation using the determinant expression in Theorem II.4 of supporting information. All the circulations of non-main cycles actually decrease with increasing [MATH] when [MATH] is large. Examples are show... |
. This large separation in the magnitudes of the weights gives rise to the stochastic synchronization, and this stochastic limit cycle can be defined as a “stable” one, whose attractive domain is global. |
As in (Zhang et al. 2006, Ge et al. 2007), we also notice that there exists an inflection point in the curve in Fig. . This implies a cooperative transition of the net circulation of the main cycle while varying the noise level [MATH] , which equivalently means some sort of “phase transition” around [MATH] from chaotic... |
IV Discussion From detailed models to simplified Boolean network approach Deterministic nonlinear mathematical models, based on the Law of Mass Action, have been traditionally used for biochemical reaction networks. Furthermore, noises are unavoidable in small biochemical reaction systems such as those inside a single ... |
In many cellular biochemical modelling, it is impossible to build a detailed, molecular number-based CME model due to a lack of quantitative experimental data. Thus, one often seeks a simplified network model based on simple binary states of the signaling molecules. This leads to the Boolean, Markov network model of th... |
Modelling the oscillatory dynamics of p53 pathways after being perturbed The realization that p53 is a common denominator in human cancer has stimulated an avalanche of research since 1989. The p53 gene can integrate numerous signals that control cell life and death, and damped oscillations for p53 and Mdm2 has been ob... |
plays a very important role in the model similar to the cell cycle model in (Ge et al. 2007), which can induce the level of p53 from a low steady state to oscillated behaviors. |
On the other hand, ideally, one should try to combine these Boolean network models of negative feedback loops together to construct a clear and integrated picture of the p53 pathways, maybe especially including several positive feedback loops (Harris et al. 2005). But unfortunately we haven’t developed a reasonable way... |
Synchronization and circulation in stochastic Boolean network Synchronization is an important characteristics of many biological networks (Strogatz 2003, Winfree 2000) whose dynamics has been modelled traditionally by deterministic, coupled nonlinear ordinary differential equations in terms of regulatory mechanisms and... |
As we know, the occurrence of a deterministic limit cycle in an ODE model is the hallmark of a synchronization phenomenon, while such a definite concept no longer holds in a stochastic system. It is observed that in our present model of p53 pathways, the trajectory concentrates around a main cycle, which we call stocha... |
Stability and robustness of the p53 genetic networks Biological functions in living cells are controlled by protein interaction and genetic networks, and have to be robust to function in complex (and noisy) environments. More robustness would also mean being more evolvable, and thus more likely to survive. |
More precisely, these molecular networks should be dynamically stable against various fluctuations which are inevitable in the living world. Therefore, the stability and robustness of Boolean network model can not be determined if the noise hasn’t been introduced into the model, since it is not reasonable to simply app... |
Acknowledgement The authors would like to thank Professor Minping Qian in Peking University for calling our attention to the p53 network. |
# Source: arxiv 0904.4218 # Title: Discrete- versus continuous-state descriptions of the F1-ATPase molecular motor # Sections: all # Downloaded: 2026-03-03T05:14:37.317107+00:00 |
Discrete- versus continuous-state descriptions of the F -ATPase molecular motor Abstract A discrete-state model of the F -ATPase molecular motor is developed which describes not only the dependences of the rotation and ATP consumption rates on the chemical concentrations of ATP, ADP, and inorganic phosphate, but also o... |
Keywords: molecular motor, F -ATPase, mechanochemical coupling, stochastic process, nonequilibrium thermodynamics. Introduction -ATPase is a ubiquitous protein producing adenosine triphosphate (ATP) in mitochondria Alberts oster In vivo , the F part of this protein is embedded in the inner membrane of mitochondria and ... |
In our previous work pgeg , we carried out a theoretical study of the stochastic chemomechanics of the F -ATPase molecular motor on the basis of the experimental observations reported in Ref. N410 . The stochasticity of the motion is the consequence of the nanometric size of the F motor making it sensitive to the therm... |
However, coarser descriptions are often considered in which the angle (or the position in the case of linear motors) performs discrete jumps instead of varying continuously. The discretization is considered not only because it is always required in order to simulate the random process of diffusion for the angle XWO05 ,... |
The comparison between the two levels of description is interesting not only for the methodology, but also because it can reveal important properties of the molecular motor such as the nature of the coupling between their chemistry and mechanics, i.e., the question whether the mechanochemical coupling is tight or loose... |
The paper is organized as follows. In section II , we present in detail the discrete-state description of the F molecular motor. In section III , this description is compared with our previous model with a continuous angle variable. In section IV , we discuss the properties of the F motor in the light of the comparison... |
II Discrete-state description II.1 Chemistry of the F motor In the discrete-state description, the discrete values of the angle of the [MATH] -shaft correspond to the chemical states so that the mechanical motion of the motor is directly controlled by its chemistry. |
The motor is powered by the hydrolysis of ATP into ADP and P [EQUATION] This reaction is driven by the difference of chemical potential [MATH] between the three species (ATP, ADP, and P ): |
[EQUATION] where the chemical potential [MATH] is equal to the corresponding Gibbs free energy per molecule defined as [EQUATION] |
with X = ATP, ADP, or P , the absolute temperature [MATH] , Boltzmann’s constant [MATH] J/K, and the reference concentration [MATH] mole per liter at which the chemical potential of species X takes its standard value [MATH] |
The standard Gibbs free energy of ATP hydrolysis takes the value [MATH] kJ/mol [MATH] kcal/mol [MATH] pN nm at the temperature of [MATH] C, the external pressure of [MATH] atm, and pH [MATH] |
KAI04 . We notice that ATP hydrolysis provides a significant amount of free energy of [MATH] above the thermal energy [MATH] pN nm. Since the chemical potential difference ( ) vanishes at equilibrium, the equilibrium concentrations of ATP, ADP, and P satisfy |
[EQUATION] showing that ATP tends to hydrolyze into its products. In Eq. ( ) and the following, we no longer write the reference concentration [MATH] , assuming that the concentrations are counted in mole per liter (M). |
The kinetic scheme of our model is based on the phenomenological observations of 120 rotation of the [MATH] -shaft per consumed ATP molecule. In accordance with Ref. N410 , the first substep, the 90 rotation of the [MATH] -shaft, is induced by the binding of ATP to an empty catalytic site. The second substep, the 30 ro... |
[EQUATION] From left to right: In state 1, ATP can bind to an empty ( [MATH] [MATH] -catalytic site of F with the [MATH] -shaft at angular position [MATH] State 1 is thus defined by [MATH] Binding of ATP induces the [MATH] rotation of the [MATH] -shaft, which we represent by [MATH] and fills this catalytic site. ATP st... |
Notice that Eq. ( ) does not necessarily represent the same [MATH] -subunit during different real catalytic states. Following our reference data N410 , ADP and P are released together, but from a different [MATH] -subunit then the ATP binding one. In this case, Eq. ( ) represents different [MATH] -subunits; one that bi... |
Since F is a hexamer composed of three [MATH] -subunits, the reactions ( ) appear three times for the angles [MATH] with [MATH] , so that the motor has a total of six chemical states. However, by the three-fold symmetry of the F motor, these six states can be regroup three by three since the process repeats itself simi... |
According to the mass-action law of chemical kinetics, the reaction rates [MATH] in Eq. ( ) depend on the molecular concentrations in the solution surrounding the motor as follows |
[EQUATION] where the quantities [MATH] [MATH] are the constants of the forward and backward reactions of binding and unbinding of ATP or ADP with P and, [ATP], [ADP], [P ] represent the concentrations of each species. [MATH] is the constant of ATP binding often denoted [MATH] and [MATH] the ATP unbinding constant [MATH... |
An important aspect of the experiments with beads or actin filaments attached to the [MATH] -shaft is that the behavior of the molecular motor also depends on the friction [MATH] of the attached objects moving in the viscous medium surrounding F , as well as on the external torque [MATH] which is applied in some experi... |
II.2 Thermodynamics of the F motor In order for the description to be consistent with thermodynamics, let us summarize here the conditions that the chemical and mechanical properties of the motor must satisfy. |
The internal energy of the motor changes with the rotation angle [MATH] and the numbers [MATH] of molecules (X=ATP, ADP, P ) entering the catalytic sites according to the Gibbs relation |
[EQUATION] where [MATH] denotes the external torque, [MATH] the chemical potential ( ), [MATH] the temperature, and [MATH] the thermodynamic entropy. The changes in molecular numbers due to the overall reaction ( ) satisfy |
[EQUATION] where [MATH] is the number of molecules of species X entering the motor. Consequently, the Gibbs relation ( 10 ) becomes |
[EQUATION] in terms of the chemical potential ( ) of the overall reaction. In an open system such as a motor, energy and entropy vary a priori because of exchanges with the surrounding medium or possibly due to internal variations: |
[EQUATION] The laws of thermodynamics only rule the internal variations: [EQUATION] Since energy and entropy are state variables, they recover their starting value after completing a cyclic process in a motor whereupon their integration over a cycle vanishes, [MATH] and [MATH] . For the isothermal process of a nanomoto... |
[EQUATION] which is deduced by using Eqs. ( 12 )-( 16 ) for a cyclic process. Introducing the mean angular velocity of the motor in revolution per second |
[EQUATION] and the mean rate of ATP consumption [EQUATION] the entropy production is given by [EQUATION] in terms of the so-called thermodynamics forces or affinities, [MATH] and [MATH] , and the corresponding fluxes or currents, [MATH] and [MATH] |
P67 KP98 Schnak . All these quantities vanish at the thermodynamic equilibrium. Mechanical or chemical energies are provided from the exterior if either the torque [MATH] or the reaction free energy [MATH] are non vanishing. |
In general, molecular motors can be driven out of equilibrium by either the external torque [MATH] or the molecular concentrations of ATP, ADP and P entering [MATH] , possibly in combination. In this regard, the thermodynamic forces [MATH] and [MATH] are independent control parameters. However, in a regime where the ch... |
[EQUATION] In this case, the entropy production ( 20 ) becomes [EQUATION] with the chemomechanical affinity [EQUATION] which is here written in dimensionless form. In the regime of tight coupling, the mechanical and chemical thermodynamic forces are thus no longer independent control parameters but are replaced by the ... |
[EQUATION] in the presence of the external torque [MATH] . This result shows that the chemical equilibrium is displaced in the presence of an external torque. In vivo , such an external torque comes from the F part of ATPase and the transmembrane pH difference in mitochondria. This displacement of equilibrium finds its... |
In the discrete-state description, the discrete values of the angle of the [MATH] -shaft correspond to the chemical states so that the mechanical motion is tightly coupled to the chemistry of F , a feature which is characteristic of such discrete models. Therefore, the relations ( 21 ), ( 22 ), and ( 23 ) apply to such... |
II.3 Master equation description The chemistry of F is a stochastic process because the arrival of each substrate molecule (ATP or ADP with P ) is a random event in time. Accordingly, the time evolution of the molecular motor is described in terms of the probabilities [MATH] to find it in one or the other of the two st... |
[EQUATION] with a sum over the two reactions [MATH] and the two states [MATH] before the transition [MATH] or after the reverse transition [MATH] . The quantity [MATH] is the transition rate per unit time from the state [MATH] to the state [MATH] due to the reaction [MATH] , which can be identified with the rate of the... |
[EQUATION] with the normalisation condition [MATH] always satisfied. The mean consumption rates of the different species are given by |
[EQUATION] Since the [MATH] -shaft rotates by a substep of [MATH] during the first reaction [MATH] and by a substep of [MATH] during the second reaction [MATH] , the mean angle of the [MATH] -shaft evolves in time according to |
[EQUATION] in radian per second. Stationary state. The master equation admits a time-independent stationary solution such that [MATH] The stationary solution for each two states is given by |
[EQUATION] where we used the normalization condition [MATH] We notice that this state is stationary in the statistical sense because an individual motor continues to fluctuate in time. The probabilities ( 31 )-( 32 ) define the statistical distribution of its fluctuations between the states 1 and 2 after sampling over ... |
In the stationary state, the mean angular velocity of the [MATH] -shaft is thus given by [EQUATION] and the mean rates of molecular consumption by |
[EQUATION] The condition ( 21 ) of tight chemomechanical coupling is thus satisfied. Equilibrium state. At thermodynamical equilibrium, all the nonequilibrium constraints vanish. In this case, the stationary solution represents the equilibrium probability [MATH] and obeys the detailed balance conditions pierr |
[EQUATION] for all reactions [MATH] and all the transitions [MATH] . As a consequence, the mean velocity ( 33 ) and the rate ( 34 ) vanish which occurs under the condition: |
[EQUATION] Comparing with the condition obtained from the thermodynamics of the motor, we get the equilibrium constant of the reaction: |
[EQUATION] which depends on the external torque [MATH] acting on the molecular motor. We see that considerations of equilibrium thermodynamics give a constraint allowing us to fix one reaction constant if we know the three other constants. However, equilibrium thermodynamics does not provide enough relations in order t... |
II.4 Determination of the reaction constants The motor is in a nonequilibrium stationary state if the chemomechanical affinity ( 23 ) is non-vanishing, i.e., if the equilibrium condition ( 36 ) is not satisfied. In this case, the mean velocity ( 33 ) does not vanish and can be compared with experimental data N410 or wi... |
First of all, the mean velocity ( 33 ) can be rewritten in the following form: [EQUATION] in terms of the constants [EQUATION] together with the equilibrium constant [MATH] defined by Eq. ( 37 ). The knowledge of these four constants is equivalent to knowing the four reaction constants [MATH] and [MATH] . Each one of t... |
The constants ( 39 ) and ( 40 ) can in particular be determined in the absence of the products of hydrolysis, [MATH] , in which case the velocity follows a typical Michaelis-Menten kinetics: |
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