We identify two independent causes for the deviation between ODE and SDE when using the prevailing stochastic integrations: 1 the existence of a non-detailed balance part; 2 a variable-dependent diffusion matrix multiplicative noise. Both causes are related to the freedom of choosing a stochastic interpretation for SDE.
Therefore, the prevailing stochastic simulations could not achieve the consistency for systems without detailed balance. More details are given in Sect. I of Supplemental Information. The consistency is particularly necessary in a scenario where an ODE model is properly constructed and quantitatively correspond to experimental data on average, but are invalidated by the usual way of simulating SDE in reconstituting the original stochastic process and vice versa.
The results below are valid for multiplicative noise. For Eq. Even so, a decomposed dynamics equivalent to Eq. A - type integration is defined as the connection between the SDE 2 and the FPE 5 , and realized by two explicit limiting procedures: first the usual integration limit and then the zero mass limit A - type simulation can thus be implemented as follows: Eq. Thus, one can simulate Eq. An advantage led by the A-type simulation is that for arbitrary noise strength the sampled steady-state distribution of Eq.
Then, the deterministic part of Eq. A-type simulation also reserves topology of the landscape for arbitrary noise strength 31 , 47 , as exemplified in Fig. Note that two numerical methods have been used to estimate the steady-state distribution or potential function in Eq. To implement these two methods, the matrix Q x needs to be solved. The essential information for multi-stable systems is the relative stability between stable states, which can be extracted from the potential difference 1 , 2.
The detail of finding the least action path is given in Methods. The path integral formulation for Eq. The formulation needs to be consistent with the stochastic integration used As we have transformed Eqs 2 to 6 , it is more convenient to use the equivalent path integral formulation:. The action function:. By using the decomposition in Eq. For clarity, we ignore the symbol x for functions of x in the derivation.
The inequality in Eq. Thus, Eq. Then, minimization of the action function. With Eq. The significance of Eq. As the steady state obeys the Boltzmann-Gibbs distribution, the probability ratio between stable states is:. Equation 15 is valid under arbitrary noise strength, and thus can show variation of the ratio with different noise intensities. It provides the probability ratios of quantities such as the number of different cell states. Specifically, we apply it to analyze tumor heterogeneity by large noise in the section of Application. Different from 51 , in our framework the non-detailed balance part does not provide correction terms that explicitly appear in the pre-factor of the rate formula as analyzed in Sect.
IV of Supplemental Information. Computational costs of methods mentioned above for systems with respect to dimension N are analyzed here. For stochastic simulation of Eq. This method also has the problem of slow convergence when noise strength is small. Both of the two methods need to solve the matrix Q for A-type interpretation. However, information critical for many stochastic models is mainly obtained from estimating the relative probabilities between stable states.
Under such circumstances, it is sufficient to find the potential differences between fixed points, without the necessity of obtaining the whole steady-state distribution. As a result, the least action method is efficient in high dimensional systems. We list the results in the table of Fig. Protocol I can be extended to obtain a global landscape for systems with multiple stable states Protocol II :.
Find the fixed points under consideration from solving the ODE counterpart, such as by Newton iteration method Classify all the fixed points into two groups by calculating the eigenvalues of the linearized Jacobian matrix in their neighborhood: stable fixed points no eigenvalue with positive real part and unstable points at least one eigenvalue with positive real part.
Choose a saddle point as reference. Start from the points in a small neighborhood of the saddle point, and simulate the ODE to find all the stable fixed points reached. Calculate potential difference between the saddle point and the stable fixed points by the least action method in Eq.
The detail of the minimization procedure in Eq. Repeat step 3 for all saddle points. Assign relative potential difference between the saddle points if they reach a common stable fixed point. For any other points in state space, simulate the ODE to find the fixed point it reaches. Obtain their potential difference by the least action method.
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The total computational cost depends on the potential value of how many points are calculated. With the calculated potential values, extract the relative probabilities between the states by Eq. The consistency between ODE and SDE enables to utilize information of fixed points and basins of attraction from ODE, which greatly improves the efficiency of our algorithm. As each point in state space reaches a single fixed point, its potential value is uniquely determined if the ideal least action method is found numerically during minimization procedure, which leads to a global landscape without ambiguity.
Specifically, the probability ratio between a fixed and a point within the potential well is calculated by Eq. For dynamical systems with complex attractors such as limit cycle 33 , 47 , 53 , the point on the stable limit cycle can be treated similarly as the stable fixed point, and thus our method can be generalized. Heterogeneity of cell populations is widely observed in biological systems such as cancer 54 , where different cell phenotypes emerge in tumor tissues 34 , It is proposed that an underlying regulation network and quantification on the network dynamics by SDE models can describe various cell states and transitions between them 6 , Starting from the SDE model, the calculated potential landscape provides an integrated picture to study heterogeneity.
Specifically, valleys in the landscape correspond to different cell states, and the potential barrier separating them quantifies the transition rates. This approach of landscape is helpful to understand systematically the effect of perturbations on cell state interconversions. Here, we investigate whether the variation of noise strength leads to changes of cell states, as noise plays a crucial role in biological processes, for example, it drives the cell fate decision 20 , The previous methods 24 , 25 , 26 , 27 , 28 can not be applied to study the function of large noise, because identification on valleys of landscape and calculation on potential barrier by these methods are restricted to the zero noise limit.
Now, we are able to quantify the role of large noise on heterogeneity in high dimensional network dynamics, because the present calculation on landscape is robust under arbitrary noise strength.
From our method, the ratios of cell states can be controlled by manipulating noise strength, which allows the cell-to-cell variability under the same gene regulation network. As an illustrative example, we demonstrate the effectiveness of our approach by applying it to a network model for the prostate cancer The network dynamics is modeled by a dimensional SDE. Each dynamical variable represents the expression or activity level of a gene. The dynamics of each component is written as a sum of the generation and degradation, including the activation or inhibition by the other genes.
We use the standard Hill equation to model such interactions 56 , The ODE counterpart and the parameters are given in Sect. VII of the Supplemental Information. From the analysis on the ODE, we know the system has 10 stable fixed points and 16 saddle points. We find that four stable fixed points correspond to various cell states shown in Fig. For clarity, we consider additive noise case with diffusion matrix D x as an identity matrix in this example. It should be emphasized that the deviation between ODE and SDE appears even with additive noise, because this system does not obey the detailed balance condition.
As a result, considering additive noise is sufficient to demonstrate the advantage of our method based on A-type integration compared with the prevailing methods based on other stochastic integrations. Potential barriers calculated by Protocol II in the prostate cancer model Left panel: The chosen four cell states: differentiated D , proliferating P , cancer C , inflammation I. They are stable fixed points obtained from ODE of the dimensional system.
Right panel: the heights of potential barriers between stable fixed points connected by saddles. The lengths of arrows are proportional to barrier heights listed in the table below. Table: potential barriers between stable fixed points are calculated by the least action method. The parameters of the system chosen here are for typical cancer patients 11 , where cancer and inflammation states are more stable.
We use the least action method to calculate heights of the potential barriers between stable states, as shown in Fig. We note that in order to have the global landscape, we use the continuous condition to set the potential value of P to D specifically. According to Eq. When noise is small, e. This demonstrates the emergence of tumor heterogeneity with respect to increasing the noise strength.
The probability ratios between D, P, C, I states of the prostate cancer model. The ratios are calculated by Eq. The tumor heterogeneity emerges when noise strength becomes large where the four types of cells are almost equally distributed. We elucidate more on the application of controlling the ratios of cell states through varying the noise intensity, which can be implemented by tuning temperature. First, it was demonstrated that an therapy with combination of hyperthermia and other treatments, such as immunotherapy and radiotherapy, can improve the efficiency of cancer cure In those cases, temperature plays the role of enhancer to switch on and off the effectiveness of other therapies.
Specifically, drug cytotoxicity triggered by temperature variation leads to the death of tumor cells, and therefore combination of hyperthermia and chemotherapy is regarded as an effective treatment of cancer Second, as different levels of heating were found to bring distinct modulatory effect on tumor targets, our method valid for arbitrary noise strength may be applied to study sensitivity of thermal treatment regulated by temperature, which will provide new designs on clinical trials. Third, the regional hyperthermia to radiotherapy 60 shows an improvement on survival rates of cancer patients, because hyperthermia can guide the action of chemotherapy to specific heated tumor region.
This can be modeled by multiplicative noise, as exemplified in Sect. I of the Supplemental Information. Therefore, the present approach could form the theoretical basis for hyperthermia that employs effect of temperature in tumor treatment. With the obtained potential barrier, transition rates of cell state interconversions are given by Eq.
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Under the given parameters and noise strength, the result provides a set of predictions: 1 the cancer and the inflammation states are more stable than the proliferating and the differentiated states; 2 transitions from the cancer state to the proliferating state and from the inflammation state to the differentiated state are difficult than the other way around; 3 transition to the cancer state from the inflammation state is more frequent than from the proliferating state. These suggest that the model may describe a cancer patient, and new strategies for medical treatments should be designed to raise the potential energy of the cancer and the inflammation states.
The above results depend on a specific choice on the stochastic integration. From the mathematical aspect, different stochastic integrations are equivalent, and can be transformed to each other by modifying the drift term f x correspondingly. From the physical aspect, for a system with clearly separated sources for the deterministic force and the stochastic force, i. There are particular scenarios in which such measurements are possible, for example, the experiments 61 , 62 have shown that A-type is chosen for a class of systems, which demonstrates that the A-type integration is not only a theoretical treatment.
However, in many extended applications such as phenomenological models for biological systems, separating deterministic and stochastic forces would not even be meaningful, and we would rather consider stochastic integrations as mathematically equivalent tools that is chosen by modeling. Under the situation, SDE model set by a combination of the drift term f x and stochastic integration can be non-unique, and each is an effective description for the system. If the deterministic rate equations are properly constructed and quantitatively correspond to the experimental measurements on average, e.
The cancer model discussed above belongs to this category, as its ODE part was demonstrated to match the experimental observation on average In biology, noise has a variety of sources 63 , such as locations of molecules, micro-environmental fluctuations, gene expression noise, and cellular processes like cell growth. For complex systems like cancer, noise may come from different sources. SDE model reconstitutes the random fluctuations into a single noise term, which reflects the various sources of noise 19 , Therefore, several experimental operations can implement the change of the noise strength discussed here in real biological systems.
Our method can be applied to systems that are modeled by master equation CME with discrete dynamical variable First, CME can be transformed to be the chemical Langevin equation with continuous variable 40 , which can be cast into the form of Eq. Then, our method is applicable to improve efficiency. The approximation is tolerably accurate when the copy number of variables are large, and it also requires that the dynamical process has a time scale during which multiple reactions occur and the reaction rate does not change dramatically These conditions are expected to hold for the present high dimensional cancer dynamics 10 , 11 , where the proteins usually has high copy numbers.
II of Supplemental Information. Third, for systems with low copy numbers, SDE can still provide an appropriate description on the effect of noises Fourth, for stochastic processes on the level of single molecules, such as gene burst process 65 , CME is a more proper approximation to capture the discrete nature of species 9. Nevertheless, this kind of noise will diminish by accumulation of proteins with long lifetime Mathematically, SDE and CME are two independent modeling methodologies, and are on an equal footing to describe the stochastic dynamics. From computational side, a whole set of CME to describe the stochastic dynamics in detail is typically high dimensional, and the Gillespie algorithm 67 to simulate CME is time consuming.
Thus, the present method handling SDE valid for arbitrary noise strength is practically useful to investigate high dimensional systems with large fluctuations, particularly when the ODE counterpart is properly constructed and quantitatively correspond to the average experimental data. Several other remarks are in order.
First, our calculation on the potential difference is applicable to systems both with and without detailed balance condition 14 , i. For such cases, the least action path also differ from the deterministic saddle-node trajectories Second, there are many efficient numerical methods to calculate fixed points of ODEs in high dimension 52 , such as Newton iteration method. Third, the present action function has the dimension of energy, and the conventional action in classical physics has the dimension of energy multiplied by time.
Fourth, positions for the locally most probable states in Eq. Fifth, the constructed potential function is also useful to extract thermodynamical free energy for non-equilibrium systems 68 , We next compare our framework with the previous works. However, computational cost of solving this partial differential equation increases exponentially. II of Supplemental Information is a question without a definite answer. These effects in general defy the use of dynamical information from the ODE counterpart, and our method provide a possibility to reserve useful results by ODE analysis for SDE with arbitrary noise strength.
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For general D x , the minimization of action function Eq. A more detailed comparison is given in Sect. V of Supplemental Information. Recently, the existence of decomposition Eq. Here, we provide a framework to directly extend the potential function to the cases with finite noise. In this paper, we find that a unique potential function can be obtained by finding the ideal least action path for the minimization in Eq.
Even Q in Eq. In order to be more efficient in finding the least action path as given by the time-reversal adjoint dynamics, here we provide a strategy of minimization called ODE-based-adaptive-time method. First, we simulate ODE, Eq. Since the least action path is time-reversal of the adjoint dynamics, its duration and length should be the same as that of the corresponding trajectory for the original ODE. To conclude, we have identified two independent causes for the deviation between the SDE and its ODE counterpart when using the prevailing stochastic integrations: the existence of a non-detailed balance component or a variable-dependent diffusion matrix multiplicative noise.
We have developed a new numerical approach to study multi-stability and stochastic transitions between stable states for the SDE modeling. We are able to compute efficiently probability ratios between stable states in high dimensional nonequilibrium systems subject to large noise, through a least action method under the A-type stochastic interpretation.
The modeling results of a prostate cancer network reveal a new mechanism to control the ratios of cell states by manipulating the noise intensity. Our approach should also be practically useful to study the role of the noise in the dynamical modeling of many other real-world high dimensional stochastic processes. We have demonstrated in Sect. Therefore, we can choose action with specific stochastic integration for the convenience of numerical calculations.
Numerically, we use discretized form of Eq. Thus, we minimize the discretized action:. Besides, Eq. The minimization procedure requires to calculate gradient of the action function for the optimization step, and each step has a cost linearly proportional to dimension. Thus, in numerical realization the computational cost may become superlinear to dimension, and is at most quadratic to dimension. Such increasing demand on computational power is also encountered for the gradient expansion method. Therefore, both the stochastic simulation and the gradient expansion method are more computationally expensive than the present method.
Then, the number of points K is required to be large enough such that the minimization procedure can find out the long least action path. Besides, we find in the numerical experiment that if T is too large, the least action path may pass an additional saddle point limit cycle with higher potential energy before going through the expected saddle point. Special care is needed to choose suitable K and T in these cases, and the ODE-based-adaptive-time method proposed in Discussion is a candidate to improve the efficiency.
There are also numerical methods to adjust the grid points on the least action path, such as the adaptive minimum action method They can be applied to optimize our current numerical code. An amendment to this paper has been published and can be accessed via a link at the top of the paper. Kramers, H. Brownian motion in a field of force and the diffusion model of chemical reactions.
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Ao, P. Potential in stochastic differential equations: novel construction. David N. Yanheng Ding. Jinqiao Duan. Dirk Blomker. Tusheng Zhang. Zhen-qing Chen. Changpin Li.
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