1,720,970 research outputs found
Krylov and steady-state techniques for the solution of the chemical master equation for the mitogen-activated protein kinase cascade
No full textMultiscale modeling of chemical kinetics via the master equation
Get PDFWe present numerical methods for both the direct solution and simulation of the chemical master equation (CME), and, compared to popular methods in current use, such as the Gillespie stochastic simulation algorithm (SSA) and τ-Leap approximations, this new approach has the advantage of being able to detect when the system has settled down to equilibrium. This improved performance is due to the incorporation of information from the associated CME, a valuable complementary approach to the SSA that has often been felt to be too computationally inefficient. Hybrid methods, that combine these complementary approaches and so are able to detect equilibrium while maintaining the efficiency of the leap methods, are also presented. Amongst CME-solvers the recently suggested finite state projection algorithm is especially well suited to this purpose and has been adapted here for the task, leading to a type of “exact τ-Leap.” It is also observed that a CMEsolver is often more efficient than an SSA or even a τ-Leap approach for computing moments of the solution such as the mean and variance. These techniques are demonstrated on a test suite of five biologically inspired models, namely, stochastic models of the genetic toggle, receptor oligomerization, the Schl¨ogl reactions, Goutsias’ model of regulated gene transcription, and a decaying-dimerizing reaction set. For the gene toggle it is observed that important experimentally measurable traits such as the percentage of cells that undergo so-called switching may also be more efficiently approximated via a CME-based approach
An improved dynamic Finite State Projection algorithm for the numerical solution of the chemical master equation with applications
No full textRecently, Munsky and Khammash suggested the Finite State Projection (FSP) algorithm for the numerical solution of the Chemical Master Equation, which provides a discrete and stochastic modelling framework for chemical kinetics. The important question of whether or not the algorithm is guaranteed to terminate is not addressed in the original work. We show that the well-known explosive birth process provides a counter example. We also give sufficient criteria for a model to be suitable for the FSP technique. We demonstrate the FSP technique on three novel applications. Results are presented for: (i) the Schlogl reactions; (ii) another example from Gillespie's celebrated book; and (iii) models for the role that dimerization plays in reducing noise in simple gene regulatory networks. Finally, we augment the dimerization model to include tetramers and show that this enhances the noise reduction properties of the network
Application of the strang splitting to the chemical master equation for simulating biochemical kinetics
No full textRecently the Finite State Projection (FSP) algorithm was suggested for the numerical solution of the chemical master equation (CME). The CME is a discrete partial differential equation that describes the probability density function associated with stochastic biochemical kinetic processes. We show that the FSP approximation is an example of operator splitting, giving insight into why the method performs so well for biochemical applications. Furthermore we describe the application of the Strang splitting, in combination with the FSP approximation, to chemical reactions with disparate rates and show how this relates to recent applications of the quasi-steady-state approximation (QSSA) to the CME. The application of the Strang splitting to the direct solution of the CME is demonstrated to be extremely effective on four examples commonly used as test problems in the literature: Michaelis-Menten enzyme kinetics, the Goldbeter-Koshland switch, dual phosphorylation, and a decaying-dimerizing model. Finally, the framework gives a multiscale approach for combining the Strang splitting and QSSA approximations, leading to greater overall computational efficiency
Inexact uniformization method for computing transient distributions of Markov chains
Get PDFThe uniformization method (also known as randomization) is a numerically stable algorithm for computing transient distributions of a continuous time Markov chain. When the solution is needed after a long run or when the convergence is slow, the uniformization method involves a large number of matrix-vector products. Despite this, the method remains very popular due to its ease of implementation and its reliability in many practical circumstances. Because calculating the matrix-vector product is the most time-consuming part of the method, overall efficiency in solving large-scale problems can be significantly enhanced if the matrix-vector product is made more economical. In this paper, we incorporate a new relaxation strategy into the uniformization method to compute the matrix-vector products only approximately. We analyze the error introduced by these inexact matrix-vector products and discuss strategies for re. ning the accuracy of the relaxation while reducing the execution cost. Numerical experiments drawn from computer systems and biological systems are given to show that significant computational savings are achieved in practical applications
A Krylov-based finite state projection algorithm for solving the chemical master equation arising in the discrete modelling of biological systems
No full textBiochemical reactions underlying genetic regulation are often modelled as a continuous-time, discrete-state, Markov process, and the evolution of the associated probability density is described by the so-called chemical master equation (CME). However the CME is typically difficult to solve, since the state-space involved can be very large or even countably infinite. Recently a finite state projection method (FSP) that truncates the state-space was suggested and shown to be effective in an example of a model of the Pap-pili epigenetic switch. However in this example, both the model and the final time at which the solution was computed, were relatively small. Presented here is a\ud
Krylov FSP algorithm based on a combination of state-space truncation and inexact matrix-vector product routines. This allows larger-scale models to be studied and solutions for larger final times to be computed in a realistic execution time. Additionally the new method computes the solution\ud
at intermediate times at virtually no extra cost, since it is derived from Krylov-type methods for computing matrix exponentials. For the purpose of comparison the new algorithm is applied to the model of the Pap-pili epigenetic switch, where the original FSP was first demonstrated. Also\ud
the method is applied to a more sophisticated model of regulated transcription. Numerical results indicate that the new approach is significantly faster and extendable to larger biological models
Stochastic modelling of T cell homeostasis for two competing clonotypes via the master equation
No full textStochastic models for competing clonotypes of T cells by multivariate, continuous-time, discrete state, Markov processes have been proposed in the literature by Stirk, Molina-París and van den Berg (2008). A stochastic modelling framework is important because of rare events associated with small populations of some critical cell types. Usually, computational methods for these problems employ a trajectory-based approach, based on Monte Carlo simulation. This is partly because the complementary, probability density function (PDF) approaches can be expensive but here we describe some efficient PDF approaches by directly solving the governing equations, known as the Master Equation. These computations are made very efficient through an approximation of the state space by the Finite State Projection and through the use of Krylov subspace methods when evolving the matrix exponential. These computational methods allow us to explore the evolution of the PDFs associated with these stochastic models, and bimodal distributions arise in some parameter regimes. Time-dependent propensities naturally arise in immunological processes due to, for example, age-dependent effects. Incorporating time-dependent propensities into the framework of the Master Equation significantly complicates the corresponding computational methods but here we describe an efficient approach via Magnus formulas. Although this contribution focuses on the example of competing clonotypes, the general principles are relevant to multivariate Markov processes and provide fundamental techniques for computational immunology
Cauchy integrals for computational solutions of master equations
Get PDFCauchy contour integrals are demonstrated to be effective in computationally solving master equations. A fractional generalization of a bimolecular master equation is one interesting application.
References A. Andreychenko, L. Mikeev, D. Spieler, and V. Wolf. Approximate maximum likelihood estimation for stochastic chemical kinetics. EURASIP J. Bioinf. Sys. Biol., 2012:9, 2012. doi:10.1186/1687-4153-2012-9 C. N. Angstmann, I. C. Donnelly, B. I. Henry, and J. A. Nichols. A discrete time random walk model for anomalous diffusion. J. Comput. Phys., 293:53–69, 2014. doi:10.1016/j.jcp.2014.08.003 Y. Berkowitz, Y. Edery, H. Scher, and B. Berkowitz. Fickian and non-Fickian diffusion with bimolecular reactions. Phys. Rev. E, 87:032812, 2013. doi:10.1103/PhysRevE.87.032812 J. C. Butcher. On the numerical inversion of Laplace and Mellin transforms. Conference on Data Processing and Automatic Computing Machines, 117:1–8, 1957. D. Ding, D. A. Benson, A. Paster, and D. Bolster. Modeling bimolecular reactions and transport in porous media via particle tracking. Adv. Water Resour., 53:56–65, 2013. doi:10.1016/j.advwatres.2012.11.001 B. Drawert, S. Engblom, and A. Hellander. URDME: a modular framework for stochastic simulation of reaction-transport processes in complex geometries. BMC Syst. Biol., 6:76, 2012. doi:10.1186/1752-0509-6-76 T. A Driscoll, N. Hale, and L. N. Trefethen. Chebfun Guide. Pafnuty Publications, 2014. http://www.chebfun.org/docs/guide/ N. Dunford and J. Schwartz. Linear Operators I, II, III. Wiley New York, 1971. http://au.wiley.com/WileyCDA/WileyTitle/productCd-0471608483.html, http://au.wiley.com/WileyCDA/WileyTitle/productCd-0471608475.html, http://au.wiley.com/WileyCDA/WileyTitle/productCd-0471608467.html D. Fulger, E. Scalas, and G. Germano. Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation. Phys. Rev. E, 77:021122, 2008. doi:10.1103/PhysRevE.77.021122 D. Gillespie. Markov Processes: An Introduction for Physical Scientists. Academic Press, 1991. http://www.elsevier.com/books/markov-processes/gillespie/978-0-12-283955-9 M. Hegland, C. Burden, L. Santoso, S. MacNamara, and H. Booth. A solver for the stochastic master equation applied to gene regulatory networks. J. Comput. Appl. Math., 205(2):708–724, 2006. doi:10.1016/j.cam.2006.02.053 B. I. Henry, T. A. M. Langlands, and S. L. Wearne. Anomalous diffusion with linear reaction dynamics: From continuous time random walks to fractional reaction-diffusion equations. Phys. Rev. E, 74(3):031116, 2006. doi:10.1103/PhysRevE.74.031116 N. J. Higham. Functions of Matrices. SIAM, 2008. doi:10.1137/1.9780898717778 R. Hilfer and L. Anton. Fractional master equations and fractal time random walks. Phys. Rev. E, 51:R848, 1995. doi:10.1103/PhysRevE.51.R848 K. J. in 't Hout and J. A. C. Weideman. A contour integral method for the Black–Scholes and Heston equations. SIAM J. Sci. Comput., 33:763–785, 2011. doi:10.1137/090776081 T. Jahnke and D. Altintan. Efficient simulation of discrete stochastic reaction systems with a splitting method. BIT, 50:797–822, 2010. doi:doi:10.1007/s10543-010-0286-0 T. Kato. Perturbation theory for linear operators. Springer-Verlag, 1976. http://link.springer.com/book/10.1007%2F978-3-642-66282-9 V. M. Kenkre, E. W. Montroll, and M. F. Shlesinger. Generalized master equations for continuous-time random walks. J. Stat. Phys., 9(1):45, 1973. doi:10.1007/BF01016796 M. Lopez-Fernandez and C. Palencia. On the numerical inversion of the Laplace transform in certain holomorphic mappings. Appl. Numer. Math., 51:289–303, 2004. doi:10.1016/j.apnum.2004.06.015 S. MacNamara, K. Burrage, and R. B. Sidje. Multiscale modeling of chemical kinetics via the master equation. SIAM Multiscale Model. Simul., 6(4):1146–1168, 2008. doi:10.1137/060678154 M. Magdziarz, A. Weron, and K. Weron. Fractional Fokker–Planck dynamics: Stochastic representation and computer simulation. Phys. Rev. E, 75:016708, 2007. doi:10.1103/PhysRevE.75.016708 F. Mainardi, R. Gorenflo, and A. Vivoli. Beyond the Poisson renewal process: A tutorial survey. J. Comput. Appl. Math., 2007. doi:10.1016/j.cam.2006.04.060 W. McLean. Regularity of solutions to a time-fractional diffusion equation. ANZIAM J., 52(2):123–138, 2010. doi:10.1017/S1446181111000617 W. McLean and V. Thomee. Time discretization of an evolution equation via Laplace transforms. IMA J. Numer. Anal., 24:439–463, 2004. doi:10.1093/imanum/24.3.439 R. Metzler and J. Klafter. The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep., 339:1–77, 2000. doi:10.1016/S0370-1573(00)00070-3 C. Moler and C. Van Loan. Nineteen Dubious Ways to Compute the Exponential of a Matrix, 25 Years Later. SIAM Rev., 45(1):3–49, 2003. doi:10.1137/S00361445024180 E. W. Montroll and G. H. Weiss. Random Walks on Lattices. II. J. Math. Phys., 6(2):167–181, 1965. doi:10.1063/1.1704269 I. Moret and P. Novati. On the Convergence of Krylov Subspace Methods for Matrix Mittag–Leffler Functions. SIAM J. Numer. Anal., 49(5):2144–2164, 2011. doi:10.1137/080738374 I. Podlubny. Fractional Differential Equations. Academic Press, San Diego, 1999. http://www.elsevier.com/books/fractional-differential-equations/podlubny/978-0-12-558840-9 M. Raberto, F. Rapallo, and E. Scalas. Semi-Markov Graph Dynamics. PLoS ONE, 6(8):e23370, 2011. doi:10.1371/journal.pone.0023370 S. C. Reddy and L. N. Trefethen. Pseudospectra of the convection-diffusion operator. SIAM J. Appl. Math., 54(6):1634–1649, 1994. doi:10.1137/S0036139993246982 E. B. Saff and A. D. Snider. Fundamentals of complex analysis with applications to engineering and science. Pearson Education, 2003. http://www.pearsonhighered.com/educator/product/Fundamentals-of-Complex-Analysis-with-Applications-to-Engineering-Science-and-Mathematics/9780139078743.page E. Scalas, R. Gorenflo, and F. Mainardi. Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation. Phys. Rev. E, 69:011107, 2004. doi:10.1103/PhysRevE.69.011107 D. Sheen, I. H. Sloan, and V. Thomee. A parallel method for time discretization of parabolic equations based on Laplace transformation and quadrature. IMA J. Numer. Anal., 23:269–299, 2003. doi:10.1093/imanum/23.2.269 M. J. Shon and A. E. Cohen. Mass action at the single-molecule level. J. Am. Chem. Soc., 134(35):14618–14623, 2012. doi:10.1021/ja3062425 G. Strang and S. MacNamara. Functions of difference matrices are Toeplitz plus Hankel. SIAM Rev., 56(3):525–546, 2014. doi:10.1137/120897572 A. Talbot. The accurate numerical inversion of Laplace transforms. J. Inst. Math. Appl., 23:97–120, 1979. doi:10.1093/imamat/23.1.97 L. N. Trefethen. Approximation Theory and Approximation Practice. SIAM, Philadelphia, 2013. http://bookstore.siam.org/ot128/ L. N. Trefethen and M. Embree. Spectra and pseudospectra: the behavior of nonnormal matrices and operators. Princeton University Press, 2005. http://press.princeton.edu/titles/8113.html L. N. Trefethen and J. A. C. Weideman. The exponentially convergent trapezoidal rule. SIAM Rev., 56(3):385–458, 2014. doi:10.1137/130932132 N. G. van Kampen. Stochastic Processes in Physics and Chemistry. Elsevier Science, 2001. http://www.elsevier.com/books/stochastic-processes-in-physics-and-chemistry/van-kampen/978-0-444-52965-7 J. A. C. Weideman. Improved contour integral methods for parabolic PDEs. IMA J. Numer. Anal., 30:334–350, 2010. doi:10.1093/imanum/drn074 J. A. C. Weideman and L. N. Trefethen. Parabolic and hyperbolic contours for computing the bromwich integral. Math. Comput., 76:1341–1356, 2007. doi:10.1090/S0025-5718-07-01945-X T. G. Wright. Eigtool, 2002. http://www.comlab.ox.ac.uk/pseudospectra/eigtool/. Q. Yang, T. Moroney, K. Burrage, I. Turner, and F. Liu. Novel numerical methods for time-space fractional reaction diffusion equations in two dimensions. ANZIAM J., 52:395–409, 2011. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/3791/146
Stochastic modeling of naïve T cell homeostasis for competing clonotypes VIA the master equation
Get PDFStochastic models for competing clonotypes of T cells by multivariate, continuoustime, discrete state, Markov processes have recently been proposed in the literature. A stochastic modeling framework is important because of rare events associated with small populations of some critical cell types. Usually, computational methods for these problems employ a trajectory-based approach, based on Monte Carlo simulation. This is partly because the complementary, probabilitydensity function (PDF) approaches can be expensive, but here we describe some efficient PDF approaches by directly solving the governing equations, known as the Master Equation. These computations are made very efficient through an approximation of the state space by projections and through the use of Krylov subspace methods when evolving the matrix exponential. These computational methods allow us to explore the evolution of the PDFs associated with these stochastic models, and bimodal distributions arise. Both experimental and theoretical investigations have emphasized the need to take into account effects due to aging. Thus time-dependent propensities naturally arise in immunological processes. Incorporating time-dependent propensities into the framework of the Master Equation significantly complicates the corresponding computational methods, but here we describe an efficient approach via Magnus formulas. Although this contribution focuses on the example of competing clonotypes, the general principles are relevant to multivariate Markov processes and provide fundamental techniques for computational immunology. © 2010 Society for Industrial and Applied Mathematics
Accelerated leap methods for simulating discrete stochastic chemical kinetics
No full textBiologists are increasingly conscious of the critical role that noise plays in cellular functions such as genetic regulation, often in connection with fluctuations in small numbers of key regulatory molecules. This has inspired the development of models that capture this fundamentally discrete and stochastic nature of cellular biology - most notably the Gillespie stochastic simulation algorithm (SSA). The SSA simulates a temporally homogeneous, discrete-state, continuous-time Markov process, and of course the corresponding probabilities and numbers of each molecular species must all remain positive. While accurately serving this purpose, the SSA can be computationally inefficient due to very small time stepping so faster approximations such as the Poisson and Binomial τ-leap methods have been suggested. This work places these leap methods in the context of numerical methods for the solution of stochastic differential equations (SDEs) driven by Poisson noise. This allows analogues of Euler-Maruyuma, Milstein and even higher order methods to be developed through the Itô-Taylor expansions as well as similar derivative-free Runge-Kutta approaches. Numerical results demonstrate that these novel methods compare favourably with existing techniques for simulating biochemical reactions by more accurately capturing crucial properties such as the mean and variance than existing methods
- …
