Stochastic modelling of T cell homeostasis for two competing clonotypes via the master equation

Stochastic 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.

Stochastic chemical kinetics and the quasi-steady-state assumption : application to the stochastic simulation algorithm and chemical master equation

Recently the application of the quasi-steady-state approximation (QSSA) to the stochastic simulation algorithm (SSA) was suggested for the purpose of speeding up stochastic simulations of chemical systems that involve both relatively fast and slow chemical reactions [Rao and Arkin, J. Chem. Phys. 118, 4999 (2003)] and further work has led to the nested and slow-scale SSA. Improved numerical efficiency is obtained by respecting the vastly different time scales characterizing the system and then by advancing only the slow reactions exactly, based on a suitable approximation to the fast reactions. We considerably extend these works by applying the QSSA to numerical methods for the direct solution of the chemical master equation (CME) and, in particular, to the finite state projection algorithm [Munsky and Khammash, J. Chem. Phys. 124, 044104 (2006)], in conjunction with Krylov methods. In addition, we point out some important connections to the literature on the (deterministic) total QSSA (tQSSA) and place the stochastic analogue of the QSSA within the more general framework of aggregation of Markov processes. We demonstrate the new methods on four examples: Michaelis–Menten enzyme kinetics, double phosphorylation, the Goldbeter–Koshland switch, and the mitogen activated protein kinase cascade. Overall, we report dramatic improvements by applying the tQSSA to the CME solver.

A Krylov-based finite state projection algorithm for solving the chemical master equation arising in the discrete modelling of biological systems

Biochemical 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 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 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 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.

Further development of discrete computational techniques for calculation of restricted diffusion propagators in porous media

Magnetic resonance is a well-established tool for structural characterisation of porous media. Features of pore-space morphology can be inferred from NMR diffusion-diffraction plots or the time-dependence of the apparent diffusion coefficient. Diffusion NMR signal attenuation can be computed from the restricted diffusion propagator, which describes the distribution of diffusing particles for a given starting position and diffusion time. We present two techniques for efficient evaluation of restricted diffusion propagators for use in NMR porous-media characterisation. The first is the Lattice Path Count (LPC). Its physical essence is that the restricted diffusion propagator connecting points A and B in time t is proportional to the number of distinct length-t paths from A to B. By using a discrete lattice, the number of such paths can be counted exactly. The second technique is the Markov transition matrix (MTM). The matrix represents the probabilities of jumps between every pair of lattice nodes within a single timestep. The propagator for an arbitrary diffusion time can be calculated as the appropriate matrix power. For periodic geometries, the transition matrix needs to be defined only for a single unit cell. This makes MTM ideally suited for periodic systems. Both LPC and MTM are closely related to existing computational techniques: LPC, to combinatorial techniques; and MTM, to the Fokker-Planck master equation. The relationship between LPC, MTM and other computational techniques is briefly discussed in the paper. Both LPC and MTM perform favourably compared to Monte Carlo sampling, yielding highly accurate and almost noiseless restricted diffusion propagators. Initial tests indicate that their computational performance is comparable to that of finite element methods. Both LPC and MTM can be applied to complicated pore-space geometries with no analytic solution. We discuss the new methods in the context of diffusion propagator calculation in porous materials and model biological tissues.

Stochastic analysis of the VEGF receptor response curve

Recently, an analysis of the response curve of the vascular endothelial growth factor (VEGF) receptor and its application to cancer therapy was described in [T. Alarcón, and K. Page, J. R. Soc. Lond. Interface 4, 283–304 (2007)]. The analysis is significantly extended here by demonstrating that an alternative computational strategy, namely the Krylov FSP algorithm for the direct solution of the chemical master equation, is feasible for the study of the receptor model. The new method allows us to further investigate the hypothesis of symmetry in the stochastic fluctuations of the response. Also, by augmenting the original model with a single reversible reaction we formulate a plausible mechanism capable of realizing a bimodal response, which is reported experimentally but which is not exhibited by the original model. The significance of these findings for mechanisms of tumour resistance to antiangiogenic therapy is discussed.

Gas-phase reactions of aryl radicals with 2-butyne : an experimental and theoretical investigation employing the N-methyl-pyridinium-4-yl radical cation

Aromatic radicals form in a variety of reacting gas-phase systems, where their molecular weight growth reactions with unsaturated hydrocarbons are of considerable importance. We have investigated the ion-molecule reaction of the aromatic distonic N-methyl-pyridinium-4-yl (NMP) radical cation with 2-butyne (CH3C CCH3) using ion trap mass spectrometry. Comparison is made to high-level ab initio energy surfaces for the reaction of NMP and for the neutral phenyl radical system. The NMP radical cation reacts rapidly with 2-butyne at ambient temperature, due to the apparent absence of any barrier. The activated vinyl radical adduct predominantly dissociates via loss of a H atom, with lesser amounts of CH3 loss. High-resolution Fourier transform ion cyclotron resonance (FT-ICR) mass spectrometry allows us to identify small quantities of the collisionally deactivated reaction adduct. Statistical reaction rate theory calculations (master equation/RRKM theory) on the NMP + 2-butyne system support our experimental findings, and indicate a mechanism that predominantly involves an allylic resonance-stabilized radical formed via H atom shuttling between the aromatic ring and the C-4 side-chain, followed by cyclization and/or low-energy H atom beta-scission reactions. A similar mechanism is demonstrated for the neutral phenyl radical (Ph center dot)+2-butyne reaction, forming products that include 3-methylindene. The collisionally deactivated reaction adduct is predicted to be quenched in the form of a resonance-stabilized methylphenylallyl radical. Experiments using a 2,5-dichloro substituted methyl-pyridiniumyl radical cation revealed that in this case CH3 loss from the 2-butyne adduct is favoured over H atom loss, verifying the key role of ortho H atoms, and the shuttling mechanism, in the reactions of aromatic radicals with alkynes. As well as being useful phenyl radical analogues, pyridiniumyl radical cations may form in the ionosphere of Titan, where they could undergo rapid molecular weight growth reactions to yield polycyclic aromatic nitrogen hydrocarbons (PANHs).

Breaking time-inversion invariance through decoherence : energetic consequences for attosecond neutron scattering

Nuclei and electrons in condensed matter and/or molecules are usually entangled, due to the prevailing (mainly electromagnetic) interactions. However, the "environment" of a microscopic scattering system (e.g. a proton) causes ultrafast decoherence, thus making atomic and/or nuclear entanglement e®ects not directly accessible to experiments. However, our neutron Compton scattering experiments from protons (H-atoms) in condensed systems and molecules have a characteristic collisional time about 100|1000 attoseconds. The quantum dynamics of an atom in this ultrashort, but ¯nite, time window is governed by non-unitary time evolution due to the aforementioned decoherence. Unexpectedly, recent theoretical investigations have shown that decoherence can also have the following energetic consequences. Disentangling two subsystems A and B of a quantum system AB is tantamount to erasure of quantum phase relations between A and B. This erasure is widely believed to be an innocuous process, which e.g. does not a®ect the energies of A and B. However, two independent groups proved recently that disentangling two systems, within a su±ciently short time interval, causes increase of their energies. This is also derivable by the simplest Lindblad-type master equation of one particle being subject to pure decoherence. Our neutron-proton scattering experiments with H2 molecules provide for the first time experimental evidence of this e®ect. Our results reveal that the neutron-proton collision, leading to the cleavage of the H-H bond in the attosecond timescale, is accompanied by larger energy transfer (by about 2|3%) than conventional theory predicts. Preliminary results from current investigations show qualitatively the same e®ect in the neutron-deuteron Compton scattering from D2 molecules. We interpret the experimental findings by treating the neutron-proton (or neutron-deuteron) collisional system as an entangled open quantum system being subject to fast decoherence caused by its "environment" (i.e., two electrons plus second nucleus of H2 or D2). The presented results seem to be of generic nature, and may have considerable consequences for various processes in condensed matter and molecules, e.g. in elementary chemical reactions.

Concerted HO2 elimination from alpha-Aminoalkylperoxyl free radicals : Experimental and theoretical evidence from the gas phase NH2 center dot CHCO2- + O-2 reaction

We have investigated the gas-phase reaction of the alpha-aminoacetate (glycyl) radical anion (NH2(sic)CHCO2-) with O-2 using ion trap mass spectrometry, quantum chemistry, and statistical reaction rate theory. This radical is found to undergo a remarkably rapid reaction with O-2 to form the hydroperoxyl radical (HO2(sic)) and an even-electron imine (NHCHCO2-), with experiments and master equation simulations revealing that reaction proceeds at the ion molecule collision rate. This reaction is facilitated by a low-energy concerted HO2(sic) elimination mechanism in the NH2CH(OO(sic))CO2- peroxyl radical. These findings can explain the widely observed free-radical-mediated oxidation of simple amino acids to amides plus alpha-keto acids (their imine hydrolysis products). This work also suggests that imines will be the main intermediates in the atmospheric oxidation of primary and secondary amines, including amine carbon capture solvents such as 2-aminoethanol (commonly known as monoethanolamine, or MEA), in a process that avoids the ozone-promoting conversion of (sic)NO to (sic)NO2 commonly encountered in peroxyl radical chemistry.

Interacting motile agents : taking a mean-field approach beyond monomers and nearest-neighbor steps

We consider a discrete agent-based model on a one-dimensional lattice, where each agent occupies L sites and attempts movements over a distance of d lattice sites. Agents obey a strict simple exclusion rule. A discrete-time master equation is derived using a mean-field approximation and careful probability arguments. In the continuum limit, nonlinear diffusion equations that describe the average agent occupancy are obtained. Averaged discrete simulation data are generated and shown to compare very well with the solution to the derived nonlinear diffusion equations. This framework allows us to approach a lattice-free result using all the advantages of lattice methods. Since different cell types have different shapes and speeds of movement, this work offers insight into population-level behavior of collective cellular motion.

Collective motion of dimers

We consider a discrete agent-based model on a one-dimensional lattice and a two-dimensional square lattice, where each agent is a dimer occupying two sites. Agents move by vacating one occupied site in favor of a nearest-neighbor site and obey either a strict simple exclusion rule or a weaker constraint that permits partial overlaps between dimers. Using indicator variables and careful probability arguments, a discrete-time master equation for these processes is derived systematically within a mean-field approximation. In the continuum limit, nonlinear diffusion equations that describe the average agent occupancy of the dimer population are obtained. In addition, we show that multiple species of interacting subpopulations give rise to advection-diffusion equations. Averaged discrete simulation data compares very well with the solution to the continuum partial differential equation models. Since many cell types are elongated rather than circular, this work offers insight into population-level behavior of collective cellular motion.