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.

Accelerated leap methods for simulating discrete stochastic chemical kinetics

Biologists 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.

Supplement : Efficient weak second-order stochastic Runge-Kutta methods for non-commutative Stratonvich stochastic differential equations

This paper gives a modification of a class of stochastic Runge–Kutta methods proposed in a paper by Komori (2007). The slight modification can reduce the computational costs of the methods significantly.

Evolving genetic regulatory networks performing as stochastic switches

Recent studies have shown that small genetic regulatory networks (GRNs) can be evolved in silico displaying certain dynamics in the underlying mathematical model. It is expected that evolutionary approaches can help to gain a better understanding of biological design principles and assist in the engineering of genetic networks. To take the stochastic nature of GRNs into account, our evolutionary approach models GRNs as biochemical reaction networks based on simple enzyme kinetics and simulates them by using Gillespie’s stochastic simulation algorithm (SSA). We have already demonstrated the relevance of considering intrinsic stochasticity by evolving GRNs that show oscillatory dynamics in the SSA but not in the ODE regime. Here, we present and discuss first results in the evolution of GRNs performing as stochastic switches.

Activity modelling in crowded environments : a soft decision approach

Probabilistic topic models have recently been used for activity analysis in video processing, due to their strong capacity to model both local activities and interactions in crowded scenes. In those applications, a video sequence is divided into a collection of uniform non-overlaping video clips, and the high dimensional continuous inputs are quantized into a bag of discrete visual words. The hard division of video clips, and hard assignment of visual words leads to problems when an activity is split over multiple clips, or the most appropriate visual word for quantization is unclear. In this paper, we propose a novel algorithm, which makes use of a soft histogram technique to compensate for the loss of information in the quantization process; and a soft cut technique in the temporal domain to overcome problems caused by separating an activity into two video clips. In the detection process, we also apply a soft decision strategy to detect unusual events.We show that the proposed soft decision approach outperforms its hard decision counterpart in both local and global activity modelling.

Modelling word meaning using efficient tensor representations

Models of word meaning, built from a corpus of text, have demonstrated success in emulating human performance on a number of cognitive tasks. Many of these models use geometric representations of words to store semantic associations between words. Often word order information is not captured in these models. The lack of structural information used by these models has been raised as a weakness when performing cognitive tasks. This paper presents an efficient tensor based approach to modelling word meaning that builds on recent attempts to encode word order information, while providing flexible methods for extracting task specific semantic information.

The geometric modelling of social frames and contexts

How do humans respond to their social context? This question is becoming increasingly urgent in a society where democracy requires that the citizens of a country help to decide upon its policy directions, and yet those citizens frequently have very little knowledge of the complex issues that these policies seek to address. Frequently, we find that humans make their decisions more with reference to their social setting, than to the arguments of scientists, academics, and policy makers. It is broadly anticipated that the agent based modelling (ABM) of human behaviour will make it possible to treat such social effects, but we take the position here that a more sophisticated treatment of context will be required in many such models. While notions such as historical context (where the past history of an agent might affect its later actions) and situational context (where the agent will choose a different action in a different situation) abound in ABM scenarios, we will discuss a case of a potentially changing context, where social effects can have a strong influence upon the perceptions of a group of subjects. In particular, we shall discuss a recently reported case where a biased worm in an election debate led to significant distortions in the reports given by participants as to who won the debate (Davis et al 2011). Thus, participants in a different social context drew different conclusions about the perceived winner of the same debate, with associated significant differences among the two groups as to who they would vote for in the coming election. We extend this example to the problem of modelling the likely electoral responses of agents in the context of the climate change debate, and discuss the notion of interference between related questions that might be asked of an agent in a social simulation that was intended to simulate their likely responses. A modelling technology which could account for such strong social contextual effects would benefit regulatory bodies which need to navigate between multiple interests and concerns, and we shall present one viable avenue for constructing such a technology. A geometric approach will be presented, where the internal state of an agent is represented in a vector space, and their social context is naturally modelled as a set of basis states that are chosen with reference to the problem space.

Modelling and performance evaluation of the IEEE 802.11 DCF for real-time control

Popular wireless networks, such as IEEE 802.11/15/16, are not designed for real-time applications. Thus, supporting real-time quality of service (QoS) in wireless real-time control is challenging. This paper adopts the widely used IEEE 802.11, with the focus on its distributed coordination function (DCF), for soft-real-time control systems. The concept of the critical real-time traffic condition is introduced to characterize the marginal satisfaction of real-time requirements. Then, mathematical models are developed to describe the dynamics of DCF based real-time control networks with periodic traffic, a unique feature of control systems. Performance indices such as throughput and packet delay are evaluated using the developed models, particularly under the critical real-time traffic condition. Finally, the proposed modelling is applied to traffic rate control for cross-layer networked control system design.

Modelling contractor's and subcontractor's trust : a system dynamics approach

A general mistrust within the contactor and subcontractor companies has identified one of the significant barriers to derive benefits from true downstream supply chain integration. Using the general theory of trust in inter-organizational relations and conducting interviews, this research discusses factors that influence development of trust and cooperation in contractor– subcontractor relationships in construction projects. System dynamics is the simulation method is selected in this theory-building effort, based on qualitative data collected from two projects of a construction company in Thailand. Performance, permeability and system based trust are found to make significant contributions toward parties’ trust level. Three strategic policies such as best value contracting, management of subcontractors as internal team and semi project partnering approach are recommended to stimulate the trust factors as well as cooperative long term relationship.

A Popperian consilience : modelling mathematical knowledge and understanding

Goldin (2003) and McDonald, Yanchar, and Osguthorpe (2005) have called for mathematics learning theory that reconciles the chasm between ideologies, and which may advance mathematics teaching and learning practice. This paper discusses the theoretical underpinnings of a recently completed PhD study that draws upon Popper’s (1978) three-world model of knowledge as a lens through which to reconsider a variety of learning theories, including Piaget’s reflective abstraction. Based upon this consideration of theories, an alternative theoretical framework and complementary operational model was synthesised, the viability of which was demonstrated by its use to analyse the domain of early-number counting, addition and subtraction.

Data modelling in the beginning school years

This paper argues for a renewed focus on statistical reasoning in the beginning school years, with opportunities for children to engage in data modelling. Some of the core components of data modelling are addressed. A selection of results from the first data modelling activity implemented during the second year (2010; second grade) of a current longitudinal study are reported. Data modelling involves investigations of meaningful phenomena, deciding what is worthy of attention (identifying complex attributes), and then progressing to organising, structuring, visualising, and representing data. Reported here are children's abilities to identify diverse and complex attributes, sort and classify data in different ways, and create and interpret models to represent their data.

An approximate Bayesian computation approach for estimating parameters of complex environmental processes in a cellular automata

Modelling an environmental process involves creating a model structure and parameterising the model with appropriate values to accurately represent the process. Determining accurate parameter values for environmental systems can be challenging. Existing methods for parameter estimation typically make assumptions regarding the form of the Likelihood, and will often ignore any uncertainty around estimated values. This can be problematic, however, particularly in complex problems where Likelihoods may be intractable. In this paper we demonstrate an Approximate Bayesian Computational method for the estimation of parameters of a stochastic CA. We use as an example a CA constructed to simulate a range expansion such as might occur after a biological invasion, making parameter estimates using only count data such as could be gathered from field observations. We demonstrate ABC is a highly useful method for parameter estimation, with accurate estimates of parameters that are important for the management of invasive species such as the intrinsic rate of increase and the point in a landscape where a species has invaded. We also show that the method is capable of estimating the probability of long distance dispersal, a characteristic of biological invasions that is very influential in determining spread rates but has until now proved difficult to estimate accurately.

Closing the loop of design and analysis : Parametric modelling tools for early decision support

There is a growing need for parametric design software that communicates building performance feedback in early architectural exploration to support decision-making. This paper examines how the circuit of design and analysis process can be closed to provide active and concurrent feedback between architecture and services engineering domains. It presents the structure for an openly customisable design system that couples parametric modelling and energy analysis software to allow designers to assess the performance of early design iterations quickly. Finally, it discusses how user interactions with the system foster information exchanges that facilitate the sharing of design intelligence across disciplines.

Engineering-based modelling experiences in the elementary classroom

Introducing engineering-based model-eliciting experiences in the elementary curriculum is a new and increasingly important domain of research by mathematics, science, technology, and engineering educators. Recent research has raised questions about the context of engineering problems that are meaningful, engaging, and inspiring for young students. In the present study an environmental engineering activity was implemented in two classes of 11-year-old students in Cyprus. The problem required students to develop a procedure for selecting among alternative countries from which to buy water. Students created a range of models that adequately solved the problem although not all models took into account all of the data provided. The models varied in the number of problem factors taken into consideration and also in the different approaches adopted in dealing with the problem factors. At least two groups of students integrated into their models the environmental aspect of the problem (energy consumption, water pollution) and further refined their models. Results indicate that engineering model-eliciting activities can be introduced effectively into the elementary curriculum, providing rich opportunities for students to deal with engineering contexts and to apply their learning in mathematics and science to solving real-world engineering problems.

Complex modelling in the primary/middle school years

The world’s increasing complexity, competitiveness, interconnectivity, and dependence on technology generate new challenges for nations and individuals that cannot be met by continuing education as usual (Katehi, Pearson, & Feder, 2009). With the proliferation of complex systems have come new technologies for communication, collaboration, and conceptualisation. These technologies have led to significant changes in the forms of mathematical and scientific thinking that are required beyond the classroom. Modelling, in its various forms, can develop and broaden children’s mathematical and scientific thinking beyond the standard curriculum. This paper first considers future competencies in the mathematical sciences within an increasingly complex world. Next, consideration is given to interdisciplinary problem solving and models and modelling. Examples of complex, interdisciplinary modelling activities across grades are presented, with data modelling in 1st grade, model-eliciting in 4th grade, and engineering-based modelling in 7th-9th grades.