Mathematical approaches to differentiation and gene regulation

We consider some mathematical issues raised by the modelling of gene networks. The expression of genes is governed by a complex set of regulations, which is often described symbolically by interaction graphs. Once such a graph has been established, there remains the difficult task to decide which dynamical properties of the gene network can be inferred from it, in the absence of precise quantitative data about their regulation. In this paper we discuss a rule proposed by R.Thomas according to which the possibility for the network to have several stationary states implies the existence of a positive circuit in the corresponding interaction graph. We prove that, when properly formulated in rigorous terms, this rule becomes a theorem valid for several different types of formal models of gene networks. This result is already known for models of differential or boolean type. We show here that a stronger version of it holds in the differential setup when the decay of protein concentrations is taken into account. This allows us to verify also the validity of Thomas' rule in the context of piecewise-linear models and the corresponding discrete models. We discuss open problems as well.Comment: To appear in Notes Comptes-Rendus Acad. Sc. Paris, Biologi

Noise shaping Asynchronous SAR ADC based time to digital converter

Time-to-digital converters (TDCs) are key elements for the digitization of timing information in modern mixed-signal circuits such as digital PLLs, DLLs, ADCs, and on-chip jitter-monitoring circuits. Especially, high-resolution TDCs are increasingly employed in on-chip timing tests, such as jitter and clock skew measurements, as advanced fabrication technologies allow fine on-chip time resolutions. Its main purpose is to quantize the time interval of a pulse signal or the time interval between the rising edges of two clock signals. Similarly to ADCs, the performance of TDCs are also primarily characterized by Resolution, Sampling Rate, FOM, SNDR, Dynamic Range and DNL/INL. This work proposes and demonstrates 2nd order noise shaping Asynchronous SAR ADC based TDC architecture with highest resolution of 0.25 ps among current state of art designs with respect to post-layout simulation results. This circuit is a combination of low power/High Resolution 2nd Order Noise Shaped Asynchronous SAR ADC backend with simple Time to Amplitude converter (TAC) front-end and is implemented in 40nm CMOS technology. Additionally, special emphasis is given on the discussion on various current state of art TDC architectures.Electrical and Computer Engineerin

Robust filtering for gene expression time series data with variance constraints

This is the post print version of the article. The official published version can be obtained from the link below - Copyright 2007 Taylor & Francis Ltd.In this paper, an uncertain discrete-time stochastic system is employed to represent a model for gene regulatory networks from time series data. A robust variance-constrained filtering problem is investigated for a gene expression model with stochastic disturbances and norm-bounded parameter uncertainties, where the stochastic perturbation is in the form of a scalar Gaussian white noise with constant variance and the parameter uncertainties enter both the system matrix and the output matrix. The purpose of the addressed robust filtering problem is to design a linear filter such that, for the admissible bounded uncertainties, the filtering error system is Schur stable and the individual error variance is less than a prespecified upper bound. By using the linear matrix inequality (LMI) technique, sufficient conditions are first derived for ensuring the desired filtering performance for the gene expression model. Then the filter gain is characterized in terms of the solution to a set of LMIs, which can easily be solved by using available software packages. A simulation example is exploited for a gene expression model in order to demonstrate the effectiveness of the proposed design procedures.This work was supported in part by the Engineering and Physical Sciences Research Council (EPSRC) of the UK under Grants GR/S27658/01 and EP/C524586/1, the Biotechnology and Biological Sciences Research Council (BBSRC) of the UK under Grants BB/C506264/1 and 100/EGM17735, the Nuffield Foundation of the UK under Grant NAL/00630/G, and the Alexander von Humboldt Foundation of Germany

Molecular communication networks with general molecular circuit receivers

In a molecular communication network, transmitters may encode information in concentration or frequency of signalling molecules. When the signalling molecules reach the receivers, they react, via a set of chemical reactions or a molecular circuit, to produce output molecules. The counts of output molecules over time is the output signal of the receiver. The aim of this paper is to investigate the impact of different reaction types on the information transmission capacity of molecular communication networks. We realise this aim by using a general molecular circuit model. We derive general expressions of mean receiver output, and signal and noise spectra. We use these expressions to investigate the information transmission capacities of a number of molecular circuits

Time-delayed models of gene regulatory networks

We discuss different mathematical models of gene regulatory networks as relevant to the onset and development of cancer. After discussion of alternativemodelling approaches, we use a paradigmatic two-gene network to focus on the role played by time delays in the dynamics of gene regulatory networks. We contrast the dynamics of the reduced model arising in the limit of fast mRNA dynamics with that of the full model. The review concludes with the discussion of some open problems