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International audience; Delays appear in the dynamics of many systems due to non-vanishing reaction times of control systems. In biochemical systems, long sequences of repeated steps, especially in biopolymerization processes, can be modeled by delays. However, modelling systems with delays is often complicated by physical constraints, such as the requirement that solutions representing concentrations of chemical species remain positive. In this work, we consider a model for a detoxifying enzyme whose synthesis is controlled by its substrate. The model includes bindingsite clearance delays, caused by the time required for an RNA polymerase or ribosome to clear its binding site before another such machine can bind. The existence of a positive equilibrium and the positivity and boundedness of solutions of the corresponding delay-differential equations are proven. In addition, the stability of the model is studied using the "small-gain" theorem.

The expression of the TGF-β protein Nodal on the left side of vertebrate embryos is a determining event in the development of internal-organ asymmetry. We present a mathematical model for the control of the expression of Nodal and its antagonist Lefty consisting entirely of realistic elementary reactions. We analyze the model in the absence of Lefty and find a wide range of parameters over which bistability (two stable steady states) is observed, with one stable steady state a low-Nodal state corresponding to the right-hand developmental fate, and the other a high-Nodal state corresponding to the left. We find that bistability requires a transcription factor containing two molecules of phosphorylated Smad2. A numerical survey of the full model, including Lefty, shows the effects of Lefty on the potential for bistability, and on the conditions that lead to the system reaching one or the other steady state.

During the templated biopolymerization processes of transcription and translation, a macromolecular machine, either an RNA polymerase or a ribosome, binds to a specific site on the template. Due to the sizes of these enzymes, there is a waiting time before one clears the binding site and another can bind. These clearance delays are relatively short, and one might think that they could be neglected. However, in the case of transcription, these clearance delays are associated with conservation laws, resulting in surprisingly large effects on the bifurcation diagrams in models of gene expression networks. We study an example of this phenomenon in a model of a gene regulated by a non-coding RNA displaying bistability. Neglecting the binding-site clearance delays in this model can only be compensated for by making ad hoc, unphysical adjustments to the model's kinetic constants.

Volume 1, Issue 1 of Mathematical Biosciences was the venue for a now-classic paper on the application of singular perturbation theory in enzyme kinetics, "On the mathematical status of the pseudo-steady state hypothesis of biochemical kinetics" by F. G. Heineken, H. M. Tsuchiya and R. Aris. More than 50 years have passed, and yet this paper continues to be studied and mined for insights. This perspective discusses both the strengths and weaknesses of the work presented in this paper. For many, the justification of the pseudo-steady-state approximation using singular perturbation theory is the main achievement of this paper. However, there is so much more material here, which laid the foundation for a great deal of research in mathematical biochemistry in the intervening decades. The parameterization of the equations, construction of the first-order uniform singular-perturbation solution, and an attempt to apply similar principles to the pseudo-equilibrium approximation are discussed in particular detail.

Network conditions for Turing instability in biochemical systems with two biochemical species are well known and involve autocatalysis or self-activation. On the other hand general network conditions for potential Turing instabilities in large biochemical reaction networks are not well developed. A biochemical reaction network with any number of species where only one species moves is represented by a simple digraph and is modeled by a reaction–diffusion system with non-mass action kinetics. A graph-theoretic condition for potential Turing-Hopf instability that arises when a spatially homogeneous equilibrium loses its stability via a single pair of complex eigenvalues is obtained. This novel graph-theoretic condition is closely related to the negative cycle condition for oscillations in ordinary differential equation models and its generalizations, and requires the existence of a pair of subnetworks, each containing an even number of positive cycles. The technique is illustrated with a double-cycle Goodwin type model.

Rubisco, the most abundant protein serving as the primary engine generating organic biomass on Earth, is characterized by a low catalytic constant (in higher plants approx. 3 s −1 ) and low specificity for CO 2 leading to photorespiration. We analyze here why this enzyme evolved as the main carbon fixation engine. The high concentration of Rubisco exceeding the concentration of its substrate CO 2 by 2–3 orders of magnitude makes application of Michaelis–Menten kinetics invalid and requires alternative kinetic approaches to describe photosynthetic CO 2 assimilation. Efficient operation of Rubisco is supported by a strong flux of CO 2 to the chloroplast stroma provided by fast equilibration of bicarbonate and CO 2 and forwarding the latter to Rubisco reaction centers. The main part of this feedforward mechanism is a thylakoidal carbonic anhydrase associated with photosystem II and pumping CO 2 from the thylakoid lumen in coordination with the rate of electron transport, water splitting and proton gradient across the thylakoid membrane. This steady flux of CO 2 limits photosynthesis at saturating CO 2 concentrations. At low ambient CO 2 and correspondingly limited capacity of the bicarbonate pool in the stroma, its depletion at the sites of Rubisco is relieved by utilizing O 2 instead of CO 2 , i.e. by photorespiration, a process which supplies CO 2 back to Rubisco and buffers the redox state and energy level in the chloroplast. Thus, the regulation of Rubisco function aims to keep steady non-equilibrium levels of CO 2 , NADPH/NADP and ATP/ADP in the chloroplast stroma and to optimize the condition of homeostatic photosynthetic flux of matter and energy.

Single-enzyme systems can undergo weakly damped concentration oscillations if the enzyme can be converted to an inactive form which only slowly reverts to the active form. Inactivation can take many forms, including a simple (hysteretic) conformational change of the enzyme. This observation may provide an explanation for the damped oscillations seen in photosynthetic activity following a sudden disturbance since rubisco, the enzyme responsible for fixing carbon, is a hysteretic enzyme with appropriate kinetic constants for oscillation. Copyright 1998 Academic Press