Modeling tubular shapes in the inner mitochondrial membrane.

The inner mitochondrial membrane has been shown to have a novel structure that contains tubular components whose radii are of the order of 10 nm as well as comparatively flat regions. The structural organization of mitochondria is important for understanding their functionality. We present a model that can account, thermodynamically, for the observed size of the tubules. The model contains two lipid constituents with different shapes. They are allowed to distribute in such a way that the composition differs on the two sides of the tubular membrane. Our calculations make two predictions: (1) there is a pressure difference of 0.2 atmospheres across the inner membrane as a necessary consequence of the experimentally observed tubule radius of 10 nm, and (2) migration of differently shaped lipids causes concentration variations of the order of 7% between the two sides of the tubular membrane.

Variable maturation velocity and parameter sensitivity in a model of haematopoiesis.

We analyse an age-structured model for haematopoiesis, describing the development of specialized cells in the blood from undifferentiated stem cells and including the controlling effects of hormones. Variation in the length of time for maturing of precursor cells in this model has a stabilizing influence. When the maturing process does not vary, then the age-structured model reduces to a delay differential equation. Depending on the death process considered, either a differential equation with two time delays or a differential equation with a state-dependent delay is obtained. Each of these is analysed in turn, for its linear stability. A sensitivity analysis of the parameters in this model shows which biochemical processes in the negative feedback most strongly affect the solutions.

Regulation of platelet production: the normal response to perturbation and cyclical platelet disease.

An age-structured model for the regulation of platelet production is developed, and compared with both normal and pathological platelet production. We consider the role of thrombopoietin (TPO) in this process, how TPO affects the transition between megakaryocytes of various ploidy classes, and their individual contributions to platelet production. After the estimation of the relevant parameters of the model from both in vivo and in vitro data, we use the model to numerically reproduce the normal human response to a bolus injection of TPO. We further show that our model reproduces the dynamic characteristics of autoimmune cyclical thromobocytopenia if the rate of platelet destruction in the circulation is elevated to more than twice the normal value.

Hematopoietic model with moving boundary condition and state dependent delay: applications in erythropoiesis.

An age-structured model for erythropoiesis is extended to include the active destruction of the oldest mature cells and possible control by apoptosis. The former condition, which is applicable to other population models where the predator satiates, becomes a constant flux boundary condition and results in a moving boundary condition. The method of characteristics reduces the age-structured model to a system of threshold type differential delay equations. Under certain assumptions, this model can be reduced to a system of delay differential equations with a state dependent delay in an uncoupled differential equation for the moving boundary condition. Analysis of the characteristic equation for the linearized model demonstrates the existence of a Hopf bifurcation when the destruction rate of erythrocytes is modified. The parameters in the system are estimated from experimental data, and the model is simulated for a normal human subject following a loss of blood typical of a blood donation. Numerical studies for a rabbit with an induced auto-immune hemolytic anemia are performed and compared with experimental data.

Age-structured and two-delay models for erythropoiesis.

An age-structured model is developed for erythropoiesis and is reduced to a system of threshold-type differential delay equations using the method of characteristics. Under certain assumptions, this model can be reduced to a system of delay differential equations with two delays. The parameters in the system are estimated from experimental data, and the model is simulated for a normal human subject following a loss of blood. The characteristic equation of the two-delay equation is analyzed and shown to exhibit Hopf bifurcations when the destruction rate of erythrocytes is increased. A numerical study for a rabbit with autoimmune hemolytic anemia is performed and compared with experimental data.

Variation in concentrations of RNAs and proteins involved in gene expression of Escherichia coli.

A growing cell is not a steady-state situation. At some point in the cell cycle each gene doubles its concentration, which may result in a doubling of the production rate of its mRNA. The variations in concentration of mRNAs and proteins in an individual exponentially growing cell are studied for a gene which is constitutive, autorepressed, or autoactivated. Analysis of the mathematical model for a constitutive gene shows a fixed variation in concentration of a stable RNA between its minimum and maximum concentration of approximately 6%, independent of cell cycle time or gene location. The variation in the concentrations of the mRNA and protein for a constitutive gene are studied as gene position, molecular stability, and cell cycle time are varied. One result shows that doubling the growth rate can more than double the percentage of product from a gene near the origin of replication compared to one near the terminus. Results from these theoretical studies compare favorably to experimental results for rRNAs and the fully induced lac gene. Additional studies were performed to determine the effects of autorepression and autoactivation on the variation of concentration of mRNAs and proteins. The studies show that RNA and protein products of an autorepressed gene have almost the same variation in concentration as products of a constitutively expressed gene though the absolute concentrations are decreased. It is shown that the stability of an mRNA or protein affects variation in its concentration throughout the cell cycle, much more than the type of genetic control. The strength of repression has no effect on the variation in concentration of RNA and protein gene products through a cell cycle. Studies of an autoactivated gene show that it is significantly more responsive to a shift up or down, such as those caused by nutritional changes. An example is provided where the autoactivated gene is not expressed at one growth rate, but turns on at a higher growth rate. Furthermore, it is shown that gene position may determine whether or not the autoactivated gene is expressed.

Oscillations in a model of repression with external control.

A mathematical model for control by repression by an extracellular substance is developed, including diffusion and time delays. The model examines how active transport of a nutrient can produce either oscillatory or stable responses depending on a variety of parameters, such as diffusivity, cell size, or nutrient concentration. The system of equations for the mathematical model is reduced to a system of delay differential equations and linear Volterra equations. After linearizing these equations and forming the limiting Volterra equations, the resulting linear system no longer has any spatial dependence. Local stability analysis of the radially symmetric model shows that the system of equations can undergo Hopf bifurcations for certain parameter values, while other ranges of the parameters guarantee asymptotic stability. One numerical study shows that the model can exhibit intracellular biochemical oscillations with increasing extracellular concentrations of the nutrient, which suggests a possible trigger mechanism for morphogenesis.

A model for the initiation of replication in Escherichia coli.

The role of the protein DnaA as the principal control of replication initiation is investigated by a mathematical model. Data showing that DnaA is growth rate regulated suggest that its concentration alone is not the only factor determining the timing of initiation. A mathematical model with stochastic and deterministic components is constructed from known experimental evidence and subdivides the total pool of DnaA protein into four forms. The active form, DnaA.ATP, can be bound to the origin of replication, oriC, where it is assumed that a critical level of these bound molecules is needed to initiate replication. The active form can also exist in a reserve pool bound to the chromosome or a free pool in the cytoplasm. Finally, a large inactive pool of DnaA protein completes the state variables and provides an explanation for how the DnaA.ATP form could be the principal controlling element in the timing of initiation. The fact that DnaA protein is an autorepressor is used to derive its synthesis rate. The model studies a single exponentially growing cell through a series of cell divisions. Computer simulations are performed, and the results compare favorably to data for different cell cycle times. The model shows synchrony of initiation events in agreement with experimental results.

Cellular control models with linked positive and negative feedback and delays. II. Linear analysis and local stability.

An analysis of local behavior is made of two nonlinear models which incorporate both an induction or positive feedback control mechanism and a repression or negative feedback control mechanism. The systems of differential equations with delays are linearized about their equilibria. The related characteristic equations which are exponential polynomials are studied to determine the local stability of the models. Computer studies are included to show the range of stability for different parameter values, and the biological significance is discussed briefly.

Cellular control models with linked positive and negative feedback and delays. I. The models.

Basic techniques from biochemical kinetics are used to develop models for a cellular control system with linked positive and negative feedback. The models are represented by a system of nonlinear differential equations with delays. The lac operon provides an example of a control system where the transcription of the operon is controlled by induction or positive feedback control and catabolite repression or negative feedback control. These processes are linked through the metabolism of lactose.

Models of genetic control by repression with time delays and spatial effects.

Two models for cellular control by repression are developed in this paper. The models use standard theory from compartmental analysis and biochemical kinetics. The models include time delays to account for the processes of transcription and translation and diffusion to account for spatial effects in the cell. This consideration leads to a coupled system of reaction-diffusion equations with time delays. An analysis of the steady-state problem is given. Some results on the existence and uniqueness of a global solution and stability of the steady-state problem are summarized, and numerical simulations showing stability and periodicity are presented. A Hopf bifurcation result and a theorem on asymptotic stability are given for the limiting case of the models without diffusion.