Mathematical modelling and analysis of cellular signalling in macrophages

Cell signalling pathways play a crucial role in proper cell development and behaviour, with implications to survival, chemotaxis, proliferation, and even programmed cell death known as apoptosis. In this article, we outline a mathematical model of the G-protein signalling pathway in a particular cell line of macrophages, focusing on activation of a particular G-protein-coupled receptor, P2Y(6). The model is based on the kinetics of P2Y(6) surface receptors, inositol trisphosphate, cytosolic calcium, and differential dynamics of multiple species of diacylglycerol. Insight into the dynamics of the system is given through recently available experimental results and incorporated into the model. Mathematical analysis of the model, including establishment of global existence, uniqueness, positivity, and boundedness of solutions, and global stability of a unique steady-state solution is discussed.     

Evolution of vertebrate rod and cone phototransduction genes

Vertebrate cones and rods in several cases use separate but related components for their signal transduction (opsins, G-proteins, ion channels, etc.). Some of these proteins are also used differentially in other cell types in the retina. Because cones, rods and other retinal cell types originated in early vertebrate evolution, it is of interest to see if their specific genes arose in the extensive gene duplications that took place in the ancestor of the jawed vertebrates (gnathostomes) by two tetraploidizations (genome doublings). The ancestor of teleost fishes subsequently underwent a third tetraploidization. Our previously reported analyses showed that several gene families in the vertebrate visual phototransduction cascade received new members in the basal tetraploidizations. We here expand these data with studies of additional gene families and vertebrate species. We conclude that no less than 10 of the 13 studied phototransduction gene families received additional members in the two basal vertebrate tetraploidizations. Also the remaining three families seem to have undergone duplications during the same time period but it is unclear if this happened as a result of the tetraploidizations. The implications of the many early vertebrate gene duplications for functional specialization of specific retinal cell types, particularly cones and rods, are discussed.     

Kinetics of turn-offs of frog rod phototransduction cascade

The time course of the light-induced activity of phototrandsuction effector enzyme cGMP-phosphodiesterase (PDE) is shaped by kinetics of rhodopsin and transducin shut-offs. The two processes are among the key factors that set the speed and sensitivity of the photoresponse and whose regulation contributes to light adaptation. The aim of this study was to determine time courses of flash-induced PDE activity in frog rods that were dark adapted or subjected to nonsaturating steady background illumination. PDE activity was computed from the responses recorded from solitary rods with the suction pipette technique in Ca²⁺-clamping solution.     

Autophagy supports survival and phototransduction protein levels in rod photoreceptors

Damage and loss of the postmitotic photoreceptors is a leading cause of blindness in many diseases of the eye. Although the mechanisms of photoreceptor death have been extensively studied, few studies have addressed mechanisms that help sustain these non-replicating neurons for the life of an organism. Autophagy is an intracellular pathway where cytoplasmic constituents are delivered to the lysosomal pathway for degradation. It is not only a major pathway activated in response to cellular stress, but is also important for cytoplasmic turnover and to supply the structural and energy needs of cells. We examined the importance of autophagy in photoreceptors by deleting the essential autophagy gene Atg5 specifically in rods. Loss of autophagy led to progressive degeneration of rod photoreceptors beginning at 8 weeks of age such that by 44 weeks few rods remained. Cone photoreceptor numbers were only slightly diminished following rod degeneration but their function was significantly decreased. Rod cell death was apoptotic but was not dependent on daily light exposure or accelerated by intense light. Although the light-regulated translocation of the phototransduction proteins arrestin and transducin were unaffected in rods lacking autophagy, Atg5-deficient rods accumulated transducin-α as they degenerated suggesting autophagy might regulate the level of this protein. This was confirmed when the light-induced decrease in transducin was abolished in Atg5-deficient rods and the inhibition of autophagy in retinal explants cultures prevented its degradation. These results demonstrate that basal autophagy is essential to the long-term health of rod photoreceptors and a critical process for maintaining optimal levels of the phototransduction protein transducin-α. As the lack of autophagy is associated with retinal degeneration and altered phototransduction protein degradation in the absence of harmful gene products, this process may be a viable therapeutic target where rod cell loss is the primary pathologic event.Autophagy is an intracellular pathway where cytoplasmic constituents are delivered to the lysosomes for degradation. Defective autophagy can contribute to the age-dependent accumulation of damaged proteins and organelles leading to altered cellular homeostasis and loss of function.^{1, 2, 3, 4, 5} The metabolic roles of autophagy can be classified into two types, basal and induced. In nutrient-rich conditions, autophagy is suppressed but still occurs at low levels (basal autophagy); however, when cells are subjected to stress (starvation, injury, hypoxia), autophagy is activated immediately (induced autophagy).⁶ Induced autophagy maintains the amino acid pool inside cells to adapt to starvation while constitutive autophagy has been shown to function as a cell-repair mechanism that is important for long-lived postmitotic cells.^{7, 8, 9, 10, 11} Defects in autophagy have been associated with neurodegenerative diseases,^{12, 13, 14, 15} diabetes,^{16, 17} lysosomal storage disease¹⁸ and the loss of vision.¹⁹ In addition to macroautophagy, microautophagy and chaperone-mediated autophagy (CMA) have been described. Although little is known about microautophagy in mammalian cells, macroautophagy (hereafter autophagy) is a major pathway for bulk degradation of cytoplasmic components. CMA is a more selective pathway for degradation of cytosolic proteins that can compensate for the loss of macroautophagy.^{2, 20, 21, 22}Inherited retinal degenerative diseases such as retinitis pigmentosa or Leber''s congenital amaurosis are characterized by premature and progressive death of rod and cone photoreceptor cells.²³ These diseases are characterized by the loss of night vision due to the death of rods followed by the loss of cones leading to diminished visual acuity and a reduction in the quality of life for patients. Disease is typically associated with the production of harmful gene products that promote pathology by inhibiting critical pathways resulting in cell death.^{24, 25, 26} Strategies to prevent photoreceptor death during retinal degenerative disease such as gene replacement therapies or inhibition of cell death pathways have been undertaken with some success;^{27, 28, 29} however, effective treatments for these blinding disorders are lacking.Another strategy that could be used in conjunction with other therapies might be to enhance survival by stimulating autophagy. Augmenting autophagy would increase the supply of nutrients to stressed cells and accelerate removal of damaged proteins thereby prolonging cell survival beyond what would be possible by only preventing cell death. Although canonical^{22, 30, 31, 32, 33} and noncanonical autophagic mechanisms³⁴ have been detected in the eye, our knowledge of basic autophagy functions in this organ is still limited. In order to understand how autophagy maintains retinal homeostasis and function, we undertook studies to examine the consequences of deleting the essential autophagy gene Atg5 in rod photoreceptors.     

Narrative-based computational modelling of the Gp130/JAK/STAT signalling pathway

Background  

Appropriately formulated quantitative computational models can support researchers in understanding the dynamic behaviour of biological pathways and support hypothesis formulation and selection by "in silico" experimentation. An obstacle to widespread adoption of this approach is the requirement to formulate a biological pathway as machine executable computer code. We have recently proposed a novel, biologically intuitive, narrative-style modelling language for biologists to formulate the pathway which is then automatically translated into an executable format and is, thus, usable for analysis via existing simulation techniques.     

The gain of rod phototransduction: reconciliation of biochemical and electrophysiological measurements

We have resolved a central and long-standing paradox in understanding the amplification of rod phototransduction by making direct measurements of the gains of the underlying enzymatic amplifiers. We find that under optimized conditions a single photoisomerized rhodopsin activates transducin molecules and phosphodiesterase (PDE) catalytic subunits at rates of 120-150/s, much lower than indirect estimates from light-scattering experiments. Further, we measure the Michaelis constant, Km, of the rod PDE activated by transducin to be 10 microM, at least 10-fold lower than published estimates. Thus, the gain of cGMP hydrolysis (determined by kcat/Km) is at least 10-fold higher than reported in the literature. Accordingly, our results now provide a quantitative account of the overall gain of the rod cascade in terms of directly measured factors.     

Mathematical modelling in physiology

The article illustrates the method of mathematical modelling in physiology as a unique tool to study physiological processes. A number of demonstrated examples appear as a result of long-term experience in mathematical modelling of electrical and mechanical phenomena in the heart muscle. These examples are presented here to show that the modelling provides insight into mechanisms underlying these phenomena and is capable to predict new ones that were previously unknown. While potentialities of the mathematical modelling are analyzed with regard to the myocardium, they are quite universal to deal with any physiological processes.     

Effect of 11-cis 13-demethylretinal on phototransduction in bleach-adapted rod and cone photoreceptors

We used 11-cis 13-demethylretinal to examine the physiological consequences of retinal's noncovalent interaction with opsin in intact rod and cone photoreceptors during visual pigment regeneration. 11-Cis 13-demethylretinal is an analog of 11-cis retinal in which the 13 position methyl group has been removed. Biochemical experiments have shown that it is capable of binding in the chromophore pocket of opsin, forming a Schiff-base linkage with the protein to produce a pigment, but at a much slower rate than the native 11-cis retinal (Nelson, R., J. Kim deReil, and A. Kropf. 1970. Proc. Nat. Acad. Sci. USA. 66:531-538). Experimentally, this slow rate of pigment formation should allow separate physiological examination of the effects of the initial binding of retinal in the pocket and the subsequent formation of the protonated Schiff-base linkage. Currents from solitary rods and cones from the tiger salamander were recorded in darkness before and after bleaching and then after exposure to 11-cis 13-demethylretinal. In bleach-adapted rods, 11-cis 13-demethylretinal caused transient activation of phototransduction, as evidenced by a decrease of the dark current and sensitivity, acceleration of the dim flash responses, and activation of cGMP phosphodiesterase and guanylyl cyclase. The steady state of phototransduction activity was still higher than that of the bleach-adapted rod. In contrast, exposure of bleach-adapted cones to 11-cis 13-demethylretinal resulted in an immediate deactivation of transduction as measured by the same parameters. These results extend the validity of a model for the effects of the noncovalent binding of a retinoid in the chromophore pockets of rod and cone opsins to analogs capable of forming a Schiff-base and imply that the noncovalent binding by itself may play a role for the dark adaptation of photoreceptors.     

cAMP controls rod photoreceptor sensitivity via multiple targets in the phototransduction cascade

In early studies, both cyclic AMP (cAMP) and cGMP were considered as potential secondary messengers regulating the conductivity of the vertebrate photoreceptor plasma membrane. Later discovery of the cGMP specificity of cyclic nucleotide–gated channels has shifted attention to cGMP as the only secondary messenger in the phototransduction cascade, and cAMP is not considered in modern schemes of phototransduction. Here, we report evidence that cAMP may also be involved in regulation of the phototransduction cascade. Using a suction pipette technique, we recorded light responses of isolated solitary rods from the frog retina in normal solution and in the medium containing 2 µM of adenylate cyclase activator forskolin. Under forskolin action, flash sensitivity rose more than twofold because of a retarded photoresponse turn-off. The same concentration of forskolin lead to a 2.5-fold increase in the rod outer segment cAMP, which is close to earlier reported natural day/night cAMP variations. Detailed analysis of cAMP action on the phototransduction cascade suggests that several targets are affected by cAMP increase: (a) basal dark phosphodiesterase (PDE) activity decreases; (b) at the same intensity of light background, steady background-induced PDE activity increases; (c) at light backgrounds, guanylate cyclase activity at a given fraction of open channels is reduced; and (d) the magnitude of the Ca²⁺ exchanger current rises 1.6-fold, which would correspond to a 1.6-fold elevation of [Ca²⁺]_in. Analysis by a complete model of rod phototransduction suggests that an increase of [Ca²⁺]_in might also explain effects (b) and (c). The mechanism(s) by which cAMP could regulate [Ca²⁺]_in and PDE basal activity is unclear. We suggest that these regulations may have adaptive significance and improve the performance of the visual system when it switches between day and night light conditions.     

Multi-scale computational modelling in biology and physiology

Recent advances in biotechnology and the availability of ever more powerful computers have led to the formulation of increasingly complex models at all levels of biology. One of the main aims of systems biology is to couple these together to produce integrated models across multiple spatial scales and physical processes. In this review, we formulate a definition of multi-scale in terms of levels of biological organisation and describe the types of model that are found at each level. Key issues that arise in trying to formulate and solve multi-scale and multi-physics models are considered and examples of how these issues have been addressed are given for two of the more mature fields in computational biology: the molecular dynamics of ion channels and cardiac modelling. As even more complex models are developed over the coming few years, it will be necessary to develop new methods to model them (in particular in coupling across the interface between stochastic and deterministic processes) and new techniques will be required to compute their solutions efficiently on massively parallel computers. We outline how we envisage these developments occurring.     

Mathematical modelling of host-parasitoid systems: effects of chemically mediated parasitoid foraging strategies on within- and between-generation spatio-temporal dynamics.

In this paper we develop a novel discrete, individual-based mathematical model to investigate the effect of parasitoid foraging strategies on the spatial and temporal dynamics of host-parasitoid systems. The model is used to compare na?ve or random search strategies with search strategies that depend on experience and sensitivity to semiochemicals in the environment. It focuses on simple mechanistic interactions between individual hosts, parasitoids, and an underlying field of a volatile semiochemical (emitted by the hosts during feeding) which acts as a chemoattractant for the parasitoids. The model addresses movement at different spatial scales, where scale of movement also depends on the internal state of an individual. Individual interactions between hosts and parasitoids are modelled at a discrete (micro-scale) level using probabilistic rules. The resulting within-generation dynamics produced by these interactions are then used to generate the population levels for successive generations. The model simulations examine the effect of various key parameters of the model on (i) the spatio-temporal patterns of hosts and parasitoids within generations; (ii) the population levels of the hosts and parasitoids between generations. Key results of the model simulations show that the following model parameters have an important effect on either the development of patchiness within generations or the stability/instability of the population levels between generations: (i) the rate of diffusion of the kairomones; (ii) the specific search strategy adopted by the parasitoids; (iii) the rate of host increase between successive generations. Finally, evolutionary aspects concerning competition between several parasitoid subpopulations adopting different search strategies are also examined.     

Physicochemical modelling of cell signalling pathways

Physicochemical modelling of signal transduction links fundamental chemical and physical principles, prior knowledge about regulatory pathways, and experimental data of various types to create powerful tools for formalizing and extending traditional molecular and cellular biology.     

Fidelity and spatio-temporal control in MAP kinase (ERKs) signalling.

The mitogen activated protein (MAP) kinase module: (Raf -->MEK-->ERKs) is central to the control of cell growth, cell differentiation and cell survival. The fidelity of signalling and the spatio-temporal activation are key determinants in generating precise biological responses. The fidelity is ensured by scaffold proteins - protein kinase 'insulators' - and by specific docking sites. The duration and the intensity of the response are in part controlled by the compartmentalization of the signalling molecules. Growth factors promote rapid nuclear translocation and persistent activation of p42/p44 MAP kinases, respectively and ERK2/ERK1, during the entire G1 period with an extinction during the S-phase. These features are exquisitely controlled by the temporal induction of the MAP kinase phosphatases, MKP1-3. MKP1 and 2 induction is strictly controlled by the activation of the MAP kinase module providing evidence for an auto-regulatory mechanism. This negative regulatory loop is further enhanced by the capacity of p42/p44 MAPK to phosphorylate MKP1 and 2. This action reduces the degradation rate of MKPs through the ubiquitin-proteasomal system. Whereas the two upstream kinases of the module (Raf and MEK) remain cytoplasmic, ERKs (anchored to MEK in the cytoplasm of resting cells) rapidly translocate to the nucleus upon mitogenic stimulation. This latter process is rapid, reversible and controlled by the strict activation of the MAPK cascade. Following long-term MAPK stimulation, p42/p44 MAPKs progressively accumulate in the nucleus in an inactive form. Therefore we propose that the nucleus represents a site for ERK action, sequestration and signal termination. With the generation of knockdown mice for each of the ERK isoforms, we will illustrate that besides controlling cell proliferation the ERK cascade also controls cell differentiation and cell behaviour.     

Mathematical modelling of biofilm structures

The morphology of biofilms received much attention in the last years. Several concepts to explain the development of biofilm structures have been proposed. We believe that biofilm structure formation depends on physical as well as general and specific biological factors. The physical factors (e.g. governing substrate transport) as well as general biological factors such as growth yield and substrate conversion rates are the basic factors governing structure formation. Specific strain dependent factors will modify these, giving a further variation between different biofilm systems. Biofilm formation seems to be primarily dependent on the interaction between mass transport and conversion processes. When a biofilm is strongly diffusion limited it will tend to become a heterogeneous and porous structure. When the conversion is the rate-limiting step, the biofilm will tend to become homogenous and compact. On top of these two processes, detachment processes play a significant role. In systems with a high detachment (or shear) force, detachment will be in the form of erosion, giving smoother biofilms. Systems with a low detachment force tend to give a more porous biofilm and detachment occurs mainly by sloughing. Biofilm structure results from the interplay between these interactions (mass transfer, conversion rates, detachment forces) making it difficult to study systems taking only one of these factors into account. This revised version was published online in August 2006 with corrections to the Cover Date.     

Mathematical modelling of tumour acidity

Acid-mediated tumour invasion is receiving increasing experimental and clinical attention. Previous models proposed to describe this phenomenon failed to capture key properties of the system, such as the existence of the benign steady state, or predicted incorrectly the size of the inter-tissue gap. Here we show that taking proper account of quiescence ameliorates these drawbacks as well as revealing novel behaviour. The simplicity of the model allows us to fully identify the key parameters controlling different aspects of behaviour.     

Mathematical modelling of corneal swelling

This paper presents a differential model of the corneal transport system capable of modelling thickness changes in response to osmotic perturbations applied to either limiting membrane. The work is directed towards understanding corneal behaviour in vivo. The model considers the coupled viscous flows within the corneal stroma and across the epithelial and endothelial membranes. The flows within the stroma are established based on transport theory in porous media, while the flows across the membranes are described using the phenomenological equations of irreversible thermodynamics. The ability of the numerical model to reproduce corneal thickness changes in response to endothelial perturbations was tested against available experimental data. The sensitivity of the model to changes in stromal and membrane transport coefficients was examined.     

Mathematical modelling of tissue-engineered angiogenesis

We present a mathematical model for the vascularisation of a porous scaffold following implantation in vivo. The model is given as a set of coupled non-linear ordinary differential equations (ODEs) which describe the evolution in time of the amounts of the different tissue constituents inside the scaffold. Bifurcation analyses reveal how the extent of scaffold vascularisation changes as a function of the parameter values. For example, it is shown how the loss of seeded cells arising from slow infiltration of vascular tissue can be overcome using a prevascularisation strategy consisting of seeding the scaffold with vascular cells. Using certain assumptions it is shown how the system can be simplified to one which is partially tractable and for which some analysis is given. Limited comparison is also given of the model solutions with experimental data from the chick chorioallantoic membrane (CAM) assay.