3 tera-basepairs as a fundamental limit for robust DNA replication.

In order to maintain functional robustness and species integrity, organisms must ensure high fidelity of the genome duplication process. This is particularly true during early development, where cell division is often occurring both rapidly and coherently. By studying the extreme limits of suppressing DNA replication failure due to double fork stall errors, we uncover a fundamental constant that describes a trade-off between genome size and architectural complexity of the developing organism. This constant has the approximate value N U ≈ 3 × 1012 basepairs, and depends only on two highly conserved molecular properties of DNA biology. We show that our theory is successful in interpreting a diverse range of data across the Eukaryota.

Faster growth with shorter antigens can explain a VSG hierarchy during African trypanosome infections: a feint attack by parasites.

The parasitic African trypanosome, Trypanosoma brucei, evades the adaptive host immune response by a process of antigenic variation that involves the clonal switching of variant surface glycoproteins (VSGs). The VSGs that come to dominate in vivo during an infection are not entirely random, but display a hierarchical order. How this arises is not fully understood. Combining available genetic data with mathematical modelling, we report a VSG-length-dependent hierarchical timing of clonal VSG dominance in a mouse model, consistent with an inverse correlation between VSG length and trypanosome growth-rate. Our analyses indicate that, among parasites switching to new VSGs, those expressing shorter VSGs preferentially accumulate to a detectable level that is sufficient to trigger a targeted immune response. This may be due to the increased metabolic cost of producing longer VSGs. Subsequent elimination of faster-growing parasites then allows slower-growing parasites with longer VSGs to accumulate. This interaction between the host and parasite is able to explain the temporal distribution of VSGs observed in vivo. Thus, our findings reveal a length-dependent hierarchy that operates during T. brucei infection. This represents a 'feint attack' diversion tactic utilised by these persistent parasites to out-maneuver the host adaptive immune system.

Thymic involution and rising disease incidence with age.

For many cancer types, incidence rises rapidly with age as an apparent power law, supporting the idea that cancer is caused by a gradual accumulation of genetic mutations. Similarly, the incidence of many infectious diseases strongly increases with age. Here, combining data from immunology and epidemiology, we show that many of these dramatic age-related increases in incidence can be modeled based on immune system decline, rather than mutation accumulation. In humans, the thymus atrophies from infancy, resulting in an exponential decline in T cell production with a half-life of ∼16 years, which we use as the basis for a minimal mathematical model of disease incidence. Our model outperforms the power law model with the same number of fitting parameters in describing cancer incidence data across a wide spectrum of different cancers, and provides excellent fits to infectious disease data. This framework provides mechanistic insight into cancer emergence, suggesting that age-related decline in T cell output is a major risk factor.

Insights into Biological Complexity from Simple Foundations.

We discuss an overtly "simple approach" to complex biological systems borrowing selectively from theoretical physics. The approach is framed by three maxims, and we show examples of its success in two different applications: investigating cellular robustness at the level of gene regulatory networks and quantifying rare events of DNA replication errors.

Steady-state fluctuations of a genetic feedback loop: an exact solution.

Genetic feedback loops in cells break detailed balance and involve bimolecular reactions; hence, exact solutions revealing the nature of the stochastic fluctuations in these loops are lacking. We here consider the master equation for a gene regulatory feedback loop: a gene produces protein which then binds to the promoter of the same gene and regulates its expression. The protein degrades in its free and bound forms. This network breaks detailed balance and involves a single bimolecular reaction step. We provide an exact solution of the steady-state master equation for arbitrary values of the parameters, and present simplified solutions for a number of special cases. The full parametric dependence of the analytical non-equilibrium steady-state probability distribution is verified by direct numerical solution of the master equations. For the case where the degradation rate of bound and free protein is the same, our solution is at variance with a previous claim of an exact solution [J. E. M. Hornos, D. Schultz, G. C. P. Innocentini, J. Wang, A. M. Walczak, J. N. Onuchic, and P. G. Wolynes, Phys. Rev. E 72, 051907 (2005), and subsequent studies]. We show explicitly that this is due to an unphysical formulation of the underlying master equation in those studies.

Emergent cell and tissue dynamics from subcellular modeling of active biomechanical processes.

Cells and the tissues they form are not passive material bodies. Cells change their behavior in response to external biochemical and biomechanical cues. Behavioral changes, such as morphological deformation, proliferation and migration, are striking in many multicellular processes such as morphogenesis, wound healing and cancer progression. Cell-based modeling of these phenomena requires algorithms that can capture active cell behavior and their emergent tissue-level phenotypes. In this paper, we report on extensions of the subcellular element model to model active biomechanical subcellular processes. These processes lead to emergent cell and tissue level phenotypes at larger scales, including (i) adaptive shape deformations in cells responding to slow stretching, (ii) viscous flow of embryonic tissues, and (iii) streaming patterns of chemotactic cells in epithelial-like sheets. In each case, we connect our simulation results to recent experiments.

A 'chemotactic dipole' mechanism for large-scale vortex motion during primitive streak formation in the chick embryo.

Primitive streak formation in the chick embryo involves significant coordinated cell movement lateral to the streak, in addition to the posterior-anterior movement of cells in the streak proper. Cells lateral to the streak are observed to undergo 'polonaise movements', i.e. two large counter-rotating vortices, reminiscent of eddies in a fluid. In this paper, we propose a mechanism for these movement patterns which relies on chemotactic signals emitted by a dipolar configuration of cells in the posterior region of the epiblast. The 'chemotactic dipole' consists of adjacent regions of cells emitting chemo-attractants and chemo-repellents. We motivate this idea using a mathematical analogy between chemotaxis and electrostatics, and test this idea using large-scale computer simulations. We implement active cell response to both neighboring mechanical interactions and chemotactic gradients using the Subcellular Element Model. Simulations show the emergence of large-scale vortices of cell movement. The length and time scales of vortex formation are in reasonable agreement with experimental data. We also provide quantitative estimates for the robustness of the chemotaxis dipole mechanism, which indicate that the mechanism has an error tolerance of about 10% to variation in chemotactic parameters, assuming that only 1% of the cell population is involved in emitting signals. This tolerance increases for larger populations of cells emitting signals.

Spatiotemporal dynamics in marginal populations.

Population dynamics across a mortality gradient at an ecological margin are investigated using a novel modeling approach that allows direct comparison of stochastic spatially explicit simulation results with deterministic mean field models. The results show that demographic stochasticity has a large effect at population margins such that density profiles fall off more sharply than predicted by mean field models. Substantial spatial structure emerges at the margin, and spatial correlations (measured parallel to the margin) exhibit a sharp maximum in the tail of the density profile, indicating that spatial substructuring is greatest at an intermediate point across the ecological gradient. Such substructuring may have a substantial impact on Allee effects and evolutionary processes in marginal populations.

Predator-prey cycles from resonant amplification of demographic stochasticity.

We present the simplest individual level model of predator-prey dynamics and show, via direct calculation, that it exhibits cycling behavior. The deterministic analogue of our model, recovered when the number of individuals is infinitely large, is the Volterra system (with density-dependent prey reproduction) which is well known to fail to predict cycles. This difference in behavior can be traced to a resonant amplification of demographic fluctuations which disappears only when the number of individuals is strictly infinite. Our results indicate that additional biological mechanisms, such as predator satiation, may not be necessary to explain observed predator-prey cycles in real (finite) populations.

Modeling multicellular systems using subcellular elements.

We introduce a model for describing the dynamics of large numbers of interacting cells. The fundamental dynamical variables in the model are subcellular elements, which interact with each other through phenomenological intra- and intercellular potentials. Advantages of the model include i) adaptive cell-shape dynamics, ii) flexible accommodation of additional intracellular biology, and iii) the absence of an underlying grid. We present here a detailed description of the model, and use successive mean-field approximations to connect it to more coarse-grained approaches, such as discrete cell-based algorithms and coupled partial differential equations. We also discuss efficient algorithms for encoding the model, and give an example of a simulation of an epithelial sheet. Given the biological flexibility of the model, we propose that it can be used effectively for modeling a range of multicellular processes, such as tumor dynamics and embryogenesis.

Stochastic models in population biology and their deterministic analogs.

We introduce a class of stochastic population models based on "patch dynamics." The size of the patch may be varied, and this allows one to quantify the departures of these stochastic models from various mean-field theories, which are generally valid as the patch size becomes very large. These models may be used to formulate a broad range of biological processes in both spatial and nonspatial contexts. Here, we concentrate on two-species competition. We present both a mathematical analysis of the patch model, in which we derive the precise form of the competition mean-field equations (and their first-order corrections in the nonspatial case), and simulation results. These mean-field equations differ, in some important ways, from those which are normally written down on phenomenological grounds. Our general conclusion is that mean-field theory is more robust for spatial models than for a single isolated patch. This is due to the dilution of stochastic effects in a spatial setting resulting from repeated rescue events mediated by interpatch diffusion. However, discrete effects due to modest patch sizes lead to striking deviations from mean-field theory even in a spatial setting.

Many-body theory of chemotactic cell-cell interactions.

We consider an individual-based stochastic model of cell movement mediated by chemical signaling fields. This model is formulated using Langevin dynamics, which allows an analytic study using methods from statistical and many-body physics. In particular we construct a diagrammatic framework within which to study cell-cell interactions. In the mean-field limit, where statistical correlations between cells are neglected, we recover the deterministic Keller-Segel equations. Within exact perturbation theory in the chemotactic coupling epsilon , statistical correlations are non-negligible at large times and lead to a renormalization of the cell diffusion coefficient D(R)--an effect that is absent at mean-field level. An alternative closure scheme, based on the necklace approximation, probes the strong coupling behavior of the system and predicts that D(R) is renormalized to zero at a critical value of epsilon, indicating self-localization of the cell. Stochastic simulations of the model give very satisfactory agreement with the perturbative result. At higher values of the coupling simulations indicate that D(R) approximately epsilon(-2) , a result at odds with the necklace approximation. We briefly discuss an extension of our model, which incorporates the effects of short-range interactions such as cell-cell adhesion.

Accurate discretization of advection-diffusion equations.

We present an exact mathematical transformation which converts a wide class of advection-diffusion equations into a form allowing simple and direct spatial discretization in all dimensions, and thus the construction of accurate and more efficient numerical algorithms. These discretized forms can also be viewed as master equations which provide an alternative mesoscopic interpretation of advection-diffusion processes in terms of diffusion with spatially varying hopping rates.

Population dynamics with global regulation: the conserved Fisher equation.

We introduce and study a conserved version of the Fisher equation. Within a population biology context, this model describes spatially extended populations in which the total number of individuals is fixed due to either biotic or environmental factors. We find a rich spectrum of dynamical phases including a pseudotraveling wave and, in the presence of the Allee effect, a phase transition from a locally constrained high density state to a low density fragmented state.

Extinction times and moment closure in the stochastic logistic process.

We investigate the statistics of extinction times for an isolated population, with an initially modest number M of individuals, whose dynamics are controlled by a stochastic logistic process (SLP). The coefficient of variation in the extinction time V is found to have a maximum value when the death and birth rates are close in value. For large habitat size K we find that Vmax is of order K1/4 / M1/2, which is much larger than unity so long as M is small compared to K1/2. We also present a study of the SLP using the moment closure approximation (MCA), and discuss the successes and failures of this method. Regarding the former, the MCA yields a steady-state distribution for the population when the death rate is low. Although not correct for the SLP model, the first three moments of this distribution coincide with those calculated exactly for an adjusted SLP in which extinction is forbidden. These exact calculations also pinpoint the breakdown of the MCA as the death rate is increased.

Population dynamics with a refuge: fractal basins and the suppression of chaos.

We consider the effect of coupling an otherwise chaotic population to a refuge. A rich set of dynamical phenomena is uncovered. We consider two forms of density dependence in the active population: logistic and exponential. In the former case, the basin of attraction for stable population growth becomes fractal, and the bifurcation diagrams for the active and refuge populations are chaotic over a wide range of parameter space. In the case of exponential density dependence, the dynamics are unconditionally stable (in that the population size is always positive and finite), and chaotic behavior is completely eradicated for modest amounts of dispersal. We argue that the use of exponential density dependence is more appropriate, theoretically as well as empirically, in a model of refuge dynamics.

Critical dimensions of the diffusion equation.

We study the evolution of a random initial field under pure diffusion in various space dimensions. From numerical calculations we find that the persistence properties of the system show sharp transitions at critical dimensions d(1) approximately 26 and d(2) approximately 46. We also give refined measurements of the persistence exponents for low dimensions.