Nonlinear Growth Dynamics Research Articles

Nonlinear Growth Dynamics of Neuronal Cells Cultured on Directional Surfaces.

During the development of the nervous system, neuronal cells extend axons and dendrites that form complex neuronal networks, which are essential for transmitting and processing information. Understanding the physical processes that underlie the formation of neuronal networks is essential for gaining a deeper insight into higher-order brain functions such as sensory processing, learning, and memory. In the process of creating networks, axons travel towards other recipient neurons, directed by a combination of internal and external cues that include genetic instructions, biochemical signals, as well as external mechanical and geometrical stimuli. Although there have been significant recent advances, the basic principles governing axonal growth, collective dynamics, and the development of neuronal networks remain poorly understood. In this paper, we present a detailed analysis of nonlinear dynamics for axonal growth on surfaces with periodic geometrical patterns. We show that axonal growth on these surfaces is described by nonlinear Langevin equations with speed-dependent deterministic terms and gaussian stochastic noise. This theoretical model yields a comprehensive description of axonal growth at both intermediate and long time scales (tens of hours after cell plating), and predicts key dynamical parameters, such as speed and angular correlation functions, axonal mean squared lengths, and diffusion (cell motility) coefficients. We use this model to perform simulations of axonal trajectories on the growth surfaces, in turn demonstrating very good agreement between simulated growth and the experimental results. These results provide important insights into the current understanding of the dynamical behavior of neurons, the self-wiring of the nervous system, as well as for designing innovative biomimetic neural network models.

Steps behaving badly: Nonlinear dynamics and a terrace-loading instability during solution growth of lysozyme crystals

A detailed finite-element model for step motion during solution crystal growth is applied to analyze the nonlinear growth dynamics of lysozyme. Insights from this model are applied to develop a much simpler, lumped-parameter model to describe step motion for this growth system. Bifurcation analysis applied to the lumped-parameter model demonstrates a critical bulk supersaturation at which an equidistant step train loses stability, via a supercritical Hopf bifurcation, to a stable, time-periodic system that exhibits a single step bunch. As supersaturation is further increased, additional bifurcations lead to more complicated behaviors, such as quasi-periodic patterns, multiple step bunches, and period doubling. The initial instability of the step train arises from the dynamics of terrace loading due to slow adsorption and step incorporation kinetics under high bulk supersaturation. These effects cause a time lag in the surface supersaturation level with respect to changes in terrace width, producing a phase shift of the surface supersaturation field in the direction of step motion and driving the formation of step bunches.

Transition Therapy: Tackling the Ecology of Tumor Phenotypic Plasticity

Phenotypic switching in cancer cells has been found to be present across tumor types. Recent studies on Glioblastoma report a remarkably common architecture of four well-defined phenotypes coexisting within high levels of intra-tumor genetic heterogeneity. Similar dynamics have been shown to occur in breast cancer and melanoma and are likely to be found across cancer types. Given the adaptive potential of phenotypic switching (PHS) strategies, understanding how it drives tumor evolution and therapy resistance is a major priority. Here we present a mathematical framework uncovering the ecological dynamics behind PHS. The model is able to reproduce experimental results, and mathematical conditions for cancer progression reveal PHS-specific features of tumors with direct consequences on therapy resistance. In particular, our model reveals a threshold for the resistant-to-sensitive phenotype transition rate, below which any cytotoxic or switch-inhibition therapy is likely to fail. The model is able to capture therapeutic success thresholds for cancers where nonlinear growth dynamics or larger PHS architectures are in place, such as glioblastoma or melanoma. By doing so, the model presents a novel set of conditions for the success of combination therapies able to target replication and phenotypic transitions at once. Following our results, we discuss transition therapy as a novel scheme to target not only combined cytotoxicity but also the rates of phenotypic switching.

Polysaccharide-Bacteria Interactions From the Lens of Evolutionary Ecology.

Microbes have the unique ability to break down the complex polysaccharides that make up the bulk of organic matter, initiating a cascade of events that leads to their recycling. Traditionally, the rate of organic matter degradation is perceived to be limited by the chemical and physical structure of polymers. Recent advances in microbial ecology, however, suggest that polysaccharide persistence can result from non-linear growth dynamics created by the coexistence of alternate degradation strategies, metabolic roles as well as by ecological interactions between microbes. This complex “landscape” of degradation strategies and interspecific interactions present in natural microbial communities appears to be far from evolutionarily stable, as frequent gene gain and loss reshape enzymatic repertoires and metabolic roles. In this perspective, we discuss six challenges at the heart of this problem, ranging from the evolution of genetic repertoires, phenotypic heterogeneity in clonal populations, the development of a trait-based ecology, and the impact of metabolic interactions and microbial cooperation on degradation rates. We aim to reframe some of the key questions in the study of polysaccharide-bacteria interactions in the context of eco-evolutionary dynamics, highlighting possible research directions that, if pursued, would advance our understanding of polysaccharide degraders at the interface between biochemistry, ecology and evolution.

Where and how do localized perturbations affect stream and coastal ocean populations with nonlinear growth dynamics?

Chaotic systems are sensitive to small changes in parameters and initial conditions, but are spatially distributed chaotic populations sensitive to the location of perturbations within their domain? Here, we examine the transient responses to perturbation in a density-dependent population model with asymmetrical dispersal, where offspring are moved some distance downstream of the parent population and diffusively spread, an appropriate approximation for species in streams, rivers, and ocean currents. We find that over a portion of the studied parameter space, our system displays extreme convective instability; that is, an arbitrarily small point-source perturbation to the upstream region grows and propagates downstream (in our finite one-dimensional domain, the current moves offspring preferentially in the downstream direction). We can then divide our domain into distinct upstream and downstream regions based on the response to perturbation. A perturbation to the upstream region will create transient downstream instability, and a perturbation to the downstream region has no qualitative effect on the population. We investigate the factors affecting the existence/size of the sensitive upstream region, discuss the ecologically relevant time scales of the transient dynamics following a perturbation, and consider the ecological implications of managing and predicting a population with transient amplification of perturbations.

Modeling the Dichotomy of the Immune Response to Cancer: Cytotoxic Effects and Tumor-Promoting Inflammation.

Although the immune response is often regarded as acting to suppress tumor growth, it is now clear that it can be both stimulatory and inhibitory. The interplay between these competing influences has complex implications for tumor development, cancer dormancy, and immunotherapies. In fact, early immunotherapy failures were partly due to a lack in understanding of the nonlinear growth dynamics these competing immune actions may cause. To study this biological phenomenon theoretically, we construct a minimally parameterized framework that incorporates all aspects of the immune response. We combine the effects of all immune cell types, general principles of self-limited logistic growth, and the physical process of inflammation into one quantitative setting. Simulations suggest that while there are pro-tumor or antitumor immunogenic responses characterized by larger or smaller final tumor volumes, respectively, each response involves an initial period where tumor growth is stimulated beyond that of growth without an immune response. The mathematical description is non-identifiable which allows an ensemble of parameter sets to capture inherent biological variability in tumor growth that can significantly alter tumor-immune dynamics and thus treatment success rates. The ability of this model to predict non-intuitive yet clinically observed patterns of immunomodulated tumor growth suggests that it may provide a means to help classify patient response dynamics to aid identification of appropriate treatments exploiting immune response to improve tumor suppression, including the potential attainment of an immune-induced dormant state.

Nonlinear tree growth dynamics predict resilience to disturbance

Following a disturbance, why does one tree survive while another dies? Physiological mechanisms may explain varying responses to disturbance between different tree species, but fewer studies have investigated conspecific variation in resilience to forest disturbance. We propose that a dynamic signal found in trees may provide clues to their post‐disturbance fate. Specifically, linear versus nonlinear growth dynamics of a tree may be an indicator of its likelihood to survive a disturbance. Here, we investigate stands of red oak (Quercus rubra L.) that experienced disturbances in the form of drought and insect attack. Earlier work indicated that oaks dying during these disturbances had faster growth rates in their first years of life, but there was no obvious difference in canopy status, size, age, or microsite habitat between trees that survived and those that died. To investigate potential differences in growth dynamics between these trees, we quantified radial growth of individual trees and used two forecasting models to classify tree growth dynamics as linear or nonlinear. Trees were classified as healthy, declining, or dying based on crown cover, and dynamic patterns of growth were related to these health classifications. Contrary to expectations, we found healthy Q. rubra were significantly more likely to exhibit nonlinear growth dynamics relative to declining and dying trees. The drivers of this effect remain unclear, but nonlinear growth dynamics in healthy trees may represent an enhanced ability to benefit from resource pulses, in turn promoting greater resilience. Our work suggests that forecasting models offer a means of predicting tree survival during forest disturbances and thus represent an increasingly valuable tool as forest disturbances increase in frequency and severity.

Are financial activities harmful for regional growth? Contradictory evidence from the Indonesian panel data

Using high-frequency panel data for Indonesian provinces, we find that financial activities are associated with slower growth rates of productivity and output per capita. In particular, bank credit alone appears to inhibit growth more than the impact of credit and financial savings combined. This result contrasts starkly with the evidence from low-frequency panel, where the estimates suggest that bank credit promotes faster productivity and higher growth of per capita output. The contradictory evidence is attributable to the nonlinear growth dynamics of the finance-output nexus, where credit inhibits growth in the short run and promotes growth over the longer haul.

Viscosity Effects on Explosive Growth Dynamics of the Nonlinear Resistive Tearing Mode

A collapse of the X-point occurs above a critical island width, Δ'wc , in the resistive tearing mode for large instability parameter, Δ', leading to current sheet formation [N.F. Loureiro et al. Phys. Rev. Lett. 95, 235003 (2005)]. In this study, we analyze this problem by including viscosity effects on the onset of the X-point collapse and the explosive nonlinear growth dynamics of the reconnected flux. While explosive growth seems to be independent of viscosity in the magnetic Prandtl number regime Pr < 1, a transition behavior is revealed at Pr ≈ 1 for the viscosity dependence of Δ'wc , for the X-point collapse as well as the linear tearing instability. A secondary instability analysis, which included quasi-linear modifications of the equilibrium current profile due to the zonal current, shows that current peaking is plausibly responsible for the onset of the X-point collapse and the explosive growth of reconnected flux, which leads to the current sheet formation.

Nonlinear growth dynamics of Langmuir monolayers limited by both surface and bulk diffusion

A theory of growth of Langmuir monolayers limited by both surface and bulk impurity diffusion is developed. It is shown that unlike the traditional situation where only surface diffusion is present---leading to similarity solution where a straight front advances with time as $\sqrt{t}$ which is at the basis of dendritic growth and ``fractal-like'' morphologies---bulk diffusion leads to the existence of a straight front moving at a constant speed. This is interpreted in terms of dimensional considerations: bulk diffusion introduces a new length scale, making this solution possible. As a consequence, the growth morphology must be dense. This is what is observed experimentally. An exact solution for a straight front and its stability is provided analytically. The straight front is unstable above a critical speed (or critical supersaturation). The nonlinear dynamics are tackled by means of a gauge-field-invariant geometrical formulation. It is shown that the existence of a straight front solution moving steadily also implies that a circular front solution moving at a constant speed exists as well. For a nearly straight geometry (but deformed) dynamics falls into a Kuramoto-Sivashinsky one where spatiotemporal chaos is expected. For a more curved front (such as the one generated initially from a circle instability), numerical analysis reveals a variety of compact patterns.

Nonlinear growth dynamics and the origin of fluctuating asymmetry

The nonlinear, complex nature of biosynthesis magnifies the impacts of small, random perturbations on organism growth, leading to distortions in adaptive allometries and, in particular, to fluctuating asymmetry. These distortions can be partly checked by cell-cell and inter-body part feedback during growth and development, though the latter mechanism also may lead to complex patterns in right-left asymmetry. Stress can be expected to increase the degree to which random growth perturbations are magnified and may also result in disruption of the check mechanisms, thus exaggerating fluctuating asymmetry.