Coarse-grained models of protein folding: toy models or predictive tools?

Coarse-grained models are emerging as a practical alternative to all-atom simulations for the characterization of protein folding mechanisms over long time scales. While a decade ago minimalist toy models were mainly designed to test general hypotheses on the principles regulating protein folding, the latest coarse-grained models are increasingly realistic and can be used to characterize quantitatively the detailed folding mechanism of specific proteins. The ability of such models to reproduce the essential features of folding dynamics suggests that each single atomic degree of freedom is not by itself particularly relevant to folding and supports a statistical mechanical approach to characterize folding transitions. When combined with more refined models and with experimental studies, the systematic investigation of protein systems and complexes using coarse-grained models can advance our theoretical understanding of the actual organizing principles that emerge from the complex network of interactions among protein atomic constituents.     

Leslie matrix models as “stroboscopic snapshots” of McKendrick PDE models

 High dimensional Leslie matrix models have long been viewed as discretizations of McKendrick PDE models. However, these two fundamental classes of models can be linked in a completely different way. For populations with periodic birth pulses, Leslie models of any dimension can be viewed as “stroboscopic snapshots” (in time) of an associated impulsive McKendrick model; that is, the solution of the discrete model matches the solution of the corresponding continuous model at every discrete time step. In application, McKendrick models of populations with birth pulses can be used to identify the state of the population between the discrete census times of the associated Leslie model. Furthermore, McKendrick models describing populations with near-synchronous birth pulses can be viewed as realistic perturbations of the associated Leslie model. Received: 7 August 1997 / Revised version: 15 January 1998     

On selection of functional response models: Holling’s models and more

Model selection is a common and established research method. Statistically rigorous model selection methods are used in a variety of research fields. In contrast, studies that characterize functional response models commonly use a model selection method that is specific to functional response studies. The specific method aims to distinguish between Holling’s type II and type III functional response models. This paper discusses problems associated with the specific method and suggests that it would be better to use general model selection methods that allow to consider a variety of models.     

How well do reduced models capture the dynamics in models of interacting neurons?

This paper introduces a class of stochastic models of interacting neurons with emergent dynamics similar to those seen in local cortical populations. Rigorous results on existence and uniqueness of nonequilibrium steady states are proved. These network models are then compared to very simple reduced models driven by the same mean excitatory and inhibitory currents. Discrepancies in firing rates between network and reduced models are investigated and explained by correlations in spiking, or partial synchronization, working in concert with “nonlinearities” in the time evolution of membrane potentials. The use of simple random walks and their first passage times to simulate fluctuations in neuronal membrane potentials and interspike times is also considered.

     

Bělehrádek-type models

Summary Square-root (or Ratkowsky) models are a special case of Blehrádek's temperature rate-relationship first published in 1926 and widely used in several fields of biology. Blehrádek-type models also describe microbial growth, and have been extended for use in food microbiology by the inclusion of terms for water activity and pH. The parameters of the square root-type models are defined and their determination described. Favorable features of square root-type models include parsimony, parameter estimation properties, and ease of use. Square root-type models have been developed for a number of organisms of concern to the food industry and have also been adopted for use in a number of electronic devices used in predictive microbiology. Criticisms of square root-type models are also considered.Mention of brand or firm names does not constitute an endorsement by the US Department of Agriculture over others of a similar nature not mentioned.     

Metabolic theory or metabolic models?

The metabolic theory of ecology (MTE) claims to derive ecological relationships from the structure of resource distribution networks, which is assumed to determine the scaling of metabolism with body mass, and from the effect of temperature on the rate of biological processes. MTE is controversial. I propose that some of the controversy stems from the implicit adoption of different views of science by the proponents and critics of MTE. The perspective of proponents is consistent with the theory-centric view of science called the received view, whereas many of the critics implicitly adopt an alternative view consistent with a model-centric view of science. I propose that adopting the model-centric view can help to settle some of the differences among proponents and critics of MTE.     

Do stroke models model stroke?

Stroke is one of the leading causes of death worldwide and the biggest reason for long-term disability. Basic research has formed the modern understanding of stroke pathophysiology, and has revealed important molecular, cellular and systemic mechanisms. However, despite decades of research, most translational stroke trials that aim to introduce basic research findings into clinical treatment strategies – most notably in the field of neuroprotection – have failed. Among other obstacles, poor methodological and statistical standards, negative publication bias, and incomplete preclinical testing have been proposed as ‘translational roadblocks’. In this article, we introduce the models commonly used in preclinical stroke research, discuss some of the causes of failed translational success and review potential remedies. We further introduce the concept of modeling ‘care’ of stroke patients, because current preclinical research models the disorder but does not model care or state-of-the-art clinical testing. Stringent statistical methods and controlled preclinical trials have been suggested to counteract weaknesses in preclinical research. We conclude that preclinical stroke research requires (1) appropriate modeling of the disorder, (2) appropriate modeling of the care of stroke patients and (3) an approach to preclinical testing that is similar to clinical testing, including Phase 3 randomized controlled preclinical trials as necessary additional steps before new therapies enter clinical testing.     

How do cross-migration models arise?

In this paper we present a general method to derive spatio-temporal population models mechanistically. We consider a system of multiple species living in a patchy habitat in which the local population of each species consists of some behavioural groups. We then formulate a continuous-time model where a small positive parameter is present, measuring the time scale of behavioural transitions relative to that of giving birth, death and migration among patches. By the singular perturbation method the model is reduced to a lower dimensional one in which the migration terms are, in general, nonlinear and related to the reaction terms describing the local dynamics. Two examples demonstrating the emergence of cross-migration models, i.e., the models in which the per-capita migration rate of one species depends on the density of some other species, are given.     

Can correct protein models be identified?

The ability to separate correct models of protein structures from less correct models is of the greatest importance for protein structure prediction methods. Several studies have examined the ability of different types of energy function to detect the native, or native-like, protein structure from a large set of decoys. In contrast to earlier studies, we examine here the ability to detect models that only show limited structural similarity to the native structure. These correct models are defined by the existence of a fragment that shows significant similarity between this model and the native structure. It has been shown that the existence of such fragments is useful for comparing the performance between different fold recognition methods and that this performance correlates well with performance in fold recognition. We have developed ProQ, a neural-network-based method to predict the quality of a protein model that extracts structural features, such as frequency of atom-atom contacts, and predicts the quality of a model, as measured either by LGscore or MaxSub. We show that ProQ performs at least as well as other measures when identifying the native structure and is better at the detection of correct models. This performance is maintained over several different test sets. ProQ can also be combined with the Pcons fold recognition predictor (Pmodeller) to increase its performance, with the main advantage being the elimination of a few high-scoring incorrect models. Pmodeller was successful in CASP5 and results from the latest LiveBench, LiveBench-6, indicating that Pmodeller has a higher specificity than Pcons alone.     

Are reproductive skew models evolutionarily stable?

Reproductive skew theory has become a popular way to phrase problems and test hypotheses of social evolution. The diversity of reproductive skew models probably stems from the ease of generating new variations. However, I show that the logical basis of skew models, that is, the way in which group formation is modelled, makes use of hidden assumptions that may be problematical as they are unlikely to be fulfilled in all social systems. I illustrate these problems by re-analysing the basic concessive skew model with staying incentives. First, the model assumes that dispersal is an all-or-nothing response: all subordinates disperse as soon as concessions drop below a certain value. This leads to a discontinuous 'cliff-edge' shape of dominant fitness, and it is not clear that selection will balance a population at such an edge. Second, it is assumed that subordinates have perfect knowledge of their benefits if they stay in the group. I examine the effects of relaxing these two assumptions. Relaxing the first one strengthens reproductive skew theory, but relaxing the latter makes evolutionary stability disappear. In cases where subordinates cannot accurately measure benefits provided by the individual dominant with which they live, so that their behaviour instead evolves as a response to population-wide average benefits, the logic of reproductive skew models does not apply. This warns against too indiscriminate an application of reproductive skew theory to problems in social evolution: for example, transactional models of extra-pair paternity assume perfect knowledge of paternity, which is unlikely to hold true in nature. It is recommended that models specify the mechanisms by which individuals can adjust their behaviour to that of others, and pay attention to changes that occur in evolutionary versus behavioural time.     

Do consensus models outperform individual models? Transferability evaluations of diverse modeling approaches for an invasive moth

Transferability is key to many of the most novel and interesting applications of ecological niche models, such that maximizing predictive power of model transfers is crucial. Here, we explored consensus methods as a means of reducing uncertainty and improving model transferability in anticipating the potential distribution of an invasive moth (Hyphantria cunea). Individual native-range niche models were calibrated using seven modelling algorithms and four environmental datasets, representing different degrees of dimensionality, spatial correlation, and ecological relevance, and showing different degrees of climate niche expansion. Four consensus methods were used to combine individual niche models; we assessed transferability of consensus models and the individual models used to generate them. The results suggested that ideal criteria for environmental variable selection vary among algorithms, as different algorithms showed different sensitivities to spatial dimensionality and correlation. Consensus models reflected the central tendency of individual models, and reduced uncertainty by consolidating consistency across individual models, but did not outperform individual models. The question of whether interpolation accuracy comes at the expense of transferability suggests caution in planning methodologies for processing niche models to predict invasive potential. These explorations outline approaches by which to reduce uncertainty and improve niche model transferability with vital implications for ensemble forecasting.     

Scaling up: how do exogenous fluctuations in individual-based resource competition models re-emerge in aggregated stochastic population models?

In applied population dynamics the choice of stochastic per capita growth function has implications for population viability analyses, management recommendations, and pest control. This model choice is often based on statistical criteria, mathematical tractability or personal preferences, and general ecological guidelines are either too vague or entirely missing. To identify such guidelines, it is important to understand how exogenous and endogenous factors interact at the individual level and re-emerge at the aggregated population level. We therefore study different types of resource competition (contest vs. scramble competition) and different types of exogenous fluctuations (food and weather fluctuations) at the individual level in a simple individual-based simulation model. We statistically fit the resulting time series to find out (1) which functional form of the growth function (‘hyperbolic’ or ‘exponential’) better describes contest and scramble competition and (2) whether the pattern of population fluctuations resulting from the simulations can be assigned to vertical, lateral or nonlinear perturbations in the stochastic growth function (a classification scheme suggested by Royama 1992, Analytical Population Dynamics, Chapman and Hall, London). We found that the same type of competition can result in ‘hyperbolic’ or ‘exponential’ functional forms, depending on the type of exogenous fluctuations. So it is the interplay between exogenous variability and endogenous resource competition that affects model performance. In contrast to the widespread assumption of vertical (additive) perturbations, our findings highlight the importance of (non-additive) lateral and nonlinear perturbations and their combinations with vertical perturbations. The choice of the stochastic growth function should therefore consider not only statistical criteria but also ecological guidelines. We derive such ecological guidelines from our analysis.     

Chemoautotrophic endosymbioses: contemporary models for symbiogenesis?

Oxygen appears to be one of the key factors in understanding the evolution of life on Earth. Almost absent during more than 2 billion years, its subsequent increase is correlated with the emergence of oxygenic photosynthesis by Cyanobacteria, followed by aerobic Prokaryotes and eventually Eukaryotes, all primitively aerobic, and more recently, the development of complex multicellular organisms. However, in some reduced environments, still present at the surface of the Earth and even more so in ocean depths (hydrothermal vents, cold seeps, massive organic falls,...), anaerobic or micro-aerobic Prokaryotes continue to grow, including some chemoautotrophic bacteria deriving energy from sulfide oxidation for instance. A few Metazoa have managed to collaborate with such chemoautotroph Prokaryotes, the most abundant species forming endosymbiotic associations. The most studied of these endosymbioses (the mussels Bathymodiolus, the vestimentiferan tubeworm Riftia pachyptila, or the clams Calyptogena) have revealed important differences in the degree of interdependence between host and symbionts, and in the mode of symbiont transmission. The evolutive process of these symbioses is reminiscent of the primary endosymbioses which have given rise to the organelles of heterotrophic Eukaryotes (mitochondria) and phototrophic Eukaryotes (chloroplasts). The study of these modern days biological models could shed light on symbiogenesis itself and also potentially reveal thiotrophic Eukaryotes as a new lineage.     

Mathematical models of α-synuclein transport in axons

To investigate possible effects of diffusion on α-synuclein (α-syn) transport in axons, we developed two models of α-syn transport, one that assumes that α-syn is transported only by active transport, as part of multiprotein complexes, and a second that assumes an interplay between motor-driven and diffusion-driven α-syn transport. By comparing predictions of the two models, we were able to investigate how diffusion could influence axonal transport of α-syn. The predictions obtained could be useful for future experimental work aimed at elucidating the mechanisms of axonal transport of α-syn. We also attempted to simulate possible defects in α-syn transport early in Parkinson's disease (PD). We assumed that in healthy axons α-syn localizes in the axon terminal while in diseased axons α-syn does not localize in the terminal (this was simulated by postulating a zero α-syn flux into the terminal). We found that our model of a diseased axon predicts the build-up of α-syn close to the axon terminal. This build-up could cause α-syn accumulation in Lewy bodies and the subsequent axonal death pattern observed in PD (‘dying back’ of axons).     

Individual behavioral variation in predator–prey models

The role of individual behavioral variation in community dynamics was studied. Behavioral variation in this study does not refer to differences in average responses (e.g., average response between presence and absence of antipredator behavior). Rather it refers to the variation around the average response that is not explained by trivial experimental treatments. First, the effect of behavioral variation was examined based on Jensen’s inequality. In cases of commonly used modeling framework with type II functional response, neglecting behavioral variation (a component of encounter rate) causes overestimation of predation effects. The effect of this bias on community processes was examined by incorporating the behavioral variation in a commonly used consumer-resource model (Rosenzweig–MacArthur model). How such a consideration affects a model prediction (paradox of enrichment) was examined. The inclusion of behavioral variation can both quantitatively and qualitatively alter the model characteristics. Behavioral variation can substantially increase the stability of the community with respect to enrichment.     

Mobility unevenness in rock–paper–scissors models

We investigate a tritrophic system whose cyclic dominance is modelled by the rock–paper–scissors game. We consider that organisms of one or two species are affected by movement limitations, which unbalances the cyclic spatial game. Performing stochastic simulations, we show that mobility unevenness controls the population dynamics. In the case of one slow species, the predominant species depends on the level of mobility restriction, with the slow species being preponderant if the mobility limitations are substantial. If two species face mobility limitations, our outcomes show that being higher dispersive does not constitute an advantage in terms of population growth. On the contrary, if organisms move with higher mobility, they expose themselves to enemies more frequently, being more vulnerable to being eliminated. Finally, our findings show that biodiversity benefits in regions where species are slowed. Biodiversity loss for high mobility organisms, common to cyclic systems, may be avoided with coexistence probability being higher for robust mobility limitations. Our results may help biologists understand the dynamics of unbalanced spatial systems where organisms’ dispersal is fundamental to biodiversity conservation.     

How do species interactions affect species distribution models?

One of the most promising recent advances in biogeography has been the increased interest and understanding of species distribution models – estimates of the probability that a species is present given environmental data. Unfortunately, such analyses ignore many aspects of ecology, and so are difficult to interpret. In particular, we know that species interactions have a profound influence on distributions, but it is not usually possible to incorporate this knowledge into species distribution models. What is needed is a rigorous understanding of how unmeasured biotic interactions affect the inferences generated by species distribution models. To fill this gap, we develop a general mathematical approach that uses probability theory to determine how unmeasured biotic interactions affect inferences from species distribution models. Using this approach, we reanalyze one of the most important classes of mechanistic models of competition: models of consumer resource dynamics. We determine how measurements of one aspect of the environment – a single environmental variable – can be used to estimate the probability that an environment is suitable with species distribution models. We show that species distribution models, which ignore numerous facets of consumer resource dynamics such as the presence of a competitor or the dynamics of depletable resources, can furnish useful predictions for the probability that an environment is suitable in some circumstances. These results provide a rigorous link between complex mechanistic models of species interactions and species distribution models. In so doing they demonstrate that unmeasured biotic interactions can have strong and counterintuitive consequences on species distribution models.