Waddington redux: models and explanation in stem cell and systems biology

Stem cell biology and systems biology are two prominent new approaches to studying cell development. In stem cell biology, the predominant method is experimental manipulation of concrete cells and tissues. Systems biology, in contrast, emphasizes mathematical modeling of cellular systems. For scientists and philosophers interested in development, an important question arises: how should the two approaches relate? This essay proposes an answer, using the model of Waddington’s landscape to triangulate between stem cell and systems approaches. This simple abstract model represents development as an undulating surface of hills and valleys. Originally constructed by C. H. Waddington to visually explicate an integrated theory of genetics, development and evolution, the landscape model can play an updated unificatory role. I examine this model’s structure, representational assumptions, and uses in all three contexts, and argue that explanations of cell development require both mathematical models and concrete experiments. On this view, the two approaches are interdependent, with mathematical models playing a crucial but circumscribed role in explanations of cell development.     

A mathematical programming approach for gene selection and tissue classification

MOTIVATION: Extracting useful information from expression levels of thousands of genes generated with microarray technology needs a variety of analytical techniques. Mathematical programming approaches for classification analysis outperform parametric methods when the data depart from assumptions underlying these methods. Therefore, a mathematical programming approach is developed for gene selection and tissue classification using gene expression profiles. RESULTS: A new mixed integer programming model is formulated for this purpose. The mixed integer programming model simultaneously selects genes and constructs a classification model to classify two groups of tissue samples as accurately as possible. Very encouraging results were obtained with two data sets from the literature as examples. These results show that the mathematical programming approach can rival or outperform traditional classification methods.     

Estimating dinosaur maximum running speeds using evolutionary robotics

Maximum running speed is an important locomotor parameter for many animals-predators as well as prey-and is thus of interest to palaeobiologists wishing to reconstruct the behavioural ecology of extinct species. A variety of approaches have been tried in the past including anatomical comparisons, bone scaling and strength, safety factors and ground reaction force analyses. However, these approaches are all indirect and an alternative approach is to create a musculoskeletal model of the animal and see how fast it can run. The major advantage of this approach is that all assumptions about the animal's morphology and physiology are directly addressed, whereas the exact same assumptions are hidden in the indirect approaches. In this paper, we present simple musculoskeletal models of three extant and five extinct bipedal species. The models predict top speed in the extant species with reasonably good agreement with accepted values, so we conclude that the values presented for the five extinct species are reasonable predictions given the modelling assumptions made. Improved musculoskeletal models and better estimates of soft tissue parameters will produce more accurate values. Limited sensitivity analysis is performed on key muscle parameters but there is considerable scope for extending this in the future.     

Classification accuracy in multiple color fluorescence imaging microscopy

BACKGROUND: The discriminatory power and imaging efficiency of different multicolor FISH (M-FISH) analysis systems are key factors in obtaining accurate and reproducible classification results. In a recent paper, Garini et al. put forth an analytical technique to quantify the discriminatory power ("S/N ratio") and imaging efficiency ('excitation efficiency') of multicolor fluorescent karyotyping systems. METHODS: A parametric model of multicolor fluorescence microscopy, based on the Beer-Lambert law, is analyzed and reduced to a simple expression for S/N ratio. Parameters for individual system configurations are then plugged into the model for comparison purposes. RESULTS: We found that several invalid assumptions, which are used to reduce the complex mathematics of the Beer-Lambert law to a simple S/N ratio, result in some completely misleading conclusions about classification accuracy. The authors omit the most significant noise source, and consider only one highly abstract and unrepresentative situation. Unwisely chosen parameters used in the examples lead to predictions that are not consistent with actual results. CONCLUSIONS: The earlier paper presents an inaccurate view of the M-FISH situation. In this short communication, we point out several inaccurate assumptions in the mathematical development of Garini et al. and the poor choices of parameters in their examples. We show results obtained with different imaging systems that indicate that reliable and comparable results are obtained if the metaphase samples are well-hybridized. We also conclude that so-called biochemical noise, not photon noise, is the primary factor that limits pixel classification accuracy, given reasonable exposure times. Copyright Wiley-Liss, Inc.     

Evaluation of two unstructured mathematical models for the penicillin G fedbatch fermentation

The mathematical model for the penicillin G fed-batch fermentation proposed by Heijnen et al. (1979) is compared with the model of Bajpai & Reuß (1980). Although the general structure of these models is similar, the difference in metabolic assumptions and specific growth and production kinetics results in a completely different behaviour towards product optimization. A detailed analysis of both models reveals some physical and biochemical shortcomings. It is shown that it is impossible to make a reliable estimation of the model parameters, only using experimental data of simple constant glucose feed rate fermentations with low initial substrate amount. However, it is demonstrated that some model parameters might be key factors in concluding whether or not altering the substrate feeding strategy has an important influence on the final amount of product.It is illustrated that feeding strategy optimization studies can be a tool in designing experiments for parameter estimation purposes.     

An information‐theoretic approach to estimating the composite genetic effects contributing to variation among generation means: Moving beyond the joint‐scaling test for line cross analysis

The pace and direction of evolution in response to selection, drift, and mutation are governed by the genetic architecture that underlies trait variation. Consequently, much of evolutionary theory is predicated on assumptions about whether genes can be considered to act in isolation, or in the context of their genetic background. Evolutionary biologists have disagreed, sometimes heatedly, over which assumptions best describe evolution in nature. Methods for estimating genetic architectures that favor simpler (i.e., additive) models contribute to this debate. Here we address one important source of bias, model selection in line cross analysis (LCA). LCA estimates genetic parameters conditional on the best model chosen from a vast model space using relatively few line means. Current LCA approaches often favor simple models and ignore uncertainty in model choice. To address these issues we introduce Software for Analysis of Genetic Architecture (SAGA), which comprehensively assesses the potential model space, quantifies model selection uncertainty, and uses model weighted averaging to accurately estimate composite genetic effects. Using simulated data and previously published LCA studies, we demonstrate the utility of SAGA to more accurately define the components of complex genetic architectures, and show that traditional approaches have underestimated the importance of epistasis.     

The logseries distribution and Mountford's similarity index as a basis for the study of stream benthic community structure

SUMMARY. 1. Of the various approaches to studying species-abundance patterns in aquatic ecosystems (particularly streams) the logseries distribution is the most appropriate. The distribution parameter, α, is readily estimated and goodness-of-fit of the model to the data can be estimated.
2. A method exists which uses a to calculate an index of similarity from which cluster dendrograms can be constructed and the clusters tested for significant differences. Assumptions and calculations are presented.
3. Procedures are illustrated, with examples from Idaho streams, and approaches to interpreting the results are included.
4. Studies of stream macro invertebrates can fulfill the assumptions of the logseries distribution but cannot meet those of commonly used diversity indices.     

On mathematical modeling of circadian rhythms, performance, and alertness

Mathematical models of neurobehavioral performance and alertness have both basic science and practical applications. These models can be especially useful in predicting the effect of different sleep-wake schedules on human neurobehavioral objective performance and subjective alertness under many conditions. Several relevant models currently exist in the literature. In principle, the development and refinement of any mathematical model should be based on an explicit modeling methodology, such as the Box modeling paradigm, that formally defines the model structure and calculates the set of parameters. While most mathematical models of neurobehavioral performance and alertness include homeostatic, circadian, and sleep inertia components and their interactions, there may be fundamental differences in the equations included in these models. In part, these may be due to differences in the assumptions of the underlying physiology. Because the choice of model equations can have a dramatic influence on the results, it is necessary to consider these differences in assumptions when examining the results from a model and when comparing results across models. This article presents principles of mathematical modeling and examples of how such procedures can be applied to the development and refinement of mathematical models of neurobehavioral performance and alertness. This article also presents several methods of testing and comparing these models, suggests different uses of the models, and discusses problems with current models.     

Birth-death symmetry in the evolution of a social trait

Studies of the evolution of a social trait often make ecological assumptions (of population structure, life history), and thus a trait can be studied many different times with different assumptions. Here, I consider a Moran model of continuous reproduction and use an inclusive fitness analysis to investigate the relationships between fecundity or survival selection and birth-death (BD) or death-birth (DB) demography on the evolution of a social trait. A simple symmetry obtains: fecundity (respectively survival) effects under BD behave the same as survival (respectively fecundity) effects under DB. When these results are specialized to a homogeneous population, greatly simplified conditions for a positive inclusive fitness effect are obtained in both a finite and an infinite population. The results are established using the elegant formalism of mathematical group theory and are illustrated with an example of a finite population arranged in a cycle with asymmetric offspring dispersal.     

A homogenized constrained mixture (and mechanical analog) model for growth and remodeling of soft tissue

Most mathematical models of the growth and remodeling of load-bearing soft tissues are based on one of two major approaches: a kinematic theory that specifies an evolution equation for the stress-free configuration of the tissue as a whole or a constrained mixture theory that specifies rates of mass production and removal of individual constituents within stressed configurations. The former is popular because of its conceptual simplicity, but relies largely on heuristic definitions of growth; the latter is based on biologically motivated micromechanical models, but suffers from higher computational costs due to the need to track all past configurations. In this paper, we present a temporally homogenized constrained mixture model that combines advantages of both classical approaches, namely a biologically motivated micromechanical foundation, a simple computational implementation, and low computational cost. As illustrative examples, we show that this approach describes well both cell-mediated remodeling of tissue equivalents in vitro and the growth and remodeling of aneurysms in vivo. We also show that this homogenized constrained mixture model suggests an intimate relationship between models of growth and remodeling and viscoelasticity. That is, important aspects of tissue adaptation can be understood in terms of a simple mechanical analog model, a Maxwell fluid (i.e., spring and dashpot in series) in parallel with a “motor element” that represents cell-mediated mechanoregulation of extracellular matrix. This analogy allows a simple implementation of homogenized constrained mixture models within commercially available simulation codes by exploiting available models of viscoelasticity.     

Direct fitness or inclusive fitness: how shall we model kin selection?

Two standard mathematical formulations of kin-selection models can be found. Inclusive fitness is an actor-centred approach, which calculates the fitness effect on a number of recipients of the behaviour of a single actor. Direct fitness is a recipient-centred approach, which calculates the fitness effect on the recipient of the behaviour of a number of actors. Inclusive fitness offers us a powerful heuristic, of choosing behaviour to maximize fitness, but direct fitness can be mathematically easier to work with and has recently emerged as the preferred approach of theoreticians. In this paper, we explore the fundamental connection between these two approaches in both homogeneous and class-structured populations, and we show that under simple assumptions (mainly fair meiosis and weak selection) they provide equivalent formulations, which correspond to the predictions of Price's equation for allele frequency change. We use a couple of examples to highlight differences in their conception and formulation, and we briefly discuss a two-species example in which we have a class of 'actor' that is never a 'recipient', which the standard direct fitness method can handle but the usual inclusive fitness cannot.     

Model collaboration for the improved assessment of biomass supply,demand, and impacts

Existing assessments of biomass supply and demand and their impacts face various types of limitations and uncertainties, partly due to the type of tools and methods applied (e.g., partial representation of sectors, lack of geographical details, and aggregated representation of technologies involved). Improved collaboration between existing modeling approaches may provide new, more comprehensive insights, especially into issues that involve multiple economic sectors, different temporal and spatial scales, or various impact categories. Model collaboration consists of aligning and harmonizing input data and scenarios, model comparison and/or model linkage. Improved collaboration between existing modeling approaches can help assess (i) the causes of differences and similarities in model output, which is important for interpreting the results for policy‐making and (ii) the linkages, feedbacks, and trade‐offs between different systems and impacts (e.g., economic and natural), which is key to a more comprehensive understanding of the impacts of biomass supply and demand. But, full consistency or integration in assumptions, structure, solution algorithms, dynamics and feedbacks can be difficult to achieve. And, if it is done, it frequently implies a trade‐off in terms of resolution (spatial, temporal, and structural) and/or computation. Three key research areas are selected to illustrate how model collaboration can provide additional ways for tackling some of the shortcomings and uncertainties in the assessment of biomass supply and demand and their impacts. These research areas are livestock production, agricultural residues, and greenhouse gas emissions from land‐use change. Describing how model collaboration might look like in these examples, we show how improved model collaboration can strengthen our ability to project biomass supply, demand, and impacts. This in turn can aid in improving the information for policy‐makers and in taking better‐informed decisions.     

The fate of balanced,phenotypic polymorphisms in fragmented metapopulations

In large populations, genetically distinct phenotypic morphs can be maintained in equilibrium (at a 1 : 1 ratio in the simplest case) by frequency‐dependent selection, as shown by Sewall Wright. The consequences of population fragmentation on this equilibrium are not widely appreciated. Here, I use a simple computational model to emphasize that severe fragmentation biases the morph ratio towards the homozygous recessive genotype through drift in very small populations favouring the more common recessive allele. This model generalizes those developed elsewhere for heterostylous plants and major histocompatibility complex alleles, emphasizes one particular outcome and avoids the restricting assumptions of more analytical models. There are important implications for both fundamental evolutionary biology and conservation genetics. I illustrate this with a range of examples but refer particularly to shell polymorphism in snails. These examples show how habitat fragmentation could have a direct and often unappreciated effect on species at the level of their population genetics.     

Fisher's geometrical model and the mutational patterns of antibiotic resistance across dose gradients

Fisher's geometrical model (FGM) has been widely used to depict the fitness effects of mutations. It is a general model with few underlying assumptions that gives a large and comprehensive view of adaptive processes. It is thus attractive in several situations, for example adaptation to antibiotics, but comes with limitations, so that more mechanistic approaches are often preferred to interpret experimental data. It might be possible however to extend FGM assumptions to better account for mutational data. This is theoretically challenging in the context of antibiotic resistance because resistance mutations are assumed to be rare. In this article, we show with Escherichia coli how the fitness effects of resistance mutations screened at different doses of nalidixic acid vary across a dose‐gradient. We found experimental patterns qualitatively consistent with the basic FGM (rate of resistance across doses, gamma distributed costs) but also unexpected patterns such as a decreasing mean cost of resistance with increasing screen dose. We show how different extensions involving mutational modules and variations in trait covariance across environments, can be discriminated based on these data. Overall, simple extensions of the FGM accounted well for complex mutational effects of resistance mutations across antibiotic doses.     

Impact of Awareness Programs on Cholera Dynamics: Two Modeling Approaches

We propose two differential equation-based models to investigate the impact of awareness programs on cholera dynamics. The first model represents the disease transmission rates as decreasing functions of the number of awareness programs, whereas the second model divides the susceptible individuals into two distinct classes depending on their awareness/unawareness of the risk of infection. We study the essential dynamical properties of each model, using both analytical and numerical approaches. We find that the two models, though closely related, exhibit significantly different dynamical behaviors. Namely, the first model follows regular threshold dynamics while rich dynamical behaviors such as backward bifurcation may arise from the second one. Our results highlight the importance of validating key modeling assumptions in the development and selection of mathematical models toward practical application.     

Statistics of protein library construction

SUMMARY: We have investigated the statistics associated with constructing and sampling large protein-encoding libraries. Using fairly simple statistics we have written algorithms for estimating the diversity in libraries generated by the most commonly used protocols, including error-prone PCR, DNA shuffling, StEP PCR, oligonucleotide-directed randomization, MAX randomization, synthetic shuffling, DHR, ADO and SISDC. AVAILABILITY: Web interface and C++ source code available at http://guinevere.otago.ac.nz/stats.html. SUPPLEMENTARY INFORMATION: Complete mathematical notes, model assumptions and justification, users' guide and worked examples at above website.