A mathematical framework for functional mapping of complex phenotypes using delay differential equations

Funmap2: an R package for QTL mapping using longitudinal phenotypes

Quantitative trait locus (QTL) mapping has been used as a powerful tool for inferring the complexity of the genetic architecture that underlies phenotypic traits. This approach has shown its unique power to map the developmental genetic architecture of complex traits by implementing longitudinal data analysis. Here, we introduce the R package Funmap2 based on the functional mapping framework, which integrates prior biological knowledge into the statistical model. Specifically, the functional mapping framework is engineered to include longitudinal curves that describe the genetic effects and the covariance matrix of the trait of interest. Funmap2 chooses the type of longitudinal curve and covariance matrix automatically using information criteria. Funmap2 is available for download at https://github.com/wzhy2000/Funmap2.

A mapping framework of competition-cooperation QTLs that drive community dynamics

Genes have been thought to affect community ecology and evolution, but their identification at the whole-genome level is challenging. Here, we develop a conceptual framework for the genome-wide mapping of quantitative trait loci (QTLs) that govern interspecific competition and cooperation. This framework integrates the community ecology theory into systems mapping, a statistical model for mapping complex traits as a dynamic system. It can characterize not only how QTLs of one species affect its own phenotype directly, but also how QTLs from this species affect the phenotype of its interacting species indirectly and how QTLs from different species interact epistatically to shape community behavior. We validated the utility of the new mapping framework experimentally by culturing and comparing two bacterial species, Escherichia coli and Staphylococcus aureus, in socialized and socially isolated environments, identifying several QTLs from each species that may act as key drivers of microbial community structure and function.

Integrating Evolutionary Game Theory into Mechanistic Genotype-Phenotype Mapping

Natural selection has shaped the evolution of organisms toward optimizing their structural and functional design. However, how this universal principle can enhance genotype-phenotype mapping of quantitative traits has remained unexplored. Here we show that the integration of this principle and functional mapping through evolutionary game theory gains new insight into the genetic architecture of complex traits. By viewing phenotype formation as an evolutionary system, we formulate mathematical equations to model the ecological mechanisms that drive the interaction and coordination of its constituent components toward population dynamics and stability. Functional mapping provides a procedure for estimating the genetic parameters that specify the dynamic relationship of competition and cooperation and predicting how genes mediate the evolution of this relationship during trait formation.

Mapping complex traits as a dynamic system

Despite increasing emphasis on the genetic study of quantitative traits, we are still far from being able to chart a clear picture of their genetic architecture, given an inherent complexity involved in trait formation. A competing theory for studying such complex traits has emerged by viewing their phenotypic formation as a "system" in which a high-dimensional group of interconnected components act and interact across different levels of biological organization from molecules through cells to whole organisms. This system is initiated by a machinery of DNA sequences that regulate a cascade of biochemical pathways to synthesize endophenotypes and further assemble these endophenotypes toward the end-point phenotype in virtue of various developmental changes. This review focuses on a conceptual framework for genetic mapping of complex traits by which to delineate the underlying components, interactions and mechanisms that govern the system according to biological principles and understand how these components function synergistically under the control of quantitative trait loci (QTLs) to comprise a unified whole. This framework is built by a system of differential equations that quantifies how alterations of different components lead to the global change of trait development and function, and provides a quantitative and testable platform for assessing the multiscale interplay between QTLs and development. The method will enable geneticists to shed light on the genetic complexity of any biological system and predict, alter or engineer its physiological and pathological states.

A model for computing genes governing marital dissolution through sentimental dynamics

Adverse sentimental relationships that cause marital dissolution may involve a genetic component composed of genes from a couple, which interact with cultural, sociological, psychological and economic factors. However, the identification of these genes is very challenging. Here, we address this challenge by developing a computational model that can identify specific genes that impact on sentimental relationships of couples. The model was derived by implementing the second law of thermodynamics that quantifies sentimental relationships within a dynamic gene identification framework, called systems mapping. The model is equipped with a capacity to characterize and test the pattern of how genes from a couple interact with each other to determine the dynamic behavior of their marital relationships. The testing procedure is based on comparing genotypic differences in mathematical parameters of sentimental dynamics described by a group of ordinary differential equations (ODE). The model allows the test of individual parameters or a combination of parameters, addressing specific details related to martial relationships. The model may find its implications for designing an optimal effort policy and therapy to maintain a harmonic family in light of genetic blueprints of individual couples.

Systems mapping of genes controlling chemotherapeutic drug efficiency for cancer stem cells

Cancer can be controlled effectively by using chemotherapeutic drugs to inhibit cancer stem cells, but there is considerable inter-patient variability regarding how these cells respond to drug intervention. Here, we describe a statistical framework for mapping genes that control tumor responses to chemotherapeutic drugs as well as the efficacy of treatments in arresting tumor growth. The framework integrates the mathematical aspects of the cancer stem cell hypothesis into genetic association studies, equipped with a capacity to quantify the magnitude and pattern of genetic effects on the kinetic decline of cancer stem cells in response to therapy. By quantifying how specific genes and their interactions govern drug response, the model provides essential information to tailor personalized drugs for individual patients.

Systems mapping of metabolic genes through control theory

The formation of any complex phenotype involves a web of metabolic pathways in which one chemical is transformed through the catalysis of enzymes into another. Traditional approaches for mapping quantitative trait loci (QTLs) are based on a direct association analysis between DNA marker genotypes and end-point phenotypes, neglecting the mechanistic processes of how a phenotype is formed biochemically. Here, we propose a new dynamic framework for mapping metabolic QTLs (mQTLs) responsible for phenotypic formation. By treating metabolic pathways as a biological system, robust differential equations have proven to be a powerful means of studying and predicting the dynamic behavior of biochemical reactions that cause a high-order phenotype. The new framework integrates these differential equations into a statistical mixture model for QTL mapping. Since the mathematical parameters that define the emergent properties of the metabolic system can be estimated and tested for different mQTL genotypes, the framework allows the dynamic pattern of genetic effects to be quantified on metabolic capacity and efficacy across a time-space scale. Based on a recent study of glycolysis in Saccharomyces cerevisiae, we design and perform a series of simulation studies to investigate the statistical properties of the framework and validate its usefulness and utilization in practice. This framework can be generalized to mapping QTLs for any other dynamic systems and may stimulate pharmacogenetic research toward personalized drug and treatment intervention.

Stochastic modeling of systems mapping in pharmacogenomics

As a basis of personalized medicine, pharmacogenetics and pharmacogenomics that aim to study the genetic architecture of drug response critically rely on dynamic modeling of how a drug is absorbed and transported to target tissues where the drug interacts with body molecules to produce drug effects. Systems mapping provides a general framework for integrating systems pharmacology and pharmacogenomics through robust ordinary differential equations. In this chapter, we extend systems mapping to more complex and more heterogeneous structure of drug response by implementing stochastic differential equations (SDE). We argue that SDE-implemented systems mapping provides a computational tool for pharmacogenetic or pharmacogenomic research towards personalized medicine.

Systems mapping: how to map genes for biomass allocation toward an ideotype

The recent availability of high-throughput genetic and genomic data allows the genetic architecture of complex traits to be systematically mapped. The application of these genetic results to design and breed new crop types can be made possible through systems mapping. Systems mapping is a computational model that dissects a complex phenotype into its underlying components, coordinates different components in terms of biological laws through mathematical equations and maps specific genes that mediate each component and its connection with other components. Here, we present a new direction of systems mapping by integrating this tool with carbon economy. With an optimal spatial distribution of carbon fluxes between sources and sinks, plants tend to maximize whole-plant growth and competitive ability under limited availability of resources. We argue that such an economical strategy for plant growth and development, once integrated with systems mapping, will not only provide mechanistic insights into plant biology, but also help to spark a renaissance of interest in ideotype breeding in crops and trees.

Systems mapping of HIV-1 infection

Mathematical models of viral dynamics in vivo provide incredible insights into the mechanisms for the nonlinear interaction between virus and host cell populations, the dynamics of viral drug resistance, and the way to eliminate virus infection from individual patients by drug treatment. The integration of these mathematical models with high-throughput genetic and genomic data within a statistical framework will raise a hope for effective treatment of infections with HIV virus through developing potent antiviral drugs based on individual patients’ genetic makeup. In this opinion article, we will show a conceptual model for mapping and dictating a comprehensive picture of genetic control mechanisms for viral dynamics through incorporating a group of differential equations that quantify the emergent properties of a system.

Dynamic modeling of genes controlling cancer stem cell proliferation

The growing evidence that cancer originates from stem cells (SC) holds a great promise to eliminate this disease by designing specific drug therapies for removing cancer SC. Translation of this knowledge into predictive tests for the clinic is hampered due to the lack of methods to discriminate cancer SC from non-cancer SC. Here, we address this issue by describing a conceptual strategy for identifying the genetic origins of cancer SC. The strategy incorporates a high-dimensional group of differential equations that characterizes the proliferation, differentiation, and reprogramming of cancer SC in a dynamic cellular and molecular system. The deployment of robust mathematical models will help uncover and explain many still unknown aspects of cell behavior, tissue function, and network organization related to the formation and division of cancer SC. The statistical method developed allows biologically meaningful hypotheses about the genetic control mechanisms of carcinogenesis and metastasis to be tested in a quantitative manner.

Parameters in dynamic models of complex traits are containers of missing heritability

Polymorphisms identified in genome-wide association studies of human traits rarely explain more than a small proportion of the heritable variation, and improving this situation within the current paradigm appears daunting. Given a well-validated dynamic model of a complex physiological trait, a substantial part of the underlying genetic variation must manifest as variation in model parameters. These parameters are themselves phenotypic traits. By linking whole-cell phenotypic variation to genetic variation in a computational model of a single heart cell, incorporating genotype-to-parameter maps, we show that genome-wide association studies on parameters reveal much more genetic variation than when using higher-level cellular phenotypes. The results suggest that letting such studies be guided by computational physiology may facilitate a causal understanding of the genotype-to-phenotype map of complex traits, with strong implications for the development of phenomics technology.

Despite an ever-increasing number of genome locations reported to be associated with complex human diseases or quantitative traits, only a small proportion of phenotypic variations in a typical quantitative trait can be explained by the discovered variants. We argue that this problem can partly be resolved by combining the statistical methods of quantitative genetics with computational biology. We demonstrate this for the in silico genotype-to-phenotype map of a model heart cell in conjunction with publically accessible genomic data. We show that genome wide association studies (GWAS) on model parameters identify more causal variants and can build better prediction models for the higher-level phenotypes than by performing GWAS on the higher-level phenotypes themselves. Since model parameters are in principle measurable physiological phenotypes, our findings suggest that development of future phenotyping technologies could be guided by mathematical models of the biological systems being targeted.