The last decade has seen an explosion in models that describe phenomena in systems medicine. Such models are especially useful for studying signaling pathways, such as the Wnt pathway. In this chapter we use the Wnt pathway to showcase current mathem
atical and statistical techniques that enable modelers to gain insight into (models of) gene regulation, and generate testable predictions. We introduce a range of modeling frameworks, but focus on ordinary differential equation (ODE) models since they remain the most widely used approach in systems biology and medicine and continue to offer great potential. We present methods for the analysis of a single model, comprising applications of standard dynamical systems approaches such as nondimensionalization, steady state, asymptotic and sensitivity analysis, and more recent statistical and algebraic approaches to compare models with data. We present parameter estimation and model comparison techniques, focusing on Bayesian analysis and coplanarity via algebraic geometry. Our intention is that this (non exhaustive) review may serve as a useful starting point for the analysis of models in systems medicine.
Social and biological contagions are influenced by the spatial embeddedness of networks. Historically, many epidemics spread as a wave across part of the Earths surface; however, in modern contagions long-range edges -- for example, due to airline tr
ansportation or communication media -- allow clusters of a contagion to appear in distant locations. Here we study the spread of contagions on networks through a methodology grounded in topological data analysis and nonlinear dimension reduction. We construct contagion maps that use multiple contagions on a network to map the nodes as a point cloud. By analyzing the topology, geometry, and dimensionality of manifold structure in such point clouds, we reveal insights to aid in the modeling, forecast, and control of spreading processes. Our approach highlights contagion maps also as a viable tool for inferring low-dimensional structure in networks.
We discuss our outreach efforts to introduce school students to network science and explain why networks researchers should be involved in such outreach activities. We provide overviews of modules that we have designed for these efforts, comment on o
ur successes and failures, and illustrate the potentially enormous impact of such outreach efforts.
The mitogen activated protein kinase (MAPK) family of proteins is involved in regulating cellular fate activities such as proliferation, differentiation and apoptosis. Their fundamental importance has attracted considerable attention on different asp
ects of the MAPK signaling dynamics; this is particularly true for the Erk/Mek system, which has become the canonical example for MAPK signaling systems. Erk exists in many different isoforms, of which the most widely studied are Erk1 and Erk2. Until recently, these two kinases were considered equivalent as they differ only subtly at the sequence level; however, these isoforms exhibit radically different trafficking between cytoplasm and nucleus. Here we use spatially resolved data on Erk1/2 to develop and analyze spatio-temporal models of these cascades; and we discuss how sensitivity analysis can be used to discriminate between mechanisms. We are especially interested in understanding why two such similar proteins should co-exist in the same organism, as their functional roles appear to be different. Our models elucidate some of the factors governing the interplay between processes and the Erk1/2 localization in different cellular compartments, including competition between isoforms. This methodology is applicable to a wide range of systems, such as activation cascades, where translocation of species occurs via signal pathways. Furthermore, our work may motivate additional emphasis for considering potentially different roles for isoforms that differ subtly at the sequence level.
