Here we present a mathematical model of movement in an abstract space representing states of cellular differentiation. We motivate this work with recent examples that demonstrate a continuum of cellular differentiation using single cell RNA sequencing data to characterize cellular states in a high-dimensional space, which is then mapped into or with dimension reduction techniques. We represent trajectories in the differentiation space as a graph, and model directed and random movement on the graph with partial differential equations. We hypothesize that flow in this space can be used to model normal and abnormal differentiation processes. We present a mathematical model of hematopoeisis parameterized with publicly available single cell RNA-Seq data and use it to simulate the pathogenesis of acute myeloid leukemia (AML). The model predicts the emergence of cells in novel intermediate states of differentiation consistent with immunophenotypic characterizations of a mouse model of AML.
Childhood obesity is a health emergency in many parts of the world including the U.S. and, consequently, identifying local, regional or national intervention models capable, of altering the dynamics of obesity at scales that make a difference remains a challenge. The fact that consumption of healthful foods among most youth has yet to meet recommended nutritional standards highlights a lack of effective policies aimed at addressing the epidemic of obesity. Mathematical models are used to evaluate the roles of socialization and school environment on the diet dynamics of children. Data suggest that standard nutrition education programs may have, at best, minimal impact in altering diet dynamics at the population-level. Inclusion of peer influence (model as contagion) reinforced by the use of culturally-sensitive school menus (environmental disruption) may prove capable of modifying obesity enhancing diet dynamics; altering the diets of a significant (critical) proportion of youngsters. A framework is introduced to explore the value of behavior-based interventions and policies that account for the sociocultural environments of at risk communities. These models capture carefully choreographed scenarios to account for the fact that when dealing with diet-dynamics systems, thinking additively is not enough as it cannot account for the power of nonlinear effects.
Since the publication of the National Research Councils Report BIO2010, e orts have increased to better integrate mathematics and biology in undergraduate education. Unfortunately, equivalent e orts to introduce these quantitative topics at the secondary level have been seldom. This could cause differential success of undergraduate students who come from diverse secondary science backgrounds. Undergraduate courses regularly use technology to integrate these two disciplines, and we believe that technology can similarly be used at the secondary level to prevent quantitative achievement mismatch in undergraduate biology programs. In this paper, we review the current uses of technology to teach quantitative biology at the secondary and undergraduate levels, propose needs for further implementation, and address potential barriers to integrating mathematics and biology using technology.
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