Graphical Approach to Model Reduction for Nonlinear Biochemical Networks

Model reduction is a central challenge to the development and analysis of multiscale physiology models. Advances in model reduction are needed not only for computational feasibility but also for obtaining conceptual insights from complex systems. Here, we introduce an intuitive graphical approach to model reduction based on phase plane analysis. Timescale separation is identified by the degree of hysteresis observed in phase-loops, which guides a “concentration-clamp” procedure for estimating explicit algebraic relationships between species equilibrating on fast timescales. The primary advantages of this approach over Jacobian-based timescale decomposition are that: 1) it incorporates nonlinear system dynamics, and 2) it can be easily visualized, even directly from experimental data. We tested this graphical model reduction approach using a 25-variable model of cardiac β1-adrenergic signaling, obtaining 6- and 4-variable reduced models that retain good predictive capabilities even in response to new perturbations. These 6 signaling species appear to be optimal “kinetic biomarkers” of the overall β1-adrenergic pathway. The 6-variable reduced model is well suited for integration into multiscale models of heart function, and more generally, this graphical model reduction approach is readily applicable to a variety of other complex biological systems

Hidden dynamics of soccer leagues: the predictive ‘power’ of partial standings

Objectives Soccer leagues reflect the partial standings of the teams involved after each round of competition. However, the ability of partial league standings to predict end-of-season position has largely been ignored. Here we analyze historical partial standings from English soccer to understand the mathematics underpinning league performance and evaluate the predictive ‘power’ of partial standings. Methods Match data (1995-2017) from the four senior English leagues was analyzed, together with random match scores generated for hypothetical leagues of equivalent size. For each season the partial standings were computed and Kendall’s normalized tau-distance and Spearman r-values determined. Best-fit power-law and logarithmic functions were applied to the respective tau-distance and Spearman curves, with the ‘goodness-of-fit’ assessed using the R2 value. The predictive ability of the partial standings was evaluated by computing the transition probabilities between the standings at rounds 10, 20 and 30 and the final end-of-season standings for the 22 seasons. The impact of reordering match fixtures was also evaluated. Results All four English leagues behaved similarly, irrespective of the teams involved, with the tau-distance conforming closely to a power law (R2>0.80) and the Spearman r-value obeying a logarithmic function (R2>0.87). The randomized leagues also conformed to a power-law, but had a different shape. In the English leagues, team position relative to end-of-season standing became ‘fixed’ much earlier in the season than was the case with the randomized leagues. In the Premier League, 76.9% of the variance in the final standings was explained by round-10, 87.0% by round-20, and 93.9% by round-30. Reordering of match fixtures appeared to alter the shape of the tau-distance curves. Conclusions All soccer leagues appear to conform to mathematical laws, which constrain the league standings as the season progresses. This means that partial standings can be used to predict end-of-season league position with reasonable accuracy

Reachable and controllable sets for planetary entry and landing

Understanding the envelope of entry trajectories that a given planetary lander is capable of flying can be an important aid for mission analysis and design. Two characteristics of this envelope are considered: 1) the set of states reachable from a given entry state and 2) the set of entry states controllable to a given final state. Precise definitions of these sets are given and methods for computing them are presented. To illustrate their use, the sets are employed to characterize the performance of two vehicle configurations, a low lift-to-drag-ratio capsule and a mid lift-to-dragratio ellipsled, for Mars entry. Roles for the sets in evaluating entry trajectory planning algorithms, choosing a nominal entry state, and planning skip entries are described. Copyright © 2009

Reachable and controllable sets for planetary entry and landing

Understanding the envelope of entry trajectories that a given planetary lander is capable of flying can be an important aid for mission analysis and design. Two characteristics of this envelope are considered: 1) the set of states reachable from a given entry state and 2) the set of entry states controllable to a given final state. Precise definitions of these sets are given and methods for computing them are presented. To illustrate their use, the sets are employed to characterize the performance of two vehicle configurations, a low lift-to-drag-ratio capsule and a mid lift-to-dragratio ellipsled, for Mars entry. Roles for the sets in evaluating entry trajectory planning algorithms, choosing a nominal entry state, and planning skip entries are described. Copyright © 2009

Feasible trajectory generation for atmospheric entry guidance

An atmospheric entry trajectory planner is developed that generates a feasible trajectory and associated bank angle profile. Feasibility denotes that the initial and final state conditions, the path and control constraints, and the nominal equations of motion are all satisfied. Feasible trajectories are easier to track, and thus enhanced performance is expected when the trajectory planner is combined with a tracking law for entry guidance. Insights from computing maximum crossrange trajectories are factored into the design of the planner, and as a result that it can generate trajectories to most of the landing footprint Drag profile design is central in the planning approach, but in addition both longitudinal and lateral motions are accounted for, including bank reversal planning, and the assumption of zero flight path angle is not required. Comparisons of trajectories created by the new planner and optimal trajectories and guidance simulation results using an algorithm based on the new planner demonstrate the performance improvements

Feasible trajectory generation for atmospheric entry guidance

An atmospheric entry trajectory planner is developed that generates a feasible trajectory and associated bank angle profile. Feasibility denotes that the initial and final state conditions, the path and control constraints, and the nominal equations of motion are all satisfied. Feasible trajectories are easier to track, and thus enhanced performance is expected when the trajectory planner is combined with a tracking law for entry guidance. Insights from computing maximum crossrange trajectories are factored into the design of the planner, and as a result that it can generate trajectories to most of the landing footprint Drag profile design is central in the planning approach, but in addition both longitudinal and lateral motions are accounted for, including bank reversal planning, and the assumption of zero flight path angle is not required. Comparisons of trajectories created by the new planner and optimal trajectories and guidance simulation results using an algorithm based on the new planner demonstrate the performance improvements