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  • 19 Jun 2024 11:26 PM | Publications Team (Administrator)

    Dimensional Dependence of Binding Kinetics

    by Megan Dixon & James Keener 

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    Experimentally, the strength of protein-protein interactions is typically measured in solution, and dissociation constants are traditionally reported in units of volume concentration. It is often assumed that these three-dimensional dissociation constants give direct insight into how tightly the same proteins bind when they are membrane-associated. In this article, we explore and counter this notion. We demonstrate mathematically that dissociation constants are highly dependent on dimension. In both discrete and continuous space, we present and analyze stochastic models of binding kinetics in one, two, and three dimensions. Not only do dissociation constants in two dimensions have different units and forms than dissociation constants in three dimensions, the conversion between them is quite complex and requires detailed information. We present a novel formula to convert three-dimensional dissociation constants to two-dimensional dissociation constants. This conversion allows for better understanding of protein interactions on membranes and how to appropriately model them.

  • 13 Jun 2024 10:19 AM | Publications Team (Administrator)

    Measles Infection Dose Responses: Insights from Mathematical Modeling

    by Anet Anelone & Hannah Clapham 

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    The measles virus (MV) is highly contagious and affects the whole body, including the skin and the immune system. As MV infection dose increases above one infectious particle, the peak of infectious viral load occurs sooner, yet its magnitude remains constant. It is important to improve understanding of measles, in part due to the re-emergence of measles outbreak worldwide, and a lack of research. We investigated mechanisms determining the outcomes of measles infection doses. We evaluated relevant biological hypotheses, and their respective mathematical formulations, to describe and fit data on the time course of measles infectious viral load in the peripheral blood of monkeys, following experimental measles infection with different doses. When MV infection dose increases, the initial viral load, and the initial number of responding immune cells increase. This mechanism decreases the time it takes for immune cells to control and remove infectious viral load. This mechanism also underpins the dose-independent magnitudes for measles viral load, and the loss of immune cells. Together, these findings suggest that the outcome of measles depends on how the immune system responds to incoming MV right from the beginning. This work encourages prevention, vaccination, and early diagnosis of measles.

    Caption: Measles infection dose responses: insights from mathematical modeling. Top: Model-data fits for acute viremia in response to changes in MV infection doses. using model parameterizations and assumptions in Table in the paper. 104, 103, 102, 10 and 1 TCID50 correspond to red diamonds, blue stars, orange triangles, magenta dots, and green squares respectively. The solid lines represent the trajectories generated by the proposed model parameterization. The shapes represent data. The dark grey dotted dashed line represents the limit of detection < 0.3. Bottom: Cartoon illustrating that the healthy body adjusts the response of the immune system to remove measles infectious particles sooner when the measles infection dose increases; however, the peak viral load remains constant

  • 06 Jun 2024 7:47 PM | Publications Team (Administrator)

    Modelling the Impact of NETosis During the Initial Stage of Systemic Lupus Erythematosus

    by Vladimira Suvandjieva, Ivanka Tsacheva, Marlene Santos, Georgios Kararigas Peter Rashkov

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    NETosis, or formation of Neutrophil Extracellular Traps (NETs) in response to a pathogenic stimulus, is a suspected contributing factor of autoimmunogenicity in Systemic Lupus Erythematosus. Our ODE-based mathematical model studies the interaction between macrophages, neutrophils and two types of antigen with origin in apoptotic waste and NETs during the initial stage of the disease. Analytical and numerical calculations help classify the bifurcations between equilibria, in particular those with presence of autoantigen. The model predicts that even in parameter regimes with efficient clearance of NETs by immune cells, autoantigen can persist stably in tissue, causing chronic inflammation and loss of immune tolerance in the long run.

    Sketch of the interactions between cells and antigens studied by our model.

  • 21 May 2024 10:48 PM | Publications Team (Administrator)

    Morphological Stability for in silico Models of Avascular Tumors

    by Erik Blom & Stefan Engblom

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    We develop a simple, stochastic model of an avascular tumor that displays known behavior at the considered scales. In parallel we develop, analyze, and simulate a surrogate PDE model to explain the growth pattern and shape of the stochastic model. 

    We investigate the emergent tumor morphology of the stochastic model (Fig. a) through the PDE model and through comparisons of respective model’s numerical experiments. The stochastic model displays the three characteristic avascular tumor regions -- the proliferative rim, quiescent annulus, and necrotic center -- and sigmoidal volumetric growth pattern in line with the PDE model under radial symmetry (Fig. b). The analysis predicts morphological instability under low oxygen and cell-cell adhesion conditions, as well as an ever-present creeping effect where the tumor as a whole migrates towards the oxygen source -- an effect observed during simulations of both models (Fig. c). The analysis further displays a capacity to predict non-trivial morphological patterns of the model tumor (Fig. d). 

  • 14 May 2024 5:26 PM | Publications Team (Administrator)

    The Unification of Evolutionary Dynamics through the Bayesian Decay Factor in a Game on a Graph

    by Arnaud Z. Dragicevic

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    The study unifies evolutionary dynamics on graphs by employing a decaying Bayesian update in the context of strategic uncertainty. It demonstrates that the replication of strategies leading to shifts between competition and cooperation in well-mixed and Bayesian-structured populations is equivalent under certain conditions. Specifically, this equivalence holds when the rate of transition between competitive and cooperative behaviors matches the relative strength of selection pressures. Our findings help pinpoint scenarios where cooperation is favored, independent of payoff levels, expanding the application of Price's equation beyond its original intent.

    Caption: The basins of attraction of the Price-wise unstructured population replicator dynamics

  • 01 May 2024 5:52 PM | Publications Team (Administrator)

    Convex Representation of Metabolic Networks with Michaelis-Menten Kinetics

    by Josh Taylor, Alain Rapaport & Denis Dochain

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    Polyhedral models of metabolic networks are computationally tractable and provide insight into cellular functions. For example, flux balance analysis is a linear program in which reaction fluxes are optimized over polyhedral mass-balance constraints. In this paper, we augment the standard polyhedral model of a metabolic network with a new, second-order cone representation of the Michaelis-Menten kinetics. This enables us to explicitly model metabolite concentrations without losing tractability. We formulate conic flux balance analysis, a second-order cone program in which reaction fluxes are maximized while metabolite concentrations are minimized. While not as tractable as linear programming, second-order cone programs with hundreds of thousands of variables can be solved in seconds to minutes using modern solvers like Gurobi and MOSEK. In addition to predicting both fluxes and concentrations, we can use conic duality to compute sensitivities to kinetic parameters, i.e., maximum reaction rates and Michaelis constants. We also incorporate the second-order cone representation of the Michaelis-Menten kinetics into dynamic flux balance analysis and minimal cut set analysis. These tools provide new, tractable ways to analyze reaction fluxes and metabolite concentrations in metabolic networks. The Python code for each tool is available at;!!NVzLfOphnbDXSw!CB8YzXwI0ErdeBFcgljtFA36uhpf2ATRf6MEgYTiLhceaAzDS6gF7M5m047C62AYZH8xjVWlPanu7H7qQcsSzjBGP_RJ4rc$

  • 25 Apr 2024 6:51 PM | Publications Team (Administrator)

    Discretised Flux Balance Analysis for Reaction-Diffusion Simulation of Single-Cell Metabolism

    by Yin Hoon Chew and Fabian Spill

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    Metabolism comprises thousands of biochemical reactions. It is commonly modelled using Flux Balance Analysis (FBA), a method based on linear programming, because this method requires very few parameters. However, conventional FBA implicitly assumes that all enzymatic reactions are not diffusion-limited though that may not always be the case.

    To enable the exploration of diffusion effects on cellular metabolism, we present a spatial method that implements FBA on a grid-based system. The method discretises a living cell into a two-dimensional grid; creates variables that represent the rates of reactions within grid elements and diffusions between grid elements; and solves the system as a single linear programming problem.

    Simulations using the method suggest that factors such as cell shape, diffusion regime and spatial distribution of enzyme can influence the variability and robustness of metabolism at both single-cell and population levels. We propose the use of this method to explore how spatiotemporal organisation of compartments and molecules in cells affect cellular behaviour.

    Yin Hoon Chew is a Research Fellow and Fabian Spill is a Professor of Applied Mathematics at the University of Birmingham, UK. Yin Hoon designed and implemented the method, with feedback from Fabian.

    The method can simulate living cells with different shapes and heterogeneous enzyme distribution. Simulations suggest that cell shape (perimeter-to-area ratio) does not affect cellular behaviour such as biomass growth when diffusion is fast, but there is a strong effect at low diffusion.

  • 17 Apr 2024 2:47 AM | Publications Team (Administrator)

    Mathematical modelling of parasite dynamics: A stochastic simulation-based approach and parameter estimation via modified sequential-type approximate Bayesian computation

    by Clement Twumasi, Joanne Cable and Andrey Pepelyshev. 

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    In an era marked by global health challenges and re-emerging infections, the need for sophisticated and robust mathematical models to better understand infectious diseases has never been more pressing. Our impactful study focused on a biological system known as the gyrodactylid-fish system. While existing modelling studies have fallen short in capturing vital information to reflect the biological realism of this system, our research introduced a novel individual-based stochastic simulation model to realistically simulate the spread of three different strains of Gyrodactylus across three different host populations, enhancing our knowledge of this system given observed experimental data. This study contributed mathematically and biologically to the gyrodactylid-fish system, offering insights that may apply to modelling other biological systems. Expanding on the existing studies, we have added to our understanding of this system and provided answers to open biological questions for the first time through model-based Bayesian analysis. The study also led to robust extensions of likelihood-free Bayesian estimation methods, commonly known as approximate Bayesian computation (ABC), to aid in calibrating complex models with mathematically intractable likelihood. After conducting additional posterior predictive checks, we found our proposed ABC methodologies' efficiency highly compelling and can readily be adapted to fit other highdimensional multi-parameter models.

  • 09 Apr 2024 11:43 PM | Publications Team (Administrator)

    Second-Order Effects of Chemotherapy Pharmacodynamics and Pharmacokinetics on Tumor Regression and Cachexia

    by Daniel R. Bergman, Kerri-Ann Norton, Harsh Vardhan Jain & Trachette Jackson

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    This paper presents a novel computational framework that constrains high-dimensional ABM parameter space with multidimensional real-world data.  We accomplish this by extending and validating a first-of-its-kind method that leverages explicitly formulated surrogate models to bridge the computational divide between ABMs and experimental data.  We show that Surrogate Modeling for Reconstructing Parameter Spaces (SMoRe ParS) can constrain high-dimensional ABM parameter spaces using unidimensional (single time-course) data.  We then demonstrate that it can constrain parameter spaces of more complex ABMs using multidimensional data (multiple time courses at different biological scales).  To validate our method, we compared the SMoRe ParS-inferred ABM parameter space with ABM parameters inferred by an often computationally expensive direct comparison with experimental data.  A strength of SMoRe ParS is that it allows for exploring ABM parameter space even at points that are not directly sampled and where ABM output was never generated.   Computationally efficient methods to connect ABMs with multidimensional data are timely and important as ABMs are a natural platform for capturing heterogeneity and predicting emergent behavior in multiscale systems.  SMoRe ParS is a robust and scalable computational framework that can explore the uncertainty within multidimensional parameter spaces associated with ABMs representing complex biological phenomena.

    Caption: The schematic diagram for using SMoRe ParS to infer ABM input parameters from experimental data via a surrogate model. The solid arrows connecting the Experimental Data and Agent-based Model boxes to the Surrogate Model box represent the direction of information flow in the first few steps of SMoRe ParS. Green (control), yellow (0.75μM oxaliplatin), and red (7.55μM oxaliplatin) colors in the Experimental data box refer to the dosing regimens that generated the experimental data.

    Brief description of the roles of the authors (e.g. student, group-leader etc):

    Daniel R. Bergman, postdoc

    Kerri-Ann Norton,  computational modeling collaborator, and developer of SMoRe Pars

    Harsh Vardhan Jain, co-senior author and co-developer of SMoRe Pars

    Trachette Jackson, co-senior author and co-developer of SMoRe Pars

  • 03 Apr 2024 2:17 AM | Publications Team (Administrator)

    Second-Order Effects of Chemotherapy Pharmacodynamics and Pharmacokinetics on Tumor Regression and Cachexia

    by Luke Pierik, Patricia McDonald, Alexander R.A. Anderson & Jeffrey West. 

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    Second order effects describe changes in a system which result from introducing variability or fluctuations in a system’s inputQuantifying second-order effects relies on an understanding of the convexity of an underlying function determining system output, and this has been effectively used in several fields, notably financial risk management. Previously, the vocabulary of fragile or antifragile has been used: fragile systemare harmed by variability while antifragile systems benefit from variability. The key insight here is that oncologists can control the input variability of treatment schedules, and therefore it is critical to define the fragility (or antifragility) of tumors. In cancer, second-order effects have been studied through dose response curves, which are ubiquitous theoretical and clinical tools in the field. However, these curves do not incorporate knowledge about how long dosages remain near the tumor (i.e. pharmacokinetics), which influences treatment outcomes. In this paper, we explore this relation between second-order effects and pharmacokinetics through standard mathematical models as well aa previously parameterized tumor model with 5-fluorouracil. By studying second-order effects with pharmacokinetics, more efficient treatment schedules may be devised which utilize the underlying convexity of dose response to produce greater patient outcomes.

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