Mathematical Biosciences

Mitochondria are a key player in several kinds of tissue injury, and are even the ultimate cause of certain diseases. In this work we introduce new models of mitochondrial ATP generation in multiple tissues, including liver hepatocytes and the medullary thick ascending limb in the kidney. Using this model, we predict these tissues responses to hypoxia, uncoupling, ischemia-reperfusion, and oxidative phosphorylation dysfunction. Our results suggest mechanisms explaining differences in robustness of mitochondrial function across tissues.The medullary thick ascending limb and proximal tubule in the kidney both experience a high metabolic demand, while having lower baseline activity of oxidative phosphorylation relative to the liver. These factors make these tissues susceptible to dysfunction of ComplexIII. A lower baseline oxygen tension observed in the thick ascending limb makes it susceptible to Complex IV. On the other hand, since the liver lacks these risk factors, and has higher baseline rates of glycolysis, it is less susceptible to all kinds of oxidative phosphorylation dysfunction.

The cell is the power station of life. Surprisingly, to date there is still no metabolic scaling theory that links cellular respiration to organismal metabolism and predicts the mouse-to-elephant curve, also known as Kleibers law, in an approach that is consistent with physicochemical principles. This paper shows that for a consistent model, the novel concept of the optimised Metabolic Module (MM) is the missing link between cell and organism. It is shown how evolutionary selection under resource scarcity optimises the MM towards (a) lightweight design and (b) resource efficiency. Thus, Darwins evolution by natural selection is simulated by model-based optimisation. The final general model presented is complete (for the entire mass range of the organism of different taxonomic classes), concise (it uses only five scale-invariant physicochemical constants), clear (it predicts all metabolic rates within the uncertainty range of a scale model observed in measurements) and consistent with Murrays law of capillary blood flow and cell metabolism. The model features observed asymptotes for both small protists and large endotherms. It predicts the mass-dependent metabolic rate of protists, planarians, ectotherms and endotherms with the usual uncertainty of any scaling theory. It finally turns out that Kleibers law is an asymptote of the derived general model, namely for the case of diffusion-limited cell metabolism.

Advanced cancers, such as prostate and breast cancers, commonly metastasize to bone. In the bone matrix, dendritic osteocytes form a spatial network allowing communication between osteocytes and the osteoblasts located on the bone surface. This communication network facilitates coordinated bone remodelling. In the presence of a cancerous microenvironment, the morphology of this network changes. Commonly osteocytes appear to be either overdifferentiated (i.e., there are more dendrites than healthy bone) or underdeveloped (i.e., dendrites do not fully form). In addition to structural changes, histological sections from metastatic breast cancer xenografted mice show that number of osteocytes per unit area is different between healthy bone and cancerous bone. We present a stochastic agent-based model for bone formation incorporating osteoblasts and osteocytes that allows us to probe both network structure and density of osteocytes in bone. Our model both allows for the simulation of our spatial network model and analysis of mean-field equations in the form of integro-partial differential equations. We considered variations of our model to study specific physiological hypotheses related to osteoblast differentiation; for example predicting how changing biological parameters, such as rates of bone secretion, rates of cancer formation and rates of osteoblast differentiation can allow for qualitatively different network morphologies. We then used our model to explore how commonly applied therapies such as bisphosphonates (e.g. zoledronic acid) impact osteocyte network formation.

During menopause, estrogen levels decline significantly, leading to substantial physiological changes due to estrogens regulatory role in various systems. In particular, estrogen helps prevent excessive bone resorption by its impact on bone remodeling. When estrogen levels decrease, bone resorption increases, often resulting in weakened bones and osteoporosis in post-menopausal women. Experimental studies have also shown that estrogen regulates the renin-angiotensin system (RAS), a hormone system involved in many physiological processes, including blood pressure regulation. Additionally, the RAS has an interconnected relationship with calcium regulatory and bone remodeling systems. Given these dynamic interplays, how would perturbations in one system affect the others? To answer that question, we developed a physiology-based mathematical model that simulates the interactions of estrogen, key RAS components, calcium regulation, and bone remodeling. Through sensitivity analysis and model simulations, we investigated how declining estrogen levels affect the RAS and bone mineral density. Furthermore, we quantified how RAS inhibitors, specifically angiotensin-converting enzyme inhibitors and angiotensin receptor blockers, may increase bone density during post-menopausal estrogen decline.

Hormones are regulatory molecules that impact physiological functions. Much is known about individual hormones, but general rules that connect the regulatory logic of different hormone systems are limited. In this study, we analyzed a range of human hormone systems using a mathematical approach to integrate knowledge on endocrine cells, target tissues and regulation, to uncover unifying principles and regulatory circuits. We find that the number of cells in an endocrine gland is proportional to the number of cells in its target tissues, as one single endocrine cell serves approximately 2000 target cells. We identified five classes of regulatory circuits, each has specific regulatory functions such as homeostasis or allostasis. The most complex class includes an intermediate gland, the pituitary, which can otherwise be considered redundant and exposes to fragilities. We suggest a tradeoff: with the price of fragilities comes advantages -amplification, buffering of hypersecreting tumors, and faster response times. By elucidating these unifying principles and circuits, this study deepens our understanding of the control of endocrine processes and builds the foundation for systems endocrinology.

In this paper, for an infectious disease such as Covid-19, we present a SIR model which examines the impact of waning immunity, vaccination rates, vaccine efficacy, and the proportion of the susceptible population who aspire to be vaccinated. Under an assumed constant control reproduction number, we provide simple conditions for the disease to be eliminated, and conversely for it to exhibit the more likely endemic behaviour. With regard to Covid-19, it is shown that if the control reproduction number is set to the basic reproduction number (say 6) of the dominant delta (B1.617.2) variant, vaccination alone, even under the most optimistic of assumptions about vaccine efficacy and high vaccine coverage, is very unlikely to lead to elimination of the disease. The model is not intended to be predictive but more an aid to understanding the relative importance of various biological and control parameters. For example, from a long-term perspective, it may be found that in the UK, through changes in societal behaviour (such as mask use, ventilation, and level of homeworking), without formal government interventions such as on-off lockdowns, the control reproduction number can still be maintained at a level significantly below the basic reproduction number. Even so, our simulations show that endemic behaviour ensues. The model obtains equilibrium values of the state variables such as the infection prevalence and mortality rate under various scenarios.

Each mammal has a budget of approximately one billion heartbeats after birth. This is consistent with their heart rate and life span, which scale with their mass to the power -1/4 and 1/4 respectively, given a complex cardiovascular system. However, the underlying empirical law, i.e., Kleibers law, according to which the metabolic rate scales with the mass to the power of 3/4, applies to all animals: for instance, flatworms with a most simple vascular system and a size slightly above the diffusion limit on growth show the same metabolic scaling as mammals. To date, there is no concise theory that is consistent with cell metabolism and compatible with physiological laws, e.g., that the volume flow scales with the capillary diameter to the power of three (Murrays law). In this paper we present how cell metabolism determines the scaling of the organisms metabolic rate via the Metabolic Module (MM), a cylinder formed by the organisms cells with a concentric capillary - sized and shaped with scarcity. Evolutionary changes from one taxonomic class to the next led to an unsteady increase in the number of MMs: the metabolism of protists and planarians, e.g., flatworms, is given by one MM only; for ectotherms, i.e., cold-blooded organisms, one thousand and for endotherms, i.e., warm-blooded organisms, nearly one hundred million MMs working together. Special cases, such as diffusion-limited metabolism and the 3/4 power law are asymptotes of the presented general theory. The presented general theory of metabolism offers valuable insights for the targeted development of artificial tissues.

The COVID-19 pandemic is widely studied as it continues to threaten many populations of people especially in the USA, the leading country in terms of both deaths and cases. Although vaccines are being distributed, control and mitigation strategies must still be properly enforced. More and more reports show that the spread of COVID-19 involves infected individuals first passing through a pre-symptomatic infectious stage in addition to the incubation period and that many of the infectious individuals are asymptomatic. In this study, we design and use a mathematical model to primarily address the question of who are the main drivers of COVID-19 - the symptomatic infectious or the pre-symptomatic and asymptomatic infectious in the states of Florida, Arizona, New York, Wisconsin and the entire United States. We emphasize the benefit of lockdown by showing that for all four states, earlier and later lockdown dates decrease the number of cumulative deaths. This benefit of lockdown is also evidenced by the decrease in the infectious cases for Arizona and the entire US when lockdown is implemented earlier. When comparing the influence of the symptomatic infectious versus the pre-sympomatic/asymptomatic infectious, it is shown that, in general, the larger contribution comes from the latter group. This is seen from several perspectives, as follows: (1) in terms of daily cases, (2) in terms of daily cases when the influence of one group is targeted over the other by setting the effective contact rate(s) for the non-targeted group to zero, and (3) in terms of cumulative cases and deaths for the US and Arizona when the influence of one group is targeted over the other by setting the effective contact rate(s) for the non-targeted group to zero. The consequences of the difference in the contributions of the two infectious groups is simulated in terms of testing and these simulations show that an increase in testing and isolating for the pre-symptomatic and asymptomatic infectious group has more impact than an increase in testing for the symptomatic infectious. For example, for the entire US, a 50% increase in testing for the pre-symptomatic and asymptomatic infectious group results in a 25% decrease in deaths as opposed to a lower 6% decrease in deaths when a 50% increase in testing rate for the symptomatic infectious is implemented. We also see that if the testing for infectious symptomatic is kept at the baseline value and the testing for the pre-symptomatic and asymptomatic is increased from 0.2 to 0.25, then the control reproduction number falls below 1. On the other hand, to get even close to such a result when keeping the pre-symptomatic and asymptomatic at baseline fitted values, the symptomatic infectious testing rate must be increased considerably more - from 0.4 to 1.7. Lastly, we use our model to simulate an implementation of a natural herd immunity strategy for the entire U.S. and for the state of Wisconsin (the most recent epicenter) and we find that such a strategy requires a significant number of deaths and as such is questionable in terms of success. We conclude with a brief summary of our results and some implications regarding COVID-19 control and mitigation strategies.

AO_SCPLOWBSTRACTC_SCPLOWMedical practice in the intensive care unit is based on the supposition that physiological systems such as the human glucose-insulin system are reliabile. Reliability of dynamical systems refers to response to perturbation: A dynamical system is reliable if it behaves predictably following a perturbation. Here, we demonstrate that reliability fails for an archetypal physiological model, the Ultradian glucose-insulin model. Reliability failure arises because of the presence of delay. Using the theory of rank one maps from smooth dynamical systems, we precisely explain the nature of the resulting delay-induced uncertainty (DIU). We develop a recipe one may use to diagnose DIU in a general dynamical system. Guided by this recipe, we analyze DIU emergence first in a classical linear shear flow model and then in the Ultradian model. Our results potentially apply to a broad class of physiological systems that involve delay.

Mitochondria are a key player in several kinds of tissue injury, and are even the ultimate cause of certain diseases. In this work we introduce a new model of mitochondrial ATP generation in liver hepatocytes of the rat. Ischemia-reperfusion is an intriguing example of a non-equilibrium behaviour driven by a change in tissue oxygen tension. Ischemia involves prolonged hypoxia, followed by the sudden return of oxygen during reperfusion. During reperfusion, we predict that the build up of succinate causes the electron transport chain in the liver to temporarily be in a highly reduced state. This can lead to the production of reactive oxygen species. We accurately predict the timescale on which the electron transport chain is left in a reduced state, and we observe levels of reduction likely to lead to reactive oxygen species production. Aside from the above, we predict thresholds for ATP depletion from hypoxia, and we predict the consequences for oxygen consumption of uncoupling.

The functioning of kidneys on blood flow regulation was investigated, particularly under diseased conditions such as chronic kidney disease, which includes conditions of diabetic nephropathy and other glomerular damage. A mathematical model was developed to better understand how variations based on the glomerular filtration rate impact key kidney function outputs, such as afferent arteriolar diameter, smooth muscle activation, and the chloride ion concentration at the macula densa. We have analyzed these factors by considering the dynamics of the mathematical model of ordinary and partial differential equations to study blood flow control in the kidney, which has provided new insights into the maintenance of autoregulation. By simulating the processes of renal blood flow--specifically through the afferent arteriole and glomerulus, and detailing the process of chloride transport within the renal tubule--the model offers a comprehensive view of how the kidney regulates glomerular filtration rate amidst fluctuating systemic blood pressures and disease-specific changes. Central to this model are the myogenic response that adjusts afferent arteriole muscle tone in reaction to pressure changes and the tubuloglomerular feedback, which controls arteriole size based on chloride levels at the macula densa. The models simulations reveal the robustness of renal autoregulation across a spectrum of chronic kidney disease stages, showing stability under normal conditions but indicating a breakdown in regulation with advanced chronic kidney disease. This breakdown is attributed to disruptions in the vascular and feedback systems. The findings from this model shed light on the progression of renal dysfunction in chronic kidney disease and underscore the potential for developing targeted treatments to maintain kidney health.

The COVID-19 pandemic has sparked significant interest in developing mathematical models that capture more of the complexities of the dynamics of disease transmission and control. In this study, we presented a deterministic compartmental model for the transmission of COVID-19. The model has eight compartments: susceptible, exposed, asymptomatic, unreported symptomatic, reported symptomatic, hospitalised, recovered, and dead. Individuals in each compartment are discretised into age and deprivation deciles to study the combined effect of both factors on disease dynamics. We analyse the model and present the results for both the disease-free and endemic equilibrium states. We evaluate the basic reproduction number using a next-generation matrix approach. We also prove the local and global stability of the disease-free and endemic equilibria. Sensitivity indices are calculated both analytically and numerically to identify the parameters that have the greatest influence on R0. Our results suggest that transmission, recovery, treatment, and testing rates need to be closely monitored to reduce the disease burden. Specifically, prompt testing, treatment, and recovery are critical for reducing R0.

Since Dec 2019, the COVID-19 epidemic has spread over the globe creating one of the greatest pandemics ever witnessed. This epidemic wave will only begin to roll back once a critical proportion of the population is immunised, either by mounting natural immunity following infection, or by vaccination. The latter option can minimise the cost in terms of human lives but it requires to wait until a safe and efficient vaccine is developed, a period estimated to last at least 18 months. In this work, we use optimal control theory to explore the best strategy to implement while waiting for the vaccine. We seek a solution minimizing deaths and costs due to the implementation of the control strategy itself. We find that such a solution leads to an increasing level of control with a maximum reached near the 16th month of the epidemics and a steady decrease until vaccine deployment. The average containment level is approximately 50% during the 25-months period for vaccine deployment. This strategy strongly out-performs others with constant or cycling allocations of the same amount of resources to control the outbreak. This work opens new perspectives to mitigate the effects of the ongoing COVID-19 pandemics, and be used as a proof-of-concept in using mathematical modelling techniques to enlighten decision making and public health management in the early times of an outbreak.

Nonlinear feedback controllers are ubiquitous features of biological systems at different scales. A key motif arising in these systems is a sequestration-based feedback. As a physiological example of this type of feedback architecture, platelets (specialized cells involved in blood clotting) differentiate from stem cells, and this process is activated by a protein called Thrombopoietin (TPO). Platelets actively sequester and degrade TPO, creating negative feedback whereby any depletion of platelets increases the levels of freely available TPO that upregulates platelet production. We show similar examples of sequestration-based feedback in intracellular biomolecular circuits involved in heat-shock response and microRNA regulation. Our systematic analysis of this feedback motif reveals that platelets induced degradation of TPO is critical in enhancing system robustness to external disturbances. In contrast, reversible sequestration of TPO without degradation results in poor robustness to disturbances. We develop exact analytical results quantifying the limits to which the sensitivity to disturbances can be attenuated by sequestration-based feedback. Next, we consider the stochastic formulation of the circuit that takes into account low-copy number fluctuations in feedback components. Interestingly, our results show that the extent of random fluctuations are enhanced with increasing feedback strength, but can exhibit local maxima and minima across parameter regimes. In summary, our systematic analysis highlights design principles for enhancing the robustness of sequestration-based feedback mechanisms to external disturbances and inherent noise in molecular counts.

For a given running speed and for any wind speed, we calculate the slope for which the effort to overcome in the absence of wind is energetically equivalent to running with the original wind speed on a flat track. The influence of headwind and tailwind is thus made numerically comparable to the influence of a positive or negative slope. The same applies to the lack of air resistance on a treadmill and its compensation by adjusting the incline. Moreover, for turning point routes physiological corrections are considered and the impact of speed adjustments is analyzed.

The article presents a novel stochastic mathematical model of mitosis in heterogeneous (multiple-phenotype), age-dependent cell populations. The developed computational techniques involve flexible use of differentiation tree diagrams. The applicability of the model is discussed in the context of the Haeckelian (biogenetic) paradigm. In particular, the article puts forward the conjecture of generality of Haeckels recapitulation law. The conjecture is briefly collated against relevant scientific evidence and elaborated for the specific case of evolving/mutable cell phenotypes as considered by the model. The feasibility, basic regimes and the convenience of the model are tested on examples and experimental data, and the corresponding open source simulation software is described and demonstrated.

The COVID-19 pandemic has not only presented a major global public health and socio-economic crisis, but has also significantly impacted human behavior towards adherence (or lack thereof) to public health intervention and mitigation measures implemented in communities worldwide. The dynamic nature of the pandemic has prompted extensive changes in individual and collective behaviors towards the pandemic. This study is based on the use of mathematical modeling approaches to assess the extent to which SARS-CoV-2 transmission dynamics is impacted by population-level changes of human behavior due to factors such as (a) the severity of transmission (such as disease-induced mortality and level of symptomatic transmission), (b) fatigue due to the implementation of mitigation interventions measures (e.g., lockdowns) over a long (extended) period of time, (c) social peer-pressure, among others. A novel behavior-epidemiology model, which takes the form of a deterministic system of nonlinear differential equations, is developed and fitted using observed cumulative SARS-CoV-2 mortality data during the first wave in the United States. Rigorous analysis of the model shows that its disease-free equilibrium is locally-asymptotically stable whenever a certain epidemiological threshold, known as the control reproduction number (denoted by[R] C) is less than one, and the disease persists (i.e., causes significant outbreak or outbreaks) if the threshold exceeds one. The model fits the observed data, as well as makes a more accurate prediction of the observed daily SARS-CoV-2 mortality during the first wave (March 2020 -June 2020), in comparison to the equivalent model which does not explicitly account for changes in human behavior. Of the various metrics for human behavior changes during the pandemic considered in this study, it is shown that behavior changes due to the level of SARS-CoV-2 mortality and symptomatic transmission were more influential (while behavioral changes due to the level of fatigue to interventions in the community was of marginal impact). It is shown that an increase in the proportion of exposed individuals who become asymptomatically-infectious at the end of the exposed period (represented by a parameter r) can lead to an increase (decrease) in the control reproduction number ([R]C) if the effective contact rate of asymptomatic individuals is higher (lower) than that of symptomatic individuals. The study identifies two threshold values of the parameter r that maximize the cumulative and daily SARS-CoV-2 mortality, respectively, during the first wave. Furthermore, it is shown that, as the value of the proportion r increases from 0 to 1, the rate at which susceptible non-adherent individuals change their behavior to strictly adhere to public health interventions decreases. Hence, this study suggests that, as more newly-infected individuals become asymptomatically-infectious, the level of positive behavior change, as well as disease severity, hospitalizations and disease-induced mortality in the community can be expected to significantly decrease (while new cases may rise, particularly if asymptomatic individuals have higher contact rate, in comparison to symptomatic individuals).

Withdrawal StatementThe authors have withdrawn their manuscript owing to errors in the experimental design that affect the integrity of the results. Therefore, the authors do not wish this work to be cited as reference for the project. If you have any questions, please contact the corresponding author

Cancer-associated fibroblasts (CAFs) are a central component of the tumor microenvironment that facilitate a supportive niche for cancer progression and metastasis. Experimental evidence suggests that CAFs can facilitate estrogen-independent tumor growth, thereby reducing the efficacy of anti-hormonal therapies. Understanding and quantifying the complex interactions between tumor cells, hormonal signalling, and the microenvironment are crucial for designing more effective and individualized treatment strategies. We propose a mathematical framework to explore the influence of CAFs on ER+ breast cancer progression and to evaluate strategies to mitigate their impact. We develop a system of nonlinear ordinary differential equations that substantiates the experimental observations by providing a mechanistic basis for the role of CAFs in regulating estrogen-independent growth dynamics. We then employ optimal control theory to evaluate distinct therapeutic approaches involving monotherapy or combinations of: (i) inhibition of tumor-to-CAF signaling, (ii) inhibition of CAF-to-tumor proliferative signaling, and (iii) endocrine therapy. Taken together, our results demonstrate that CAF-targeted strategies can enhance treatment efficacy across various estrogen dosing regimens. Our study provides new insights into the potential of CAF as a therapeutic target that could help to improve existing approaches for endocrine therapies.

In this work, we present a novel modeling framework for understanding the dynamics of homeostatic regulation. Inspired by engineering control theory, this framework incorporates unique features of biological systems. First, biological variables often play physiological roles, and taking this functional context into consideration is essential to fully understand the goals and constraints of homeostatic regulation. Second, biological signals are not abstract variables, but rather material molecules that may undergo complex turnover processes of synthesis and degradation. We suggest that the particular nature of biological signals may condition the type of information they can convey, and their potential role in shaping the dynamics and the ultimate purpose of homeostatic systems. We show that the dynamic interplay between regulated variables and control signals is a key determinant of biological homeostasis, challenging the necessity and the convenience of strictly extrapolating concepts from engineering control theory in modeling the dynamics of homeostatic systems. This work provides an alternative, unified framework for studying biological regulation and identifies general principles that transcend molecular details of particular homeostatic mechanisms. We show how this approach can be naturally applied to apparently different regulatory systems, contributing to a deeper understanding of homeostasis as a fundamental process in living systems.