Employing observability rank conditions for taking into account experimental information a priori

by Alejandro F. Villaverde

A dynamic model is identifiable if it is possible to infer its parameters by measuring its output over time. Likewise, it is observable if it is possible to determine its state variables in the same way. Since parameters can be treated as constant state variables, identifiability can be considered as a particular case of observability. Thus, both properties can be analysed by building an observability matrix and checking whether it has full rank. This test can be performed before collecting experimental data (i.e., “a priori”), and it may reveal structural issues of the model equations. Here we explore whether such a test can be extended to assess the influence of experimental characteristics, including the number of experiments.

Left: noiseless simulation of the model output (black line) and artificial noisy data (red circles) used for parameter estimation. Right: bootstrap results for the estimation of one of the model parameters. It can be seen that the parameter can be estimated accurately, despite the difficulty of determining high order derivatives of the output measurements.

…where we talk: the Akira Okobu prize, the Fokker-Planck equation, and the movement of bears.

Mark Lewis is a mathematical ecologist at the University of Victoria. He uses mathematical models to understand the environment and our human impacts. He tries to maintain that work life balance by spending that hard-earned free time curling.

Find out more about Mark’s groups’ work on the following website: https://lewisresearchlab.org/.

Find out more about SMB on: 

• Twitter: @smb_mathbiology 
• Bluesky: @smbmathbiology.bsky.social 
• Facebook: @smb.org 
• Linkedin: @smb_mathbiology 
• The Bulletin of Mathematical Biology

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Prevalence Estimation Methods for Time-Dependent Antibody Kinetics of Infected and Vaccinated Individuals: A Markov Chain Approach

by Prajakta Bedekar, Rayanne A. Luke, and Anthony J. Kearsley

Modeling the change in antibody levels post infection or vaccination improves understanding of the time-dependent immune response. Disease or vaccination prevalence in populations and time-dependence simultaneously affect antibody levels, interact non-trivially, and pose considerable modeling challenges. We model transitions from the naïve state to either the infected or vaccinated state using a time-varying stochastic process. This is coupled with a probabilistic framework to describe post-event antibody dynamics. An important result of this work is the design of an unbiased prevalence estimation method. This is a critical step towards analyzing protection from infection or vaccination and improving booster timing recommendations.

Graphical abstract created with bioRender.

How Residual Fertility Impacts the Efficiency of Crop Pest Control by the Sterile Insect Technique

by Marine A. Courtois, Ludovic Mailleret, Suzanne Touzeau, Louise van Oudenhove, and Frédéric Grognard

The sterile insect technique (SIT) is a pest control method. It reduces pest populations through the release of sterilized males that disrupt reproduction by preventing females from producing viable offspring. Our study examines how residual fertility—when sterilized males retain some fertility—affects SIT success. Using mathematical models, we identified residual fertility thresholds: (1) below which pest eradication is achievable, and (2) below which only population reduction can be obtained. This research enhances SIT implementation strategies by providing insights into the balance between residual fertility and sterile male attractiveness. These findings help develop stronger pest control strategies for sustainable agriculture.

Optimizing sterile insect technique (SIT): managing residual fertility.

Ecosystem knowledge should replace coexistence and stability assumptions in ecological network modelling

by Sarah Vollert, Christopher Drovandi, and Matthew Adams

Ecosystem models are often used to aid conservation decision making, to help quantify the risks of management, and assess the probability of conservation success. But these models are frequently built on the assumption that an ecosystem will naturally stabilise towards a coexisting balance of species. This paper argues that this theoretical assumption is inappropriate for conservation planning, because it downplays the risks of extinction. Instead, we demonstrate how ecological field knowledge can replace this assumption without significant loss of information and show that expert knowledge leads to more realistic population predictions.

Using ecological observations as an alternative for constructing ecosystem models.

A dynamical analysis of the alignment mechanism between two interacting cells

by Vivienne Leech, Mohit P. Dalwadi, and Angelika Manhart

In biology, organisms often align their body orientation to coordinate their movement. Famous examples are schools of fish or flocks of birds. Here we focus on aligning cells, such as bacterial cells or skin cells, whose collective dynamics affects the behaviour of the bacterial colony or the properties of the skin tissue (relevant e.g. in scar formation). We zoom into the collective alignment dynamics and focus on the interactions between two cells on a 2D surface. The cells move, turn and deform in order to avoid overlapping. We thoroughly mathematically analyse the resulting non-linear system of ordinary differential equations. This allows to understand the role the model ingredients, such as self-propulsion, play for alignment dynamics.

Mathematically analysing cell alignment.

Dynamics of Antibody Binding and Neutralization during Viral Infection

by Zhenying Chen, Hasan Ahmed, Cora Hirst, and Rustom Antia

During a viral infection, virions are continuously produced by infected cells and rapidly decay. Meanwhile, neutralizing antibodies are produced and bind to virion sites, preventing them from infecting target cells. In this context, virus growth rate and decay rate have an impact on the amount of antibody binding, potentially substantially reducing the impact of Koff on antibody binding. We then show that greatly simplified neutralization models have similar virus growth dynamics to more complex model, but they are less suited for exploring how antibody affinity and the proportion of bound sites on a virion reduce virus growth rate. Thus, the choice of models should depend on the specific research question of interest. 

Graphical abstract created with bioRender.

…where we talk: society nominations, prion proteins and murderbots.

Professor Suzanne Sindi is a Mathematical Biologist studying protein aggregation, and blood coagulation through modeling and data science. She is passionate about promoting inclusion in math and STEM, and was inspired to go into science by dinosaur-related Sci-Fi.

Find out more about Suzanne’s work on her website: https://www.sindilab.com/.

Find out more about SMB on: 

• Twitter: @smb_mathbiology 
• Bluesky: @smbmathbiology.bsky.social 
• Facebook: @smb.org 
• Linkedin: @smb_mathbiology 
• The Bulletin of Mathematical Biology

Apple Link      Spotify Link     Read the full transcript

Accumulation of Oncogenic Mutations During Progression from Healthy Tissue to Cancer

by Ruibo Zhang and Ivana Bozic

Carcinogenesis is a multi-stage process in which driver gene mutations occur sequentially. Understanding the arrival times of genetically different subclones provides important insights into tumorigenesis. In this work, we establish a multi-type branching process to model the initiation of cancer that starts from a healthy tissue in homeostasis. Mutations can be either neutral or advantageous, which reflects that inactivating a single copy of a tumor suppressor gene does not directly provide a selective growth advantage. We approximate the distribution of the arrival time for each type and compare it to computer simulations of the process. The results are applied to study the initiation of colorectal cancer and chronic myeloid leukemia. 

Model illustration. The model describes an evolutionary process that starts with a large healthy population in homeostasis (blue circles). Mutations that are either neutral or advantageous occur sequentially, which causes subsequent types to be either homeostatic (yellow) or initiated (orange). The cancerous type (red) emerges only when all the required genetic alterations have taken place.

Editorial By Sara Loo

Ways to engage with SMB

This quarter’s editorial is less an editorial, and more of a spiel of a couple of ways in which you can engage with the Society and share with others in the community! We love hearing from our community and looking for ways in which to engage with one another.  As a member of SMB, you have access to our Member Forum, news items, and can share your work and interests with others through our Highlights page. If you haven't logged on for a while, now might be a good time to make sure your profile is up to date with your latest interests - click here to update and select your preferred Subgroups to receive communications from subgroup leaders.  SMB Member Forum A primary way you can do this is through the SMB Member Forum. The forum is a great platform for sharing with others in our community – be it an upcoming conference, job postings, funding opportunities, or even a call-out for like-minded members to collaborate. On the main website, recent member forum posts can be seen on the Home page, as well as through the Communications tab. If you are a member of SMB, once you are logged on to the main website, post on our Member Forum by clicking on Create Topic and including any details you want to share. Click on Subscribe to get updates on all new posts via email. You can also subscribe to a single forum post for updates to that specific post.  For any membership issues or problems with logging on to the website, contact website@smb.org

Highlights In addition to the forum, if you have a paper published in the Bulletin of Mathematical Biology that you would like to highlight, our publications team would like to hear from you! Submit the paper to the team to highlight, using the linked form in the website menu, with a brief description (max 750 words) of the highlights of your paper, along with a figure. Think of this as a brief ‘featured figure’! These get posted by the Publications Team in the News section of our website.

Bluesky Have you found yourself in the recent wave drifting over to Bluesky?  You can find us at @smbmathbiology.bsky.social where we share recent paper highlights, repost job advertisements,  and other recent news. We’ve also pulled together an SMB Starter Pack so you can follow all your favorite mathematical biologists and continue to grow our community.

Other social media Follow us on X: @SMB_MathBiology Instagram: @smbmathbio Threads: @smbmathbio Featured Figures By Thomas Woolley In this issue, we feature the work of Jana Gevertz, College of New Jersey, and Giulia Celora, University College London.  We asked Jana Gevertz to tell us a bit more about her work here: Assessing the Role of Patient Generation Techniques in Virtual Clinical Trial Outcomes Clinical trials are research studies where novel medical interventions are tested on people who volunteer to receive the treatment of interest. These studies are the primary way that researchers find out if a new treatment is safe and effective in humans. The predictions made by clinical trials are generally limited by small sample sizes and may be biased to certain demographic groups which are more inclined to enroll in these studies. Virtual clinical trials (VCTs), grounded in data-informed mathematics models, are growing in popularity as a tool for quantitatively predicting heterogeneous treatment responses across a population. They hold the promise of complementing standard clinical trials by computationally permitting the analysis of a more diverse and representative patient population. In the context of a VCT, a “plausible patient” is an instance of a mathematical model with parameter (or attribute) values chosen to reflect features of the disease and response to treatment for that particular patient. A challenging question in the design of VCTs is to determine which set of model parameterizations (that is, which “plausible patients”) should actually be included in the virtual population. The aim of our work was to rigorously quantify the impact that VCT design choices have on the heterogeneity of the virtual population, and on the predictions of a virtual clinical trial. To isolate the impact of VCT design choices, we worked with simulated patient data and a simple, toy model of tumor growth that predicted response to the treatment. In this controlled setting, we studied the impact of the following VCT design choices (see Figure): the prior distribution of each parameter that varies across patients, and the method for selecting which parameterizations are considered virtual patients and thus included in the VCT. Our analysis revealed that the prior distribution, rather than the inclusion/exclusion criteria, has a larger impact on the heterogeneity of the virtual population. Yet, the predictions of the virtual clinical trial were more sensitive to the inclusion/exclusion criteria utilized. This foundational understanding of the role of virtual clinical trial design should help inform the development of future VCTs that use more complex models and real data. We asked Giulia Celora to tell us a bit more about her work here: Characterising Cancer Cell Responses to Cyclic Hypoxia Using Mathematical Modelling In solid tumours, the presence of regions of abnormally low oxygen levels (i.e., hypoxia) is recognised as a major driver of tumour progression and therapeutic resistance. Even though in vitro models of hypoxia exist, they often fail to capture the complex and heterogeneous oxygenation dynamics of real tumours. While most experimental studies have focussed on characterising cell responses to constant hypoxic conditions, in vivo observations show that tumour oxygen levels can fluctuate on fast timescales and expose cancer cells to periodic cycles of hypoxia; a phenomenon known as cyclic hypoxia. These observations raise questions regarding the applicability of such experimental findings to the clinical understanding of hypoxia: do cyclic and constant hypoxia elicit different responses in cancer cells? If so, what features of fluctuating oxygen conditions are cancer cells sensitive to? Is the frequency of hypoxia cycles, their duration? Or both? In our recent publication, we used mechanistic mathematical modelling to quantify the impact of prolonged exposure to various cyclic hypoxia conditions on the population growth and survival of tumour cell cultures. In particular, we developed a structured stochastic individual-based cell cycle model that accounts for hypoxia-driven dysregulation of both cell cycle and cell survival. In this framework, each cell is an agent that can either proliferate or die with probabilities that depend on its internal state. The cell internal state is described by a list of categorical and continuous structure variables, that also evolve probabilistically over time. Structure variables allow us to capture the multi-layered feedback between oxygen levels, intracellular processes (such as DNA replication and repair), and cell fate – namely, proliferation and death.

Our model allows us to efficiently characterise how cancer cells respond to hypoxia cycles of varying duration and frequency. As shown in the Figure, we find that cell responses to cyclic hypoxia can be classified into four major groups depending on the extent to which population growth and cell survival are affected by periodic exposure to hypoxia. Our results highlight the multifaceted nature of cyclic hypoxia and the role of fluctuating oxygen levels in creating heterogeneous environmental conditions within tumours. You can learn more about our work and the implications of our results on the connection between cyclic hypoxia and intra-tumour heterogeneity here: