Counting Subnetworks Under Gene Duplication in Genetic Regulatory Networks

by Ashley Scruse, Jonathan Arnold, and Robert Robinson

Gene families help us understand how gene regulation evolves. While sequence evolution is well-studied, regulatory evolution is not. Members of gene families form genetic regulatory networks (GRNs) where gene A regulates gene B. We develop a model counting regulatory subnetwork motifs within gene families as they evolve through duplication. The result is a count |M(n)| of how these motifs evolve to a stage with n total genes. The key parameter is the probability that a regulatory relation is preserved when a gene is duplicated. For Full Duplication, the mean and variance of motif counts can be computed exactly, enabling significance tests to identify targets for directed evolution. Partial and Mixed Duplication models are also presented.

This graphical abstract illustrates the mathematical framework for counting subnetwork motifs in genetic regulatory networks (GRNs) under gene duplication. The left panel depicts the gene duplication process: before duplication, gene A regulates gene B; after duplication of B, the duplicate gene B' may inherit the regulatory relationship from A with probability π. The right panel presents the key analytical results for the Full Duplication model (π = 1), including closed-form expressions for the expected motif count E(|M|) and the second moment E(|M|²). The bottom section shows the statistical framework for identifying significant motifs, with formulas for variance and Z-score calculation, along with a workflow for applying these methods to identify targets for directed evolution. The paper also derives analogous results for Partial Duplication (0 ≤ π ≤ 1), where the inheritance probability vector π = (π₁, ..., πₖ) allows each gene family to have its own probability of inheriting regulatory relationships, and discusses the Mixed Duplication model, where with probability 1−q all regulatory links are retained and with probability q each link is retained independently with probability p. See the full article for these additional duplication models.

...where we talk forging cancer cells, science policy and dancing.

Ranjini is a final-year PhD student in the IMO Department at Moffitt Cancer Center. She uses evolutionary game theory to study cancer foraging. Outside of her research, Ranjini engages in science communication and science policy with non-expert audiences, and loves to engage her creative side.

You can connect with her on Linkedin.

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Immune Modulation in the Tumor Microenvironment: Bifurcation Analysis of Cancer-CTL-Monocyte Dynamics

by Eymard Hernandez-Lopez, Russell Milne, and Xiunan Wang.

Read the paper

Cancer remains a major global health challenge. To understand how tumors grow or are controlled by immunity, we develop a mathematical model describing interactions among cancer cells, cytotoxic T cells, and monocytes. The model captures two key features: monocyte-driven T cell stimulation and a threshold effect whereby very small tumors may fail to establish because of limited resources and immune pressure. By analyzing system dynamics, we identify three outcomes—tumor elimination, stable coexistence, or uncontrolled growth—and show how these depend on parameters that can be modified clinically. This framework helps predict which therapies are most likely to shift the system toward durable tumor control.

Immune Modulation in the Tumor Microenvironment

SMB is pleased to announce need-based travel grants to the attend the ECMTB/SMB joint Annual Meeting in Graz, Austria this July. The travel grants aim to provide partial support and will be typically limited to $750 USD. The application deadline is 15 March 2026.

Details:  https://smb.org/SMB-Annual-Meeting-Travel-Award

Online application form:  https://smb.org/SMB-Annual-Meeting-Travel-Grant-Application/

Defining optimal vaccine features for pandemic preparedness: An individual-based model bridging within- and between-host dynamics.

by Xia Y., Alexandre M., Thiébaut R, Maheu-Giroux M, and Prague M.

The Study Xia et al. 2026 highlights which vaccine feature can help stop the next "Disease X" respiratory pandemic before it spreads. Researchers developed a powerful simulation that links how viruses behave within-hosts to how they spread between-hosts. Technically, the model includes parameters like the antibody concentration needed for 50% protection, the effect on clinical outcome during infection and waning immunity. Results show that vaccines lasting longer and working at lower antibody levels are the most effective at cutting transmission. This work offers a roadmap for designing next-generation vaccines that could keep future outbreaks under control.

A framework to define optimal vaccine features for pandemic preparedness.

A bibliometric study on mathematical oncology: interdisciplinarity, internationality, collaboration and trending topics

by Kira Pugh, Linnéa Gyllingberg, Stanislav Stratiev, and Sara Hamis 

In this study, we use bibliometric methods to obtain a bird’s-eye view of mathematical oncology as a research field. Our data consist of articles from five leading mathematical biology journals (referred to as our focus journals), including the Bulletin of Mathematical Biology. We examine interdisciplinarity, internationality, and trending topics in the field. Our analysis reveals that, since the 1960s, mathematical oncology has become more interactive with external research fields and globally connected, with changes in research themes. Our results show that mathematical oncology benefits both the mathematical and life sciences. Insights from our study can be used to inform funding, teaching, organisation, and communication practices.

We investigate interdisciplinarity, internationality, and trending topics in mathematical oncology through discipline-based citation flows (left figure), global author connectivity (middle figure), and word frequencies in titles and abstracts (right figure).

AI or die - with Reinhard Laubenbacher

These days you can’t throw a stone without hitting some form of AI. Whether it’s to control your home sound system, or looking for a novel way to treat disease, we are still in the learning stages of how best to use these technologies.

So Biology in Numbers welcomes back Reinhard Laubenbacher, current SMB president, who talks about some uses in the field, its relevance in education, and why math biologists and the larger scientific community need to get ahead of AI to ensure it is incorporated into research in a meaningful way.

Here’s a useful and pretty primer for anyone completely new to AI: from the BBC. For those interested in that foundational paper by McCulloch and Pitts here’s the link: A logical calculus of the ideas immanent in nervous activity.

If you’d like to leave any suggestions for Reinhard on the subject, you can reach him at: Reinhard.Laubenbacher[at]medicine.ufl.edu

Find out more about SMB on: 

• Twitter: @smb_mathbiology 
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• Facebook: @smb.org 
• Linkedin: @smb_mathbiology 
• The Bulletin of Mathematical Biology

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The Society is now accepting nominations for officer positions.These include the President-Elect, three regular members of the Board of Directors, one two-year and one four-year Early Career members of the Board. The latter are for those within five years of their PhD. Nominations are due by February 27, 2026.

Please email nominations (which could be self-nominations) to nominations@smb.org. These should include a 2-page CV and a brief election statement describing the nominee's past contributions and commitment to the SMB. Information regarding the duties of the Board and President-Elect can be found on the SMB web page, www.smb.org.

Astrocyte Reprogramming Drives Tumor Progression and Chemotherapy Resistance in Agent-Based Models of Breast Cancer Brain Metastases

by Rupleen Kaur, Rowan Barker-Clarke, and Andrew Dhawan.

Patients with advanced cancer frequently develop brain metastases, resulting in poor survival rates and limited therapeutic options. Understanding the interactions between tumor cells and the brain microenvironment remains challenging. This project employs an agent-based model to demonstrate how brain-resident astrocytes, when reprogrammed by tumor cells, can enhance tumor growth and facilitate drug resistance. By simulating distinct brain regions and quantifying tumor morphology and treatment responses, the study shows how the local microenvironment influences tumor expansion and therapeutic outcomes.

Tumor cells reprogram brain-resident astrocytes from an anti-metastatic to a pro-metastatic phenotype. Agent-based modeling reveals that astrocyte density and distribution across distinct brain regions (A-C) determine tumor expansion and morphological complexity. High uniform density (B) increases lacunarity but decreases eccentricity, producing irregular yet circular tumors. Gradient distribution (C) drives directional expansion with high eccentricity and fractal complexity but reduced lacunarity. These microenvironment-dependent morphological phenotypes may influence tumor progression and treatment outcomes.

Modelling ion channels with a view towards identifiability

by Ivo Siekmann.

Can we gain insight into the molecular structure of an ion channel from observing its opening and closing over time? Intuitively, we might expect that by mimicking the architecture of the channel in a model, the time scales of transitions between different biophysical states can be inferred from these observations. But on the one hand, many models are nonidentifiable which makes them unsuitable for mechanistic modelling of ion channels. On the other hand, models that represent completely different biophysical mechanisms can nevertheless be equivalent. Motivated by these challenges we suggest alternative approaches how opening and closing as well as biophysical dynamics of ion channels can be represented in mathematical models.

The fully-connected model is always non-identifiable model and therefore unsuitable for mechanistic modelling. But for some parameter sets it can be reduced to the models CCO or COC. CCO and COC are always equivalent although both represent different biophysical processes.