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Abstract: Many biological systems are characterised by randomness. At the cellular level, genetic material is transcribed into messenger RNA (mRNA) stochastically, proceeding in bursts rather than continuously; at the population level, gene mutations occur probabilistically in individuals within a population. To understand the dynamics and steady-state behaviours of such systems, they are often modelled in terms of stochastic processes. In this talk, I will introduce the modelling frameworks we use to study mutation accumulation and stochastic gene expression, and show how the latter can be used to investigate the heterogeneity observed in cancerous tumours. I will also present statistical inference tools that can be used to compare numerical predictions with experimental data, allowing for quantitative understanding of biological systems. This talk was the supporting lecture at the 2025 Mary Cartwright Lecture, which took place on Friday 7 November 2025 online via Zoom. ========== The London Mathematical Society has, since 1865, been the UK's learned society for the advancement, dissemination and promotion of mathematical knowledge. Our mission is to advance mathematics through our members and the broader scientific community worldwide. For further information: ► Website: https://www.lms.ac.uk ► Events: https://www.lms.ac.uk/events ► Grants and Prizes: https://www.lms.ac.uk/grants-prizes ► Publications: https://www.lms.ac.uk/publications ► Membership: https://www.lms.ac.uk/membership Follow us: ► Twitter: / londmathsoc ► Facebook: / londonmathematicalsociety ► LinkedIn: / the-london-mathematical-society ► Youtube: @LondonMathematicalSociety