The dynamics of biological systems is driven by interactions between many elements at a given level of biological organisation (e.g. molecular, cellular, organismc), but also by the couplings that exist between said levels (e.g. from molecules to cells to populations). Such couplings are highly non-linear and make the analysis of complex biological systems extremely challenging . The remit of the Mathematical and Computational Biology is the development of new theory, models, techniques, and tools that are relevant to biologists and clinicians. For this purpose we use a plethora of mathematical techniques including stochastic multiscale models, dynamical systems theory, singular perturbation analysis, bifurcation analysis, morphometrics, dimensional reduction tools and efficient simulation methods, as well as statistics, machine learning or optimization. We tackle issues such as understanding how genetic variation leads to variation in the characteristics of organisms, the so-called genotype-phenotype map, the arising of such map in embryonic development, its influence in the direction of phenotypic evolution. We also formulate new models of virus evolution and therapies that account for intrinsic heterogeneity and noise, we study the design of new strategies to avoid drug resistance induced by cancer-cell heterogeneity and analyze the mechanisms of ageing. Our research is collaborative in nature and we make an effort to keep close collaborations with both biologists and medical practitioners.


Cancer Modeling | Tomás Alarcón

Most phenomena studied by the Natural Sciences, from Material Sciences to Astrophysics, are multi-scale processes, i.e. they involve the coupling of multiple different processes characterised by widely-ranging time and length scales, with the macroscopic behaviour emerging from the complex interactions between them. Whilst considerable progress has been done in dealing with such problems in the Physical Sciences, the success achieved so far in the Biological Sciences is rather more limited. This is partly due to the fact that the individual components of biological systems (e.g. cells) are much more complex than their counterparts in physical systems and, therefore, new methods and models are needed to analyse multi-scale processes in Biology. Such is the remit of the Computational & Mathematical Biology group at CRM: To propose new models relevant to experimental biologists and clinicians and develop the analytical and computational tools necessary for their analysis. We pay special attention to problems with clinical relevance, in particular those related to cancer.

The research activity of our group is developed along the following lines:

  • Multiscale modelling of tumour growth and tumour-induced angiogenesis
  • Evolutionary dynamics of populations with complex structure, in particular cell populations with hierarchical structure and genotype-phenotype map
  • Mathematical modelling of the cell-cycle
  • Stochastic modelling of receptor tyrosine kinases
  • Tumour dormancy
Mathematics of Development and Evolution | Isaac Salazar

Our group is focused in understanding the mathematical bases of evolution.

The main question we want to address is: how did complex organisms arise in evolution? Or more in general, how can complexity evolve also in other systems like culture, society and molecular pre-biotic systems. In the case of multicellular organisms, such as us, this main question translates into three other questions that we want to address:

  1. How does a fertilized egg cell transforms into a complex adult organism? This is a complex functional organism characterized by many cells, cell types and a specific distribution of those in space. Can we understand the mathematics of such a dramatic pattern formation process?
  2. How did this complexity arose by a gradual process of evolution by natural selection. This implies explaining also the evolution of the development that produces such complexity in each generation.
  3. Are there some logical or mathematical requirements or principles that gene networks need to fulfill in order to be able to produce complex morphologies during development? If so, can we approach question 1 and 2 above from understanding these principles?
    Understanding question 1 is highly non-trivial: something very complex, our body, is produced from something much simpler and small, a simple cell, in a relatively short time.

This process can not be understood by looking at single genes. Embryonic development involves the interaction between huge numbers of genes and cells. Thus, for example, to understand which morphological changes will occur from specific mutations in a gene, we need to understand how that gene is embedded in a gene network and how that affects the dynamics of signaling and mechanical interactions between cells and tissues. In other words, we have a huge number of heavily interacting elements at different levels (e.g. genes, cells, tissues) that lead to the arising and variation of a macroscopic pattern, the body’s.

To address question 1 we build multi-scale models of embryonic development. Each such models includes a set of differential equations describing how genes regulate each other’s expression and a set of differential equations describing how cells move, change shape and activate cell behaviors (cell growth, cell contraction, cell division, etc…). Each cell contains the same set of genes and equations, but, as a result of model dynamics, different cells end up expressing genes at different intensities. Genes affect the mechanical properties and behaviors of the cells in which they are expressed. As a result, cells move and rearrange themselves in space and, in their turn, affect back gene expression by differentially affecting, through cell-cell signaling, where genes get expressed. The models are, essentially, reaction-diffusion models in which the shape of the compartment in which diffusion is taking place, the embryo, is changing over time as a consequence of the reactions, i.e. gene regulation, affecting cell movement. Through specific gene networks, initial conditions and regulation of cell behaviors and mechanical properties by gene products, we simulate how the morphology of adult organs develops. In other words, the model reproduces the position in 3D space and the gene expression in each cell in the adult and how that has changed over development. We do that for specific organs in collaboration with experimentalists (e.g. teeth (Salazar-Ciudad and Jernvall, 2002, 2010; Salazar-Ciudad 2008, 2012; Järvinen et al., 2006; Harjunmaa et al., 2014; Renvoise et al., 2017), fly wings (Ray et al., 2015), turtle carapaces (Moustakas-Verho, 2014) among others) but also for the ensemble of animal development (Salazar-Ciudad, 2000, 2001a, 2001b, Marin-Riera et al., 2016).

For question 2 we take two different approaches. In a populational one we combine the models of development we build for question 1 with models of mutation, natural selection and genetic drift in populations. In a way we put together the mathematical apparatus of populational genetics with that of the multiscale models of embryonic development. This way we simulate how complex morphologies arise in evolution. In the ensemble approach we build a huge number of random networks and check which ones are able to produce complex morphologies.

From both approaches we want to understand how phenotypes and development evolve and, also, if there are some logical or mathematical requirements that gene networks need to fulfill in order to be able to produce complex morphologies during development. If that would be the case, as we think it is, the study of development and evolution would be greatly simplified. The evolution of development could then be understood, as we attempt, by looking at the networks fulfilling these requirements and exploring how likely they arise from random mutation, or how likely one can be transformed into another, as compared with the adaptive value of the morphologies they produce (Salazar-Ciudad et al., 2001a, Salazar-Ciudad 2010). This will entitle us with building a general theory of how development works, and most importantly, of how it evolves.

In parallel we also explore how our evolution and development approach can be applied to evolution in culture (Salazar-Ciudad, 2010c) and in the origins of life (Salazar-Ciudad 2008b, 2013b). In these approaches development is just replaced by the processes that lead to phenotypic variation.

Part of our research also involves discussing how the insights acquired from our models modify several aspects of evolutionary theory such as developmental constraints (Salazar-Ciudad, 2006), canalization (Salazar-Ciudad, 2007), robustness (Salazar-Ciudad, 2007), novelty (Salazar-Ciudad, 2006b) or just evolutionary theory in general (Salazar-Ciudad, 2008).

In addition, we also have data-mining approaches in what we call statistical developmental biology. This is the combined analysis of massive transcriptomic and genomic databases to  quantitatively test several long-held hypothesis on the evolution of development, this goes from an explicit measurement of how complexity increases over development and anatomy (Salvador-Martinez and Salazar-Ciudad, 2015, 2017) to an estimation of a map of acting selection over the body of the fly (Salvador-Martinez et al., 2018).

We added an image, an example of a developmental mechanism with a gene network combining different cell behaviours implemented in EmbryoMaker. The left diagram show the gene network. The boxes indicate specific cell behaviours or cell mechanical properties regulated by specific genes in the network.  a) Initial conditions, hollow spheric epithelium with a single cell (yellow) expressing gene TF1. B) Outcome, after a number of iterations, of the complex developmental mechanism applied on the initial conditions in A. The left column shows, in section, the node types. Blue for basal side of cylinders, violet for the apical side of cylinders, red for mesenchymal cells and orange for extracellular matrix nodes. Middle and right column display concentrations of GF2 and TF5 respectively (yellow for high concentration, blue for low concentration).

Nonlinear Dynamics and Evolution (NoDE) | Josep Sardanyés

Our laboratory is interested in understanding biological nonlinear phenomena. Our research is focused on Biomedicine (including cancer and viruses), in systems and synthetic Biology as well as in theoretical ecology. To do so we use the qualitative theory of dynamical systems and computer simulations (stochastic dynamics and spatially-extended systems).

We are especially interested in characterizing both asymptotic and transient dynamics of these systems and their sensitivity to parameter changes. That is, understand which bifurcations govern transitions in biological systems.


Personal Website

Tomás Alarcon


I obtained my PhD in Theoretical Physics from the University of Barcelona in 2000. After that I spent many wonderful years working as a postdoc at the University of Oxford, UK (2001-2003), University College London, UK (2003-2006), and Imperial College London, UK (2006-2009). I briefly held a senior researcher and group leader position at BCAM, Bilbao, Spain (2009-2010), after which I moved the Centre Recerca Matematica where I lead the Cancer Modelling Group. I have also held visiting fellowships at the Universidad Complutense de Madrid, IIMAS (UNAM, Mexico DF), OCCAM (University of Oxford, UK), the Mathematical Institute (University of Oxford, UK), and the Mathematical Biosciences Institute (Columbus, Ohio, USA). In October 2015, I was appointed to an ICREA Research Professorship at the Centre de Recerca Matematica. 

Personal Website

Isaac Salazar


I studied Biology for my Bachelor degree in the UAB and went on to earn a PhD in Biology from the University of Barcelona (March 2002). Upon obtention of my PhD degree, I moved to the Developmental Biology Program of the Biotechnology Institute of the University of Helsinki as a Marie-Curie postdoctoral fellow. There I joined Jukka Jernvall’s group to continue my research in theoretical models of evolution and development. I became a Finnish Academy postdoctoral fellow two years later in the same group and a Juselius Foundation postdoctoral fellow after that. In 2008 I started as a Ramón y Cajal fellow in the department of Genetics and Microbiology of the UAB. In 2011 I moved back to Helsinki as a junior group leader (Finnish academy fellow) but I kept a small group in the UAB. In 2016 I became a PI in the Institute of Biotechnology of the University of Helsinki.

Personal Website

Josep Sardanyés


I completed a BSc in Biology at University of Barcelona (2002) and went on to earn a Master and a PhD in Biomedicine (2009) at the Complex Systems Lab (CSL, Universitat Pompeu Fabra). During my PhD thesis I worked on dynamical systems and complexity in Biology. Upon obtaining my PhD degree, I moved to Valencia where I took up a postdoc position at the Institute of Molecular and Cellular Plant Biology (Consejo Superior de Investigaciones Científicas-UPV). In 2011 I moved to The David J. Gladstone Institutes for a second postdoc (University of California San Francisco, USA), where I focused on the evolutionary dynamics of RNA virus. Then, I moved again to the CSL where I completed a third postdoc (2012-2016) working on cancer evolution, theoretical ecology and systems and synthetic biology. In 2017 I became a researcher at the CRM.

Leonardo Da Cruz
Universidade de São Paulo-USP

Postdoctoral Researcher

IP: Josep Sardanyés

Giovani Dalmasso

Postdoctoral Researcher

IP: Tomás Alarcón

Filip Ivancic


Postdoctoral Researcher

IP: Josep Sardanyés

Juan Arellano

PhD Student

Supervisor: Tomás Alarcón

Stefano Pedarra

PhD Student

Supervisor: Tomás Alarcón

Kevin Martínez

PhD Student

Supervisor: Isaac Salazar


Publications from the last 5 years

D.Stepanova, H.M. Byrne, P.K. Maini, T. Alarcon. A multiscale model of complex endothelial cell dynamics in early angiogenesis. Posted in bioRxiv.

M.O. Bernabeu, J. Kory, J.A. Grogan, B. Markelc, A. Beardo-Ricol, M. d’Avezac, J. Kaeppler, N. Daly, J. Hetherington, T. Krueger, P.K. Maini, J.M. Pitt-Francis, R.J. Muschel, T. Alarcon, H.M. Byrne. Abnormal morphology biases haematocrit distribution in tumour vasculature and contributes to heterogeneity in tissue oxygenation. Posted in bioRxiv.

A.I. Rodriguez-Villarreal, L. Ortega-Tana, J. Cid, A. Hernendez-Machado, T. Alarcon, P. Miribel-Catala, J. Colomer-Farrarons. An integrated detection method for flow viscosity measurements in microdevices. IEEE Transactions on Biomedical Engineering. Accepted for publication (2020).

E. Cuyas, S. Verdura, B. Martin-Castillo, T. Alarcon, R. Lupu, J. Bosch-Barrera, J.A. Menendez. Tumor-cell-intrinsic immunometabolism and precision nutrition in cancer immunotherapy. 12, 1757 (2020). doi: 10.3390/cancers12071757

E. Cuyas, J. Gumizio, S. Verdura, J. Brunet, J. Bosch-Barrera, B. Martin-Castillo, T. Alarcon, J.A. Encinar, A. Martin, and J.A. Menendez. The LSD1 inhibitor iadademstat (ORY-1001) targets SOX2-driven breast cancer stem cells: A potential epigenetic therapy in luminal-B and HER2-positive breast cancer subtypes. Aging.  12, 4794-4814 (2020). doi: 10.18632/aging.102887

R.A. Barrio, T. Alarcon, A. Hernandez-Machado. The dynamics of shapes of vesicle membranes with time dependent spontaneous curvature. PLoS One. 15, e0227562 (2020). doi: 10.1371/journal.pone.0227562

J.A. Menendez, E. Cuyas, N. Folguera-Blasco, S. Verdura, B. Martin-Castillo, T. Alarcon. In silico clinical trials for anti-aging therapies. Aging. 11, 6591-6601 (2019). doi: 10.18632/aging.102180

A.V. Ponce-Bobadilla, T. Carraro, H.M. Byrne, P.K. Maini, T. Alarcon. Age-structure can account for delayed logistic proliferation of scratch assays. Bull. Math. Biol. 81, 2706-2724 (2019). Preprint version available from bioRxiv

Vidiella B, Valverde S, Fontich E, Sardanyés J, Transients in simple trophic chains with facilitation: the impact of habitat destruction. Theoretical Ecology (submitted)

Sardanyés J, Raich C, Alarcón T. Noise-induced stabilization of saddle-node ghosts. Physical Review E (submitted)

Solé R, Sardanyés J, Elena SF, Phase transitions in Virology. Virus Evolution (submitted)

Zaldo Q, Serra I, Alsedà Ll, Sardanyés J, Maneja R, Reviewing the reliability of Land Use and Land Cover Data in studies relating human health to the environment. Land Use Policy (submitted)

Vidiella B, Sardanyés J, Solé R, Synthetic soil crusts against green-desert transitions: a spatial model. Royal Society Open Science​ (submitted)

Penella C, Alarcón T, Sardanyés J, Spatio-temporal dynamics of cancer phenotypic quasispecies under targeted therapy. Proceedings of the 9th International Congress on Industrial and Applied Mathematics (ICIAM) (2020) Accepted​

Nurtay A, Hennessy MG, Alsedà Ll, Elena SF, Sardanyés J, Host-virus evolutionary dynamics with specialist and generalist infection strategies: bifurcations, bistability and chaos. Chaos (2020) Accepted​

Perona J, Fontich E, Sardanyés J, Dynamical effects of loss of cooperation in discrete-time hypercycles​. Physica D: Nonlinear Phenomena 406: 132425 (2020)

Conde-Pueyo N, Vidiella B, Sardanyés J, Berdugo M, Maestre FT, de Lorenzo V, Solé R, Synthetic biology for terraformation: lessons from Mars, Earth, and the microbiome​. Life 10(2): 14 (2020)

Alsedà Ll, Vidiella B, Solé R, Lázaro JT, Sardanyés J, Dynamics in a time-discrete food-chain model with strong pressure on preys​. Communications in Nonlinear Science and Numerical Simulation 84, 105187 (2020)

Sardanyés J, Piñero J, Solé R, Habitat loss-induced tipping points in metapopulations with facilitation​. Population Ecology 1-14 (2019)

Sardanyés J. Viruses: Agents of evolutionary invention​. The Quarterly Review of Biology 94(1): 90-91 (2019)

Nurtay A, Hennessy MG, Sardanyés J, Alsedà Ll, Elena SF, Theoretical conditions for the coexistence of viral strains with differences in phenotypic traits: a bifurcation analysis​. Royal Society Open Science 6: 181179 (2019)

Fornés J, Lázaro JT, Alarcón T, Elena SF, Sardanyés J, Viral replication modes in single-peak fitness landscapes: A dynamical systems analysis. Journal of Theoretical Biology 460: 170-183 (2019)

Daniel J. Toddie-Moore Martti P. Montanari Ngan Vi Tran Evgeniy M. Brik Hanna Antson Isaac Salazar-Ciudad Osamu Shimmi, Mechano-chemical feedback mediated competition for BMP signalling leads to pattern formation

Salazar-Ciudad, Isaac. Why call it developmental bias when it is just development?. Biology Direct, 2021, vol. 16, p. 1-13.

Milocco, L., & Salazar-Ciudad, I. (2022). Evolution of the G Matrix under Nonlinear Genotype-Phenotype Maps. The American Naturalist, 199(3), 420-435.

Hagolani, P. F., Zimm, R., Vroomans, R., & Salazar-Ciudad, I. (2021). On the evolution and development of morphological complexity: A view from gene regulatory networks. PLoS computational biology, 17(2), e1008570.

Coronado-Zamora, M., Salvador-Martínez, I., Castellano, D., Barbadilla, A., & Salazar-Ciudad, I. (2019). Adaptation and conservation throughout the Drosophila melanogaster life-cycle. Genome biology and evolution, 11(5), 1463-1482.

Hagolani, P. F., Zimm, R., Marin-Riera, M., & Salazar-Ciudad, I. (2019). Cell signaling stabilizes morphogenesis against noise. Development, 146(20), dev179309.


PDC2022-133020-I00 Sistemas dinámicos y matemática computacional para la optimización de partículas interferentes terapéuticas como terapia antiviral  Pruebas de Concepto Josep Sardanyés / Tomás Alarcón MICINN 01/12/2022



HENOCANDYN: Heterogeneity and noise as engines of cancer evolution: A multiscale dynamical systems approach Generación de Conocimiento Tomás Alarcón / Josep Sardanyés MICINN 01/09/2022 01/09/2025
PCI2022-132926 MPA4SUSTAINABILITY: Enhancing Marine Protected Areas’ role in restoring biodiversity while maintaining access to ecosystem services (summary) BiodivRestore COFUND Action, BiodivERsA and Water JPI Josep Sardanyés IFD, ANR, FCT, AEI, SEPA 14/03/2022 14/03/2025
MTM – RTI2018-098322-B-I00 Los puntales matemáticos de la biología integrativa de sistemas Retos Investigación Tomás Alarcón/Josep Sardanyés MINECO 01/01/2019 31/12/2021

Predicting the evolution of complex phenotypes by contrasting and combining quantitative genetics and mathematical models of development

Isaac Salazar Ministerio de Ciencia, Innovación y Universidades 2022 2025
PGC2018-096802-B-I00 Understanding the developmental bases of additive genetic variance, environmental variance and the genotype-phenotype map Isaac Salazar

Ministerio de Ciencia, Innovación y Universidades

2019 2021


Jukka Jernvall  | Helsinki University
David Houle | Florida State University
Osamu Shimmi | Helsinki University
Stuart A Newman | New York Medical College
Antonio Barbadilla | Universitat Autònoma de Barcelona
Helen Byrne | University of Oxford
Philip K. Maini | University of Oxford
Karen M Page | University College London
Juan Calvo | Universidad de Granada
Jorge das Neves Duarte | Instituto Superior de Engenharia de Lisboa (ISEL) and Instituo Superior Técnico (IST)
Cristina Januário | Instituto Superior de Engenharia de Lisboa
Jordi Canela | Universitat Jaume I
Santiago F. Elena | Instituto de Biología Integrativa de Sistemas and The Santa Fe Institute
Cristina Januário | Instituto Superior de Engenharia de Lisboa and Instituo Superior Técnico de Lisboa
Nuno Martins | Instituo Superior Técnico de Lisboa
Javier A. Menendez | Cancer Metabolism Lab, Catalan Institute for Oncology
Daniel Oro | Centre d’Estudis Avançats de Blanes – CSIC
Carlos Peña Garay | Laboratorio Subterráneo de Canfranc
Sergi Valverde | Evolution of Networks Lab, Institut de Biologia Evolutiva – CSIC
Blai Vidiella | Evolution of Networks Lab, Institut de Biologia Evolutiva – CSIC
CSIC Associated Unit (DySCoVir) between the I2SysBio and CRM

The I2SysBio and the Centre de Recerca Matemàtica (Mathematical Research Center) launch the associated unit Dynamic Systems and Computational Virology (DysCoVir) — NewsDysCoVir website