# Mathematics for Developmental Biology (17w5164)

Arriving in Banff, Alberta Sunday, December 10 and departing Friday December 15, 2017

## Organizers

Eric Mjolsness (University of California, Irvine)

Przemyslaw Prusinkiewicz (University of Calgary)

## Objectives

The workshop will bring together developmental biologists, computer scientists, mathematicians and physicists, whose research incorporates mathematical description, modeling and analysis of development. In addition, several mathematicians specializing in areas with emerging applications to developmental biology will be invited to share their expertise and exchange ideas and inspirations. The list of participants will also include young researchers, with the potential to further transform developmental biology from a descriptive to a mathematical science.

Mathematical descriptions of the development and form of organisms can be traced to the book “On Growth and Form” by d’Arcy Thomson [1], and the paper “The chemical basis of morphogenesis” by Turing [2]. Over time, these works have led to a substantial body of computational modeling techniques and mathematical analyses. The seminal ideas of Thompson have now been refined into advanced models of growth, exploiting parallels between growth and continuum mechanics [3]. Likewise, the ideas of Turing have led to a large body of theoretical results and computational simulation models of pattern formation in organisms [4].

These well-established models of growth and patterning are significantly different from each other, yet they share a common element: they describe spatio-temporal processes in continuous terms and, consequently, are expressed as partial differential equations. The quest for describing, modeling and understanding diverse developmental processes in a realistic, insightful manner, however, highlights a need for additional mathematical tools and methods. Sample problems of interest are listed below.

1) Reaction-diffusion vs. controlled-transport models of morphogenesis. In reaction-diffusion, patterning results from feedback between local concentration and production or depletion of morphogens. Their transport is passive. In contrast, a fundamental patterning process in the kingdom of plants involves a feedback between fluxes of a morphogen (the plant hormone auxin) and its active transport [5,6]. Simulations show that patterns produced by both classes of models partially overlap [7,8], but a deeper understanding of the relations between these models and the patterns they produce is needed.

2) Geometric models of morphogenesis. In a classic paper [9], Gierer and Meinhardt observed that patterns produced by reaction-diffusion can be described and explained in terms of short-distance activation and long-distance inhibition. These explanations attribute a morphogenetic role to a geometric notion – the evaluation of distances. Furthermore, models of diverse developmental processes including phyllotaxis (patterning of plant organs around their supporting axis) [10], vein pattern formation in leaves [11] and cell division patterns [12] have been formulated directly in geometric terms. These results point to a need for a theory of morphogenesis based directly on metric geometry [13]. Such a theory would represent an intermediate level of abstraction between biochemical and biomechanical models of morphogenesis, and macroscopic patterns and forms, offering an in-depth insight into the geometric essence of different morphogenetic processes.

3) The impact of the rates of information flow. In growing structures, rates of information flow may be commensurate with the rates of growth [14]. This raises the question of how the rates of information flow affect the resulting patterns and forms. This question appears with particular clarity in geometric models of morphogenesis. It appears that, mathematically, it is related to problems underlying the propagation of information (limited by the speed of light) in an expanding universe, considered in general relativity and cosmology [15]: a highly intriguing and unexpected relation between biology, geometry, and physics.

4) Discontinuous aspects of growth. Tissue growth is typically described by a growth tensor field [16], which is mathematically equivalent to the strain tensor field in the theory of elasticity. Implicit in this description is the assumption of the spatial continuity of growth rates. However, an increasing body of experimental results indicates that growth is often highly heterogeneous, and may even be spatially discontinuous. This calls for a reexamination of the assumption of continuity in growth models. What are the weakest assumptions under which growth can proceed without sliding, buckling or tearing?

5) Constraints of space. Growing structures are necessarily embedded in the ambient space, which puts constraints on possible forms. These constraints are manifest, for example, in the wrinkled shape of leaves that grow excessively along the margin [17], the decreasing size of leaflets in recursively compound leaves, and the competition for space between branches of a growing tree [18]. The resulting forms are currently characterized using independent mathematical tools: for example, wrinkled surfaces are described in terms of negative Gaussian curvature, which does not apply to branching structures. Is there a mathematical description that would capture all these manifestations of constraints of space in a unified manner?

6) Discrete models of multicellular structures. Treating multicellular structures as continua is an often-used macroscopic approximation, but does not adequately represent microscopic structures. There, growth and divisions of individual cells, and cell-to-cell communication, play an important role. In 1968, Aristid Lindenmayer proposed an original formalism, later termed L-systems, which – with subsequent extensions – makes it possible to model and reason about development of linear and branching multicellular structures in development [19]. Extensions to two- and three-dimensional tissues (discrete 2- and 3-manifolds) have been sought for a long time, but progress has only recently been made, using the topological notion of cell complexes [20]. This partial success immediately leads to further questions ranging from computational problems (e.g., whether proposed implementations are equivalent from the viewpoint of computational complexity) to mathematical extensions (e.g., how to incorporate the internal structure of biological cells, for example linear structures such as actin filaments and microtubules, into the models).

7) Discrete differential geometry for developmental biology. The well-established notions of differential geometry clash with the discrete character of cellular or modular structures. This clash often leads to circuitous model construction, in which a spatially discrete problem (e.g., diffusion in a tissue) is first stated in continuous terms (a differential diffusion equation), then re-discretized for the sake of computer simulation. A fundamental solution to this approach is to describe the geometry and processes taking place in growing discrete manifolds directly in discrete terms. Applicable methods have recently been proposed in the scope of discrete differential geometry and the calculus of discrete differential manifolds [21,22], but, due to their novelty, these methods have not yet penetrated developmental biology.

8) Grammar-based models of development. The convenience of specifying diverse processes taking place within and/or modifying a multicellular structure is of utmost importance in modeling practice. Examples include gene expression and metabolic reactions within individual cells, short- and long-distance signaling between cells, and structural changes such as cell division. In the case of L-systems, a declarative specification rooted in the notions of formal grammars proved very successful. Is there a comparably convenient method for specifying operation in cell complexes? In particular, is it possible, and would it be useful, to define grammars operating on cell complexes?

We expect that further problems linking developmental biology with diverse areas of mathematics will be identified by participants and discussed at the workshop. By exploring diverse links, the workshop will significantly contribute to the vision of developmental biology as a science with a strong mathematical foundation. A transformation of biology from a descriptive to a mathematical science has already begun, and the proposed workshop – in which many prominent researchers have already expressed interest - can contribute to this transformation in a substantial and timely manner.

References

Note: for the sake of conciseness, only representative references are given.

[1] d’Arcy Thompson. On Growth and Form, University Press, Cambridge, 1952.

[2] A Turing. The chemical basis of morphogenesis. Philosophical Transactions of the Royal Society of London B, 237:37–72, 1952.

[3] A Green et al. Genetic control of organ shape and tissue polarity. PLoS Biology, 8:e1000537, 2010.

[4] S Kondo, T Miura. Reaction-diffusion model as a framework for understanding biological pattern formation. Science 329:1616-1620, 2010.

[5] T Sachs. Pattern Formation in Plant Tissues. University Press, Cambridge, 1991.

[6] G Mitchison. A model for vein formation in higher plants. Proceedings of the Royal Society B 207:79–109, 1980.

[7] P Sahlin, B Soderberg, H. Jonsson. Regulated transport as a mechanism for pattern generation: Capabilities for phyllotaxis and beyond. Journal of Theoretical Biology 258:60-70, 2009.

[8] M Cieslak, A Runions, P Prusinkiewicz. Auxin-driven patterning with unidirectional fluxes. Journal of Experimental Botany, 66:5083-5102, 2015

[9] A Gierer, H Meinhardt, A Theory of Biological Pattern Formation, Kybernetik 12:30-39, 1972.

[10] S Hotton et al. The possible and the actual in phyllotaxis: bridging the gap between empirical observations and iterative models. Journal of Plant Growth Regulation, 25(4):313–323, 2006.

[11] A Runions et al. Modeling and visualization of leaf venation patterns. ACM Transactions on Graphics 24(3):702–711, 2005.

[12] S Besson and J Dumais. A universal rule for the symmetric division of plant cells. Proc. Natl. Acad. Sci. USA, 108:6294–6299, 2011.

[13] D Burago, Y Burago, S Ivanov. A Course in Metric Geometry. AMS, 2001.

[14] M Renton et al. Models of long-distance transport: how is carrier-dependent auxin transport regulated in the stem? New Phytologist, 194:704–715, 2012.

[15] J Jaeger, D Irons, and N Monk. Regulative feedback in pattern formation: towards a general relativistic theory of positional information. Development, 135:3175–3183, 2008.

[16] Z Hejnowicz and J Romberger. Growth tensor of plant organs. Journal of Theoretical Biology, 110:93–114, 1984

[17] E Sharon, B Roman, H Swinney. Geometrically driven wrinkling observed in free plastic sheets and leaves. Physical Review E 75:046211, 2007.

[18] P Prusinkiewicz and P Barbier de Reuille: Constraints of space in plant development. Journal of Experimental Botany 61:2117-2129, 2010.

[19] A Lindenmayer. Mathematical models of cellular interaction in development. Journal of Theoretical Biology 18:280-315,1968.

[20] B Lane. Cell Complexes: The Structure of Space and the Mathematics of Modularity". Ph.D. thesis, University of Calgary, 2015. http://algorithmicbotany.org/papers/laneb.th2015.html

[21] M Desbrun, E Kanso, and Y Tong. Discrete differential forms for computational modeling. In E. Grinspun, P. Schroeder, and M. Desbrun (Eds.): Discrete Differential Geometry. ACM SIGGRAPH Course Notes, 2006.

[22] K Crane. Digital geometry processing with discrete exterior calculus. Manuscript, 2015. http://www.cs.columbia.edu/~keenan/Projects/DGPDEC/paper.pdf

Mathematical descriptions of the development and form of organisms can be traced to the book “On Growth and Form” by d’Arcy Thomson [1], and the paper “The chemical basis of morphogenesis” by Turing [2]. Over time, these works have led to a substantial body of computational modeling techniques and mathematical analyses. The seminal ideas of Thompson have now been refined into advanced models of growth, exploiting parallels between growth and continuum mechanics [3]. Likewise, the ideas of Turing have led to a large body of theoretical results and computational simulation models of pattern formation in organisms [4].

These well-established models of growth and patterning are significantly different from each other, yet they share a common element: they describe spatio-temporal processes in continuous terms and, consequently, are expressed as partial differential equations. The quest for describing, modeling and understanding diverse developmental processes in a realistic, insightful manner, however, highlights a need for additional mathematical tools and methods. Sample problems of interest are listed below.

1) Reaction-diffusion vs. controlled-transport models of morphogenesis. In reaction-diffusion, patterning results from feedback between local concentration and production or depletion of morphogens. Their transport is passive. In contrast, a fundamental patterning process in the kingdom of plants involves a feedback between fluxes of a morphogen (the plant hormone auxin) and its active transport [5,6]. Simulations show that patterns produced by both classes of models partially overlap [7,8], but a deeper understanding of the relations between these models and the patterns they produce is needed.

2) Geometric models of morphogenesis. In a classic paper [9], Gierer and Meinhardt observed that patterns produced by reaction-diffusion can be described and explained in terms of short-distance activation and long-distance inhibition. These explanations attribute a morphogenetic role to a geometric notion – the evaluation of distances. Furthermore, models of diverse developmental processes including phyllotaxis (patterning of plant organs around their supporting axis) [10], vein pattern formation in leaves [11] and cell division patterns [12] have been formulated directly in geometric terms. These results point to a need for a theory of morphogenesis based directly on metric geometry [13]. Such a theory would represent an intermediate level of abstraction between biochemical and biomechanical models of morphogenesis, and macroscopic patterns and forms, offering an in-depth insight into the geometric essence of different morphogenetic processes.

3) The impact of the rates of information flow. In growing structures, rates of information flow may be commensurate with the rates of growth [14]. This raises the question of how the rates of information flow affect the resulting patterns and forms. This question appears with particular clarity in geometric models of morphogenesis. It appears that, mathematically, it is related to problems underlying the propagation of information (limited by the speed of light) in an expanding universe, considered in general relativity and cosmology [15]: a highly intriguing and unexpected relation between biology, geometry, and physics.

4) Discontinuous aspects of growth. Tissue growth is typically described by a growth tensor field [16], which is mathematically equivalent to the strain tensor field in the theory of elasticity. Implicit in this description is the assumption of the spatial continuity of growth rates. However, an increasing body of experimental results indicates that growth is often highly heterogeneous, and may even be spatially discontinuous. This calls for a reexamination of the assumption of continuity in growth models. What are the weakest assumptions under which growth can proceed without sliding, buckling or tearing?

5) Constraints of space. Growing structures are necessarily embedded in the ambient space, which puts constraints on possible forms. These constraints are manifest, for example, in the wrinkled shape of leaves that grow excessively along the margin [17], the decreasing size of leaflets in recursively compound leaves, and the competition for space between branches of a growing tree [18]. The resulting forms are currently characterized using independent mathematical tools: for example, wrinkled surfaces are described in terms of negative Gaussian curvature, which does not apply to branching structures. Is there a mathematical description that would capture all these manifestations of constraints of space in a unified manner?

6) Discrete models of multicellular structures. Treating multicellular structures as continua is an often-used macroscopic approximation, but does not adequately represent microscopic structures. There, growth and divisions of individual cells, and cell-to-cell communication, play an important role. In 1968, Aristid Lindenmayer proposed an original formalism, later termed L-systems, which – with subsequent extensions – makes it possible to model and reason about development of linear and branching multicellular structures in development [19]. Extensions to two- and three-dimensional tissues (discrete 2- and 3-manifolds) have been sought for a long time, but progress has only recently been made, using the topological notion of cell complexes [20]. This partial success immediately leads to further questions ranging from computational problems (e.g., whether proposed implementations are equivalent from the viewpoint of computational complexity) to mathematical extensions (e.g., how to incorporate the internal structure of biological cells, for example linear structures such as actin filaments and microtubules, into the models).

7) Discrete differential geometry for developmental biology. The well-established notions of differential geometry clash with the discrete character of cellular or modular structures. This clash often leads to circuitous model construction, in which a spatially discrete problem (e.g., diffusion in a tissue) is first stated in continuous terms (a differential diffusion equation), then re-discretized for the sake of computer simulation. A fundamental solution to this approach is to describe the geometry and processes taking place in growing discrete manifolds directly in discrete terms. Applicable methods have recently been proposed in the scope of discrete differential geometry and the calculus of discrete differential manifolds [21,22], but, due to their novelty, these methods have not yet penetrated developmental biology.

8) Grammar-based models of development. The convenience of specifying diverse processes taking place within and/or modifying a multicellular structure is of utmost importance in modeling practice. Examples include gene expression and metabolic reactions within individual cells, short- and long-distance signaling between cells, and structural changes such as cell division. In the case of L-systems, a declarative specification rooted in the notions of formal grammars proved very successful. Is there a comparably convenient method for specifying operation in cell complexes? In particular, is it possible, and would it be useful, to define grammars operating on cell complexes?

We expect that further problems linking developmental biology with diverse areas of mathematics will be identified by participants and discussed at the workshop. By exploring diverse links, the workshop will significantly contribute to the vision of developmental biology as a science with a strong mathematical foundation. A transformation of biology from a descriptive to a mathematical science has already begun, and the proposed workshop – in which many prominent researchers have already expressed interest - can contribute to this transformation in a substantial and timely manner.

References

Note: for the sake of conciseness, only representative references are given.

[1] d’Arcy Thompson. On Growth and Form, University Press, Cambridge, 1952.

[2] A Turing. The chemical basis of morphogenesis. Philosophical Transactions of the Royal Society of London B, 237:37–72, 1952.

[3] A Green et al. Genetic control of organ shape and tissue polarity. PLoS Biology, 8:e1000537, 2010.

[4] S Kondo, T Miura. Reaction-diffusion model as a framework for understanding biological pattern formation. Science 329:1616-1620, 2010.

[5] T Sachs. Pattern Formation in Plant Tissues. University Press, Cambridge, 1991.

[6] G Mitchison. A model for vein formation in higher plants. Proceedings of the Royal Society B 207:79–109, 1980.

[7] P Sahlin, B Soderberg, H. Jonsson. Regulated transport as a mechanism for pattern generation: Capabilities for phyllotaxis and beyond. Journal of Theoretical Biology 258:60-70, 2009.

[8] M Cieslak, A Runions, P Prusinkiewicz. Auxin-driven patterning with unidirectional fluxes. Journal of Experimental Botany, 66:5083-5102, 2015

[9] A Gierer, H Meinhardt, A Theory of Biological Pattern Formation, Kybernetik 12:30-39, 1972.

[10] S Hotton et al. The possible and the actual in phyllotaxis: bridging the gap between empirical observations and iterative models. Journal of Plant Growth Regulation, 25(4):313–323, 2006.

[11] A Runions et al. Modeling and visualization of leaf venation patterns. ACM Transactions on Graphics 24(3):702–711, 2005.

[12] S Besson and J Dumais. A universal rule for the symmetric division of plant cells. Proc. Natl. Acad. Sci. USA, 108:6294–6299, 2011.

[13] D Burago, Y Burago, S Ivanov. A Course in Metric Geometry. AMS, 2001.

[14] M Renton et al. Models of long-distance transport: how is carrier-dependent auxin transport regulated in the stem? New Phytologist, 194:704–715, 2012.

[15] J Jaeger, D Irons, and N Monk. Regulative feedback in pattern formation: towards a general relativistic theory of positional information. Development, 135:3175–3183, 2008.

[16] Z Hejnowicz and J Romberger. Growth tensor of plant organs. Journal of Theoretical Biology, 110:93–114, 1984

[17] E Sharon, B Roman, H Swinney. Geometrically driven wrinkling observed in free plastic sheets and leaves. Physical Review E 75:046211, 2007.

[18] P Prusinkiewicz and P Barbier de Reuille: Constraints of space in plant development. Journal of Experimental Botany 61:2117-2129, 2010.

[19] A Lindenmayer. Mathematical models of cellular interaction in development. Journal of Theoretical Biology 18:280-315,1968.

[20] B Lane. Cell Complexes: The Structure of Space and the Mathematics of Modularity". Ph.D. thesis, University of Calgary, 2015. http://algorithmicbotany.org/papers/laneb.th2015.html

[21] M Desbrun, E Kanso, and Y Tong. Discrete differential forms for computational modeling. In E. Grinspun, P. Schroeder, and M. Desbrun (Eds.): Discrete Differential Geometry. ACM SIGGRAPH Course Notes, 2006.

[22] K Crane. Digital geometry processing with discrete exterior calculus. Manuscript, 2015. http://www.cs.columbia.edu/~keenan/Projects/DGPDEC/paper.pdf