Final Report Summary - MORPHOELASTICITY (Morphoelasticity: The Mathematics and Mechanics of Biological Growth)
Growth is the hallmark of life, it is a fascinating dynamical process that has challenged our understanding of biological systems for centuries. Whether it is cell division, early pattern formation, growth of plants, or cancer spreading, there is something truly mesmerising about the creation and development of structures in nature. Increase in size and change of forms in living systems fills many purposes and functions and takes place in many different ways and guises. Surprisingly, very little is actually known about the mathematics of these processes and the goal of my research is to develop a mathematical formulation of growth by combining mechanics, physics, and many different fields of mathematics and biology.
The coupling between growth and physical forces is also particularly important to understand the regular function and pathological disorders in many physiological systems. On the one hand, forces can induce growth. On the other hand, differential growth itself can create forces when parts of the body grow faster than others. Understanding how these forces are generated and coupled with the body response is particularly important to the study of their regular function and therefore crucial for the prevention of many diseases such as the cancer and the formation of arterial aneurysms.
The goal of our research is to develop mathematical tools to understand and model a variety of problems related to growth including neurite growth, microbial growth and invasion, the development of aneurysms, the spread of solid tumours, and the regular and abnormal functions of organs such as arteries, the heart, the oesophagus, and the trachea. My research in developing understanding of the intimate coupling between the biology of growth processes, and mechanics, helps us better predict the response of many biological and physiological systems, and guide scientists in their experimental and medical studies. As an example of particula beauty, we have developed recently new mathematical models to understand morphological patterns in seashells (ribs, spikes, ornamentation) by coupling the growth mechanics of the sof-mantle with the evolution of the hard shell by deposition. These models provide a unifying view of many observed morphologies.
I mention here 4 outcomes of our research as examples of my approach:
1. Surface growth and seashell morphology
Surface growth, or accretive growth, refers to the deposition of mass on the surface of a body resulting in beautiful, intricate global structures such as seashells and horns. Based on the progress made last year, we have developed mathematical models to explain seashell morphology. . The first part was to develop a full analytical and numerical theory to describe surface growth processes for shapes that can be modelled from the evolution of curves. That is, locally, a vector field is defined in terms of the local geometry at each point on the curve. The curve then evolves according to these local laws. The theory allows for the understanding basic principle of self-similarity and spiral coiling for many structures. The second part is to couple mechanics with growth by considering the interaction between the soft mantle and the hard shell. In particular, we have shown that morphological patters such as ribs, co-marginal ribs, and spikes can be explained from a mechanistic point of view by considering the soft mantle as an elastic structure that can be subject to buckling instability when interacting with its hard shell. In this unifying approach, the mismatch between soft-material growth and hard shell deposition can explain most of the morphological patterns observed on seashells.
2. Synaptic bistability
According to current understanding, the main biophysical mechanism for storing information in the central nervous system is activity-based changes in the strengths (weights, or efficacies) of synaptic connections between neurons (synaptic plasticity). In many cases, synaptic plasticity is expressed by changes in the number of neurotransmitter protein receptors (AMPARs) within the postsynaptic membrane of a stimulated neuron. However any modification in the number of independent receptors cannot fully account for long-term memory because of the limited dwell time of individual receptors, which are recycled into and out of a synapse several times an hour. One suggested mechanism is that receptor clusters, if formed, can survive much longer than individual receptors if the rate of receptor insertion into the membrane (exocytosis) is higher in the vicinity of other receptors due to receptor-receptor interactions and the rate of receptor removal from the membrane (endocytosis) is independent of such interactions. In a recent paper (with Paul Bressloff, Nigel Emptage and Victor Burlakov), we introduced a bistability mechanism for long-term synaptic plasticity based on switching between two metastable states that contain significantly different numbers of synaptic receptors. One state is characterized by a two-dimensional gas of mobile interacting receptors and is stabilized against clustering by a high nucleation barrier. The other state contains a receptor gas in equilibrium with a large cluster of immobile receptors, which is stabilized from growing further by the turnover rate of receptors into and out of the synapse. Transitions between the two states can be initiated by either an increase (potentiation) or a decrease (depotentiation) of the net receptor flux into the synapse. This changes the saturation level of the receptor gas and triggers nucleation or evaporation of receptor clusters and provides a natural mechanisms for short and long-term memory.
3. Cancer modelling
With my student Mark Roberston-Tessi and Ardith el-Kareh, we developed a mathematical model of the interactions between a growing tumour and the immune system. The equations and parameters of the model are based on experimental and clinical results from published studies. The model includes the primary cell populations involved in effector T-cell mediated tumour killing: regulatory T cells, helper T cells, and dendritic cells. A key feature is the inclusion of multiple mechanisms of immunosuppression through the main cytokines and growth factors mediating the interactions between the cell populations. Decreased access of effector cells to the tumour interior with increasing tumour size is accounted for. The model was applied to tumours with
different growth rates and antigenicities to gauge the relative importance of various immunosuppressive
mechanisms. We found that the most important factors leading to tumour escape are TGF-b-induced immunosuppression,
conversion of helper T cells into regulatory T cells, and the limitation of immune cell access to the full tumour at large tumour sizes. The results suggest that for a given tumour growth rate, there is an optimal antigenicity maximizing the response of the immune system. Further increases in antigenicity result in increased immunosuppression, and therefore a decrease in tumour killing rate. This result may have implications for immunotherapies which modulate the effective antigenicity. Simulation of dendritic cell therapy with the model also suggests that for some tumours, there is an optimal dose of
transfused dendritic cells.
4. The growth of phycomyces.
We have considered the growth and rotation of phycomyces. The filamentary fungus (Phycomyces blakesleeanus) undergoes a series of remarkable transitions during aerial growth. During the Stage IV growth phase, the fungus extends while rotating in a counterclockwise manner when viewed from above and then, while continuing to grow, spontaneously reverses to a clockwise rotation. This phase lasts for 24 - 48 hours and is sometimes followed by yet another reversal before the overall growth ends. We have proposed a continuum mechanical model of this entire process using nonlinear, anisotropic, elasticity and show how helical anisotropy associated with the cell wall structure can induce spontaneous rotation and, under appropriate circumstances, the observed reversal of rotational handedness.
5. Mucosal folding
A cylindrical elastic tube under uniform radial external pressure will buckle circumferentially to a non-circular cross-section at a critical pressure. The buckling represents an instability of the inner or outer edge of the tube. This is a common phenomenon in biological tissues, where it is referred to as mucosal folding. We have investigated this buckling instability in a growing elastic tube. We highlighted the competition between geometric effects, i.e. the change in tube dimensions, and mechanical effects, i.e. the effect of residual stress, due to differential growth. This competition can lead to non-intuitive results, such as a tube growing to be thinner and yet buckle at a higher pressure.
The coupling between growth and physical forces is also particularly important to understand the regular function and pathological disorders in many physiological systems. On the one hand, forces can induce growth. On the other hand, differential growth itself can create forces when parts of the body grow faster than others. Understanding how these forces are generated and coupled with the body response is particularly important to the study of their regular function and therefore crucial for the prevention of many diseases such as the cancer and the formation of arterial aneurysms.
The goal of our research is to develop mathematical tools to understand and model a variety of problems related to growth including neurite growth, microbial growth and invasion, the development of aneurysms, the spread of solid tumours, and the regular and abnormal functions of organs such as arteries, the heart, the oesophagus, and the trachea. My research in developing understanding of the intimate coupling between the biology of growth processes, and mechanics, helps us better predict the response of many biological and physiological systems, and guide scientists in their experimental and medical studies. As an example of particula beauty, we have developed recently new mathematical models to understand morphological patterns in seashells (ribs, spikes, ornamentation) by coupling the growth mechanics of the sof-mantle with the evolution of the hard shell by deposition. These models provide a unifying view of many observed morphologies.
I mention here 4 outcomes of our research as examples of my approach:
1. Surface growth and seashell morphology
Surface growth, or accretive growth, refers to the deposition of mass on the surface of a body resulting in beautiful, intricate global structures such as seashells and horns. Based on the progress made last year, we have developed mathematical models to explain seashell morphology. . The first part was to develop a full analytical and numerical theory to describe surface growth processes for shapes that can be modelled from the evolution of curves. That is, locally, a vector field is defined in terms of the local geometry at each point on the curve. The curve then evolves according to these local laws. The theory allows for the understanding basic principle of self-similarity and spiral coiling for many structures. The second part is to couple mechanics with growth by considering the interaction between the soft mantle and the hard shell. In particular, we have shown that morphological patters such as ribs, co-marginal ribs, and spikes can be explained from a mechanistic point of view by considering the soft mantle as an elastic structure that can be subject to buckling instability when interacting with its hard shell. In this unifying approach, the mismatch between soft-material growth and hard shell deposition can explain most of the morphological patterns observed on seashells.
2. Synaptic bistability
According to current understanding, the main biophysical mechanism for storing information in the central nervous system is activity-based changes in the strengths (weights, or efficacies) of synaptic connections between neurons (synaptic plasticity). In many cases, synaptic plasticity is expressed by changes in the number of neurotransmitter protein receptors (AMPARs) within the postsynaptic membrane of a stimulated neuron. However any modification in the number of independent receptors cannot fully account for long-term memory because of the limited dwell time of individual receptors, which are recycled into and out of a synapse several times an hour. One suggested mechanism is that receptor clusters, if formed, can survive much longer than individual receptors if the rate of receptor insertion into the membrane (exocytosis) is higher in the vicinity of other receptors due to receptor-receptor interactions and the rate of receptor removal from the membrane (endocytosis) is independent of such interactions. In a recent paper (with Paul Bressloff, Nigel Emptage and Victor Burlakov), we introduced a bistability mechanism for long-term synaptic plasticity based on switching between two metastable states that contain significantly different numbers of synaptic receptors. One state is characterized by a two-dimensional gas of mobile interacting receptors and is stabilized against clustering by a high nucleation barrier. The other state contains a receptor gas in equilibrium with a large cluster of immobile receptors, which is stabilized from growing further by the turnover rate of receptors into and out of the synapse. Transitions between the two states can be initiated by either an increase (potentiation) or a decrease (depotentiation) of the net receptor flux into the synapse. This changes the saturation level of the receptor gas and triggers nucleation or evaporation of receptor clusters and provides a natural mechanisms for short and long-term memory.
3. Cancer modelling
With my student Mark Roberston-Tessi and Ardith el-Kareh, we developed a mathematical model of the interactions between a growing tumour and the immune system. The equations and parameters of the model are based on experimental and clinical results from published studies. The model includes the primary cell populations involved in effector T-cell mediated tumour killing: regulatory T cells, helper T cells, and dendritic cells. A key feature is the inclusion of multiple mechanisms of immunosuppression through the main cytokines and growth factors mediating the interactions between the cell populations. Decreased access of effector cells to the tumour interior with increasing tumour size is accounted for. The model was applied to tumours with
different growth rates and antigenicities to gauge the relative importance of various immunosuppressive
mechanisms. We found that the most important factors leading to tumour escape are TGF-b-induced immunosuppression,
conversion of helper T cells into regulatory T cells, and the limitation of immune cell access to the full tumour at large tumour sizes. The results suggest that for a given tumour growth rate, there is an optimal antigenicity maximizing the response of the immune system. Further increases in antigenicity result in increased immunosuppression, and therefore a decrease in tumour killing rate. This result may have implications for immunotherapies which modulate the effective antigenicity. Simulation of dendritic cell therapy with the model also suggests that for some tumours, there is an optimal dose of
transfused dendritic cells.
4. The growth of phycomyces.
We have considered the growth and rotation of phycomyces. The filamentary fungus (Phycomyces blakesleeanus) undergoes a series of remarkable transitions during aerial growth. During the Stage IV growth phase, the fungus extends while rotating in a counterclockwise manner when viewed from above and then, while continuing to grow, spontaneously reverses to a clockwise rotation. This phase lasts for 24 - 48 hours and is sometimes followed by yet another reversal before the overall growth ends. We have proposed a continuum mechanical model of this entire process using nonlinear, anisotropic, elasticity and show how helical anisotropy associated with the cell wall structure can induce spontaneous rotation and, under appropriate circumstances, the observed reversal of rotational handedness.
5. Mucosal folding
A cylindrical elastic tube under uniform radial external pressure will buckle circumferentially to a non-circular cross-section at a critical pressure. The buckling represents an instability of the inner or outer edge of the tube. This is a common phenomenon in biological tissues, where it is referred to as mucosal folding. We have investigated this buckling instability in a growing elastic tube. We highlighted the competition between geometric effects, i.e. the change in tube dimensions, and mechanical effects, i.e. the effect of residual stress, due to differential growth. This competition can lead to non-intuitive results, such as a tube growing to be thinner and yet buckle at a higher pressure.