Theoretical investigation into the origins of multicellularity : a thesis presented in partial fulfilment of the requirements for the degree of PhD in Theoretical Biology at Massey University, Albany, New Zealand

Evolution of multicellularity is a major event in the history of life. The first step is the emergence of collectives of cooperating cells. Cooperation is generally costly to cooperators, thus, non-cooperators have a selective advantage. I investigated the evolution of cooperation in a population in which cells may migrate between collectives. Four different modes of migration were considered and for each mode I identified the set of multiplayer games in which cooperation has a higher fixation probability than defection. I showed that weak altruism may evolve without coordination among cells. However, the evolution of strong altruism requires the coordination of actions among cells. The second step in the emergence of multicellularity is the transition in Darwinian individuality. A likely hallmark of the transition is fitness decoupling. In the second part of my thesis, I present a method for characterizing fitness (de-)coupling which involves an analysis of the correlation between cell and collective fitnesses. In a population with coupled fitnesses, this correlation is close to one. As a population evolves towards multicellularity, collective fitness starts to rely more on the interactions between cells rather than the individual performance of cells, so the correlation between particle and collective fitnesses decreases. This metric makes it possible to detect fitness decoupling. I used the suggested metric to investigate under which conditions fitness decoupling occurs. I constructed a model of a population defined by a linear traits-to-fitness function and used this to identify those functions that promote fitness decoupling. In this model, the fitness correlation is equal to the cosine of the angle between the gradients of fitnesses. Therefore, my results allow an estimation of the fitness (de-)coupling state before selection takes place. In the third section of my thesis, the accuracy of this estimation was tested on available experimental data and using a model simulating an experimental selection regime, which featured non-linear traits-to-fitness functions. The results obtained from the estimation of fitness correlations showed a close approximation to the fitness correlation calculated from experimental data and from simulations in a range of selection regimes

The quasi-adiabatic approximation for coupled thermoelasticity

The equations of coupled thermoelasticity are considered in the case of stationary vibrations. The dimensional and order-of-magnitude analysis of the parameters occurring within these equations prompts the introduction of the new non-dimensionalisation scheme, highlighting the nearly-adiabatic nature of the resulting motions. The departure from the purely adiabatic regime is characterised by a natural small parameter, proportional to the ratio of the mean free path of the thermal phonons to the vibration wavelength. When the governing equations are expanded in terms of the small parameter, one can formulate an equivalent “quasi-adiabatic” system of the equations of ordinary elasticity with frequency-dependent modulae, characterising the thermoelastic issipation. Unfortunately, this model lacks the degrees of freedom necessary to satisfy boundary condition(s) for the temperature. Thus, we also derive a complementary boundary layer solution and show that to the leading order it is described by thermoelastic equations in the quasi-static approximation. Further simplifications are possible for purely dilatational motions; we illustrate this point by solving a model thermoelastic problem in 1D

Anti-symmetric motion of a pre-stressed incompressible elastic layer near shear resonance

A two-dimensional model is derived for anti-symmetric motion in the vicinity of the shear resonance frequencies in a pre-stressed incompressible elastic plate. The method of asymptotic integration is used and a second-order solution, for infinitesimal displacement components and incremental pressure, is obtained in terms of the long-wave amplitude. The leading-order hyperbolic governing equation for the long-wave amplitude is observed to be not wave-like for certain pre-stressed states, with time and one of the in-plane spatial variables swapping roles. This phenomenon is shown to be intimately related to the possible existence of negative group velocity at low wave number, i.e. in the vicinity of shear resonance frequencies

A two-dimensional model for extensional motion of a pre-stressed incompressible elastic layer near cut-of frequencies

A two-dimensional model for extensional motion of a pre-stressed incompressible elastic layer near its cut-off frequencies is derived. Leading-order solutions for displacement and pressure are obtained in terms of the long wave amplitude by direct asymptotic integration. A governing equation, together with corrections for displacement and pressure, is derived from the second-order problem. A novel feature of this (two-dimensional) hyperbolic governing equation is that, for certain pre-stressed states, time and one of the two (in-plane) spatial variables can change roles. Although whenever this phenomenon occurs the equation still remains hyperbolic, it is clearly not wave-like. The second-order solution is completed by deriving a refined governing equation from the third-order problem. Asymptotic consistency, in the sense that the dispersion relation associated with the two-dimensional model concurs with the appropriate order expansion of the three-dimensional relation at each order, is verified. The model has particular application to stationary thickness vibration of, or transient response to high frequency shock loading in, thin walled bodies

A bending quasi-front generated by an instantaneous impulse loading at the edge of a semi-infinite pre-stressed incompressible elastic plate

A refined membrane-like theory is used to describe bending of a semi-infinite pre-stressed incompressible elastic plate subjected to an instantaneous impulse loading at the edge. A far-field solution for the quasi-front is obtained by using the method of matched asymptotic expansions. A leading-order hyperbolic membrane equation is used for an outer problem, whereas a refined (singularly perturbed) membrane equation of an inner problem describes a boundary layer, which smoothes a discontinuity predicted by the outer problem at the wave front. The inner problem is then reduced to one-dimensional by an appropriate choice of inner coordinates, motivated by the wave front geometry. Using the inherent symmetry of the outer problem, a solution for the quasi-front is derived that is valid in a vicinity of the tip of the wave front. Pre-stress is shown to affect geometry and type of the generated quasi-front; in addition to a classical receding quasi-front the pre-stressed plate can support propagation of an advancing quasi-front. Possible responses may even feature both types of quasi-front at the same time, which is illustrated by numerical examples. The case of a so-called narrow quasi-front, associated with a possible degeneration of contribution of singular perturbation terms to the governing equation, is studied qualitatively

An asymptotic membrane-like theory for long-wave motion in a pre-stressed elastic plate

An asymptotically consistent two-dimensional theory is developed to help elucidate dynamic response in finitely deformed layers. The layers are composed of incompressible elastic material, with the theory appropriate for long-wave motion associated with the fundamental mode and derived in respect of the most general appropriate strain energy function. Leading-order and refined higher-order equations for the mid-surface deflection are derived. In the case of zero normal initial static stress and in-plane tension, the leading-order equation reduces to the classical membrane equation, with its refined counterpart also being obtained. The theory is applied to a one-dimensional edge loading problem for a semi-infinite plate. In doing so, the leading- and higher-order governing equations are used as inner and outer asymptotic expansions, the latter valid within the vicinity of the associated quasi-front. A solution is derived by using the method of matched asymptotic expansions

Eigenvalue of a semi-infinite elastic strip

A semi-infinite elastic strip, subjected to traction free boundary conditions, is studied in the context of in-plane stationary vibrations. By using normal (Rayleigh–Lamb) mode expansion the problem of existence of the strip eigenmode is reformulated in terms of the linear dependence within infinite system of normal modes. The concept of Gram's determinant is used to introduce a generalized criterion of linear dependence, which is valid for infinite systems of modes and complex frequencies. Using this criterion, it is demonstrated numerically that in addition to the edge resonance for the Poisson ratio ν=0, there exists another value of ν≈0.22475 associated with an undamped resonance. This resonance is best explained physically by the orthogonality between the edge mode and the first Lamé mode. A semi-analytical proof for the existence of the edge resonance is then presented for both described cases with the help of the augmented scattering matrix formalism

Optimum structure to carry a uniform load between pinned supports: Exact analytical solution

This article is available open access through the publisher’s website at the link below. Copyright @ 2010 The Royal Society.Recent numerical evidence indicates that a parabolic funicular is not necessarily the optimal structural form to carry a uniform load between pinned supports. When the constituent material is capable of resisting equal limiting tensile and compressive stresses, a more efficient structure can be identified, comprising a central parabolic section and networks of truss bars emerging from the supports. In the current article, a precise geometry for this latter structure is identified, avoiding the inconsistencies that render the parabolic form non-optimal. Explicit analytical expressions for the geometry, stress and virtual-displacement fields within and above the structure are presented. Furthermore, a suitable displacement field below the structure is computed numerically and shown to satisfy the Michell–Hemp optimality criteria, hence formally establishing the global optimality of this new structural form

Interacting cells driving the evolution of multicellular life cycles

Author summary Multicellular organisms are ubiquitous. But how did the first multicellular organisms arise? It is typically argued that this occurred due to benefits coming from interactions between cells. One example of such interactions is the division of labour. For instance, colonial cyanobacteria delegate photosynthesis and nitrogen fixation to different cells within the colony. In this way, the colony gains a growth advantage over unicellular cyanobacteria. However, not all cell interactions favour multicellular life. Cheater cells residing in a colony without any contribution will outgrow other cells. Then, the growing burden of cheaters may eventually destroy the colony. Here, we ask what kinds of interactions promote the evolution of multicellularity? We investigated all interactions captured by pairwise games and for each of them, we look for the evolutionarily optimal life cycle: How big should the colony grow and how should it split into offspring cells or colonies? We found that multicellularity can evolve with interactions far beyond cooperation or division of labour scenarios. More surprisingly, most of the life cycles found fall into either of two categories: A parent colony splits into two multicellular parts, or it splits into multiple independent cells