Robustness and timing of cellular differentiation through population-based symmetry breaking

During mammalian development, cell types expressing mutually exclusive genetic markers are differentiated from a multilineage primed state. These observations have invoked single-cell multistability view as the dynamical basis of differentiation. However, the robust regulative nature of mammalian development is not captured therein. Considering the well-established role of cell-cell communication in this process, we propose a fundamentally different dynamical treatment in which cellular identities emerge and are maintained on population level, as a novel unique solution of the coupled system. Subcritical system’s organization here enables symmetry-breaking to be triggered by cell number increase in a timed, self-organized manner. Robust cell type proportions are thereby an inherent feature of the resulting inhomogeneous solution. This framework is generic, as exemplified for early embryogenesis and neurogenesis cases. Distinct from mechanisms that rely on pre-existing asymmetries, we thus demonstrate that robustness and accuracy necessarily emerge from the cooperative behaviour of growing cell populations during development.

A New Radiative Transfer Method for Solar Radiation in a Vertically Internally Inhomogeneous Medium

The problem of solar spectral radiation is considered in a layer-based model, with scattering and absorption parallel to the plane for each medium (cloud, ocean, or aerosol layer) and optical properties assumed to be vertically inhomogeneous. A new radiative transfer (RT) method is proposed to deal with the variation of vertically inhomogeneous optical properties in the layers of a model for solar spectral radiation. This method uses the standard perturbation method to include the vertically inhomogeneous RT effects of cloud and snow. The accuracy of the new inhomogeneous RT solution is investigated systematically for both an idealized medium and realistic media of cloud and snow. For the idealized medium, the relative errors in reflection and absorption calculated by applying the homogeneous solution increase with optical depth and can exceed 20%. However, the relative errors when applying the inhomogeneous RT solution are limited to 4% in most cases. Observations show that stratocumulus clouds are vertically inhomogeneous. In the spectral band of 0.25–0.69 μm, the relative error in absorption with the inhomogeneous solution is 1.4% at most, but that with the homogeneous solution can be up to 7.4%. The effective radius of snow varies vertically. In the spectral band of 0.25–0.69 μm, the relative error in absorption with the homogeneous solution can be as much as 72% but is reduced to less than 40% by using the inhomogeneous solution. At the spectral wavelength of 0.94 μm, the results for reflection and absorption with the inhomogeneous solution are also more accurate than those with the homogeneous solution.

A Feynman integral via higher normal functions

We study the Feynman integral for the three-banana graph defined as the scalar two-point self-energy at three-loop order. The Feynman integral is evaluated for all identical internal masses in two space-time dimensions. Two calculations are given for the Feynman integral: one based on an interpretation of the integral as an inhomogeneous solution of a classical Picard–Fuchs differential equation, and the other using arithmetic algebraic geometry, motivic cohomology, and Eisenstein series. Both methods use the rather special fact that the Feynman integral is a family of regulator periods associated to a family of$K3$surfaces. We show that the integral is given by a sum of elliptic trilogarithms evaluated at sixth roots of unity. This elliptic trilogarithm value is related to the regulator of a class in the motivic cohomology of the$K3$family. We prove a conjecture by David Broadhurst which states that at a special kinematical point the Feynman integral is given by a critical value of the Hasse–Weil$L$-function of the$K3$surface. This result is shown to be a particular case of Deligne’s conjectures relating values of$L$-functions inside the critical strip to periods.

BACK-REACTION IN RELATIVISTIC COSMOLOGY

We introduce the concept of back-reaction in relativistic cosmological modeling. Roughly speaking, this can be thought of as the difference between the large-scale behavior of an inhomogeneous cosmological solution of Einstein's equations, and a homogeneous and isotropic solution that is a best-fit to either the average of observables or dynamics in the inhomogeneous solution. This is sometimes paraphrased as "the effect that structure has of the large-scale evolution of the universe." Various different approaches have been taken in the literature in order to try and understand back-reaction in cosmology. We provide a brief and critical summary of some of them, highlighting recent progress that has been made in each case.

Spatial Structurization of Local Concentration of Reactants in Spatially Inhomogeneous Solutions

A two-component spatially inhomogeneous solution of reactants placed in a homogeneous continual solvent is considered. To describe it, we use the kinetic equations earlier derived by the authors for negligibly weak chemical reaction. The solution of the equations has made it possible to study the phenomenon of spatial structurization due to the encounters of reactants of different kinds. Numerical evaluation of the characteristic times of the observed effect is given.

A Mathematical Model for the Onset of Avascular Tumor Growth in Response to the Loss of P53 Function

We present a mathematical model for the formation of an avascular tumor based on the loss by gene mutation of the tumor suppressor function of p53. The wild type p53 protein regulates apoptosis, cell expression of growth factor and matrix metalloproteinase, which are regulatory functions that many mutant p53 proteins do not possess. The focus is on a description of cell movement as the transport of cell population density rather than as the movement of individual cells. In contrast to earlier works on solid tumor growth, a model is proposed for the initiation of tumor growth. The central idea, taken from the mathematical theory of dynamical systems, is to view the loss of p53 function in a few cells as a small instability in a rest state for an appropriate system of differential equations describing cell movement. This instability is shown (numerically) to lead to a second, spatially inhomogeneous, solution that can be thought of as a solid tumor whose growth is nutrient diffusion limited. In this formulation, one is led to a system of nine partial differential equations. We show computationally that there can be tumor states that coexist with benign states and that are highly unstable in the sense that a slight increase in tumor size results in the tumor occupying the sample region while a slight decrease in tumor size results in its ultimate disappearance.

Stokes-multiplier expansion in an inhomogeneous differential equation with a small parameter

Accurate approximations to the solutions of a second-order inhomogeneous equation with a small parameter ϵ are derived using exponential asymptotics. The subdominant homogeneous solutions that are switched on by an inhomogeneous solution through a Stokes phenomenon are computed. The computation relies on a resurgence relation, and it provides the ϵ -dependent Stokes multiplier in the form of a power series. The ϵ -dependence of the Stokes multiplier is related to constants of integration that can be chosen arbitrarily in the WKB-type construction of the homogeneous solution. The equation under study governs the evolution of special solutions of the Boussinesq equations for rapidly rotating, strongly stratified fluids. In this context, the switching on of subdominant homogeneous solutions is interpreted as the generation of exponentially small inertia–gravity waves.

COSMOLOGICAL SINGULARITIES, EINSTEIN BILLIARDS AND LORENTZIAN KAC–MOODY ALGEBRAS

The structure of the general, inhomogeneous solution of (bosonic) Einstein-matter systems in the vicinity of a cosmological singularity is considered. We review the proof (based on ideas of Belinskii–Khalatnikov–Lifshitz and technically simplified by the use of the Arnowitt–Deser–Misner Hamiltonian formalism) that the asymptotic behavior, as one approaches the singularity, of the general solution is describable, at each (generic) spatial point, as a billiard motion in an auxiliary Lorentzian space. For certain Einstein-matter systems, notably for pure Einstein gravity in any spacetime dimension D and for the particular Einstein-matter systems arising in String Theory, the billiard tables describing asymptotic cosmological behavior are found to be identical to the Weyl chambers of some Lorentzian Kac–Moody algebras. In the case of the bosonic sector of supergravity in 11 dimensional spacetime the underlying Lorentzian algebra is that of the hyperbolic Kac–Moody group E10, and there exists some evidence of a correspondence between the general solution of the Einstein-three-form system and a null geodesic in the infinite dimensional coset space E10/K(E10), where K(E10) is the maximal compact subgroup of E10.

INHOMOGENEOUS de SITTER SOLUTION WITH SCALAR FIELD AND PERTURBATIONS SPECTRUM

We provide an inhomogeneous solution concerning the dynamics of a real self-interacting scalar field minimally coupled to gravity in a region of the configuration space where it performs a slow rolling on a plateau of its potential. During the inhomogeneous de Sitter phase the scalar field dominant term is a function of the spatial coordinates only. This solution is near to the FLRW model allows a classical origin for the inhomogeneous perturbations spectrum.