Cost-precision trade-off relation determines the optimal morphogen gradient for accurate biological pattern formation

Spatial boundaries formed during animal development originate from the pre-patterning of tissues by signaling molecules, called morphogens. The accuracy of boundary location is limited by the fluctuations of morphogen concentration that thresholds the expression level of target gene. Producing more morphogen molecules, which gives rise to smaller relative fluctuations, would better serve to shape more precise target boundaries; however, it incurs more thermodynamic cost. In the classical diffusion-depletion model of morphogen profile formation, the morphogen molecules synthesized from a local source display an exponentially decaying concentration profile with a characteristic length λ. Our theory suggests that in order to attain a precise profile with the minimal cost, λ should be roughly half the distance to the target boundary position from the source. Remarkably, we find that the profiles of morphogens that pattern the Drosophila embryo and wing imaginal disk are formed with nearly optimal λ. Our finding underscores the thermodynamic cost as a key physical constraint in the morphogen profile formation in Drosophila development.

Influence of Errors in Known Constants and Boundary Conditions on Solutions of Inverse Heat Conduction Problem

This work examines the effects of the known boundary conditions on the accuracy of the solution in one-dimensional inverse heat conduction problems. The failures in many applications of these problems are attributed to inaccuracy of the specified constants and boundary conditions. Since the boundary conditions and material properties in most thermal problems are imposed with uncertainty, the effects of their inaccuracy should be understood prior to the inverse analyses. The deviation from the exact solution has been examined for each case according to the errors in material properties, boundary location, and known boundary conditions. The results show that the effects of such errors are dramatic. Based on these results, the applicability and limitations of the inverse heat conduction analyses have been evaluated and discussed.

Errors and uncertainties in simulation of steady, viscous flow past a circular cylinder at Re = 20 using a systematic approach

The use of Computational Fluid Dynamics as a tool for design and analysis of aerospace systems is well established. Since the results generated by a CFD solver are numerical approximations, the solution is inherently produced with errors and uncertainties. In this paper, a simple fluid flow problem of laminar, incompressible flow past a circular cylinder at Reynolds number of 20 is allowed to be solved by the well-known finite-volume solver ANSYS Fluent. The effect of variations in mesh resolution, domain boundary location and residual criteria settings is investigated. For all the cases, finite, structured meshes of acceptable quality are used. The influence of variables on the cylinder’s drag results is analyzed and discussed. An interesting pattern in results has been observed. The study on the variation in mesh resolution showed no presence of mesh independent solution. The study on the variation of the domain distance showed that it is necessary to increase the diameter of the circle several thousand times to obtain a domain independent solution.

Relative importance of convective uncertainties in massive stars

ABSTRACT In this work, we investigate the impact of uncertainties due to convective boundary mixing (CBM), commonly called ‘overshoot’, namely the boundary location and the amount of mixing at the convective boundary, on stellar structure and evolution. For this we calculated two grids of stellar evolution models with the MESA code, each with the Ledoux and the Schwarzschild boundary criterion, and vary the amount of CBM. We calculate each grid with the initial masses of 15, 20, and $25\, \rm {M}_\odot$. We present the stellar structure of the models during the hydrogen and helium burning phases. In the latter, we examine the impact on the nucleosynthesis. We find a broadening of the main sequence with more CBM, which is more in agreement with observations. Furthermore, during the core hydrogen burning phase there is a convergence of the convective boundary location due to CBM. The uncertainties of the intermediate convective zone remove this convergence. The behaviour of this convective zone strongly affects the surface evolution of the model, i.e. how fast it evolves redwards. The amount of CBM impacts the size of the convective cores and the nucleosynthesis, e.g. the 12C to 16O ratio and the weak s-process. Lastly, we determine the uncertainty that the range of parameter values investigated introduces and we find differences of up to $70{{\ \rm per\ cent}}$ for the core masses and the total mass of the star.

Energy Release Rate Approximation for Small Surface Cracks in Three-Dimensional Domains Using the Topological Derivative

The topological derivative describes the variation of a response functional with respect to infinitesimal changes in topology, such as the introduction of an infinitesimal crack or hole. In this three-dimensional fracture mechanics work, we propose an approximation of the energy release rate field associated with a small surface crack of any boundary location, direction, and orientation combination using the topological derivative. This work builds on the work of Silva et al. (“Energy Release Rate Approximation for Small Surface-Breaking Cracks Using the Topological Derivative,” J. Mech. Phys. Solids 59(5), pp. 925–939), in which the authors proposed an approximation of the energy release rate field which was limited to two-dimensional domains. The proposed method is computationally advantageous because it only requires a single analysis. By contrast, current boundary element and finite element-based methods require an analysis for each crack length-location-direction-orientation combination. Furthermore, the proposed method is evaluated on the non-cracked domain, obviating the need for refined meshes in the crack tip region.

Effects of Cut Boundary Location on Finite Element Submodels With Contacts

Submodeling is a method used in Finite Element Modeling in order to evaluate features of interest, such as fillets or contact interfaces at a reduced cost. Submodeling can be done by creating a full-structure, coarse-mesh, “global” model and solving it. Once the solutions for this model are solved, a model just of the feature of interest, or “submodel”, can be solved using boundary conditions estimated based on global model results. The location of the submodel boundary has large effects on the accuracy of the solution and has been examined by the authors previously, but not for models with contact interfaces. This work uses the submodeling procedure on a model with two cantilever beams sandwiched together with a bolt close to the free end (dubbed “sandwich beam”). Numerous models are produced with different submodel boundary locations in order to better understand how those locations affect the solutions of a model with contact interfaces. The maximum contact pressure was the main metric used to examine the effects and the feature of interest was the bolt. In previous works, it was determined that the global model mesh size and submodel boundary locations are the main sources of error in submodeling [15]. From this work, it was concluded that the inaccuracies of the global model mesh size are magnified when the submodel mesh is too close to the feature of interest. Furthermore, contact pressure tends to be overestimated as the submodel is refined, but it tends to converge as the global model is converged. This work also demonstrated that errors in global model solution (due to meshing) are mitigated when the entire feature of interest is included in the submodel and the submodel boundary locations are far enough away from that feature.