Geometric Diversity of Living Organisms and Viruses

Abstract

Recently, it has been celebrated a hundred year anniversary of Einstein's theory of general relativity. Since, underlying geometry of space and time became main source of inspiration of new theories. Though, much attention was paid to the geometric construction of the universe, geometric diversity of living organisms got largely omitted. Of course such reluctant approach has fundamental explanation. General relativity is formulated to explain the gravity, while the role of gravitation on the molecular level in the formation of living organisms is negligibly small. However, mathematical approach used in general relativity can be carried out for two dimensional surfaces. Therefore, scientists started to develop analogical differential geometric approach to predict static shapes of living organisms. As a result associated Euler-Lagrange equations, so called “shape equations” have been formulated. But, shape equations turned to be fourth order partial nonlinear differential equations, and finding a general analytical solution is a difficult problem, even though it has been numerically solved for some specific cases. The numerical solutions generated a large body of beautiful explanation of experimental results especially observed in diversity of lipid membranes shapes. Because of shape equations as well as numerical calculations are not exact equations of motion and analytical solutions, we have decided to employ different approach to the problem. Namely, we have used tensor calculus of moving surfaces to deduce exact equations of surface motion in electromagnetic field and then to analytically solve it for some simplified cases. Simplifications have produced such outstanding equations as Maxwell equations for electrodynamics and Navier-Stokes equations for dynamic fluid are (D. V. Svintradze, (2016), arXiv:1609.07765 [cond-mat.soft]). Also, analytic solutions to simplified equations for homogeneous surfaces, in equilibrium with environment, (D. V. Svintradze, (2016), arXiv:1608.01491 [physics.bio-ph]) have explained why constant mean curvature (CMC) surfaces are such abundant shapes among living organisms and viruses.