Diatom Morphogenesis. Группа авторов

is masked then flipped along a bisecting horizontal axis, and then flipped along a bisecting vertical axis, illustrating the procedure for measurement of dihedral symmetry. The method of rotation and flipping along a bisecting axis enables entropy measurement of any organism with reflective, glide, or dihedral symmetry including pennate diatoms which develop via a sternum or sternum-raphe system [2.119]. Biddulphia sp. [2.145] was used for illustration (Figure 2.5).

Measurement error of entropy values for masked image matching after rotation and overlaying was calculated to determine accuracy and validity as From this, variation bias due to measurement error expressed as a percentage is based on the root mean square error, RMSE to image matching variance ratio.

      For analysis of all images and valve formation simulation, entropy values were characterized by their statistical distributions. A histogram of final entropies was plotted, and a probability density function (PDF) and a cumulative density function (CDF) were determined. Maximum entropy is determined from the PDF [2.65]. For a uniform PDF, maximum entropy indicates that every symmetry state has an equal chance of occurring. For a Gaussian PDF, the least amount of information available means having the lowest probability and is maximum entropy; the highest amount of information available means the lowest entropy and the highest probability of occurring. Maximum entropy is also determined via standardization of the PDF so that a prior probability distribution results [2.65, 2.67].

      2.2.7 Entropy, Symmetry, and Stability

      During morphogenesis, changes in symmetry may occur. These changes may be thought of as morphogenetic states with respect to the spatial and temporal displacement of cell components and morphological shape and surface structures. These changes occur over time and comprise a dynamical system. Symmetry is important in pattern formation in linear and non-linear dynamical systems [2.130]. The behavior of symmetry changes over time may be characterized in centric diatoms by step-wise valve formation patterns as a growth of distances [2.27] from central area to valve margin and may be construed to be characterizations of a dynamical system. In this system, stability at equilibrium with respect to symmetry state behavior may be assessed in a diatom morphogenetic dynamical system.

      To model such a system, valve formation simulation as accretionary growth behavior is used. Centric diatom valve formation on the valve surface commences at an annulus and proceeds roughly as a sequential addition of surface ornamentation to the valve margin. This is consistent with the diffusion limited aggregation model in which most deposition occurs in a narrow zone behind the leading silica growth (Figure 9 in [2.47]). Although the initial state of the system may change slightly, valve formation proceeds to produce the same species-specific valve surface. More generally, changes in this valve formation system exhibit ergodic behavior, and the stability of such behavior in a dynamical system is measurable as Lyapunov exponents.

      In a dynamical system, different states occupy different spaces that exist at different times. Changes from state to state can be delineated using a system of differential equations derived from the original relation among the variables of interest. A Lyapunov function of the general form x i = fi (x) has first derivatives that form the elements of the Jacobian matrix (i.e., Jacobian) where x(t) is x₀ at equilibrium [2.78]. The Jacobian represents the change in tangent vectors over time t as a linearization of the tangent space system [2.39].

The relation between entropy and symmetry is of the general form of xi = f(xi), and from the Boltzmann entropy equation, xi = Si and f(xi) = -ln wi A Lyapunov function, Si = - ln w for each ith state of n-symmetry states at S = 0, is a system of homogeneous differential equations representing the change in symmetry states over time as a dynamical morphogenetic system. In matrix form, S = AS is a time evolution equation where A is the matrix of constant coefficients from consisting of first partial derivatives comprising A, the Jacobian.

      To determine the behavior characteristics of symmetry states, solution to the Lyapunov function of the form ATS + SA = -Q is evaluated, where Q is the resultant matrix. If the left side of the equation is determined to be positive definite when the right side of the equation is negative definite, then the symmetry states system is stable; otherwise, the system is unstable [2.23].

      Specific characteristics of instability are evaluated via A. At equilibrium, (A - 9I) = 0 and for det(A - 9I) = 0, 9 are eigenvalues of A, and tr . For all symmetry states, ln [2.23]. Transformation of produces of A, and diagonalization and a change of basis leads to a decoupled system with the general solution of the time evolution matrix [2.22].

      Lyapunov exponents [2.88, 2.101] are meaningfully characterized by the real parts of the complex valued eigenvalues of A [2.29] and are where log is any base. Lyapunov exponents measure fluctuation as average rates of divergence (expansion and instability) or convergence (contraction and stability) and represent expected values [2.29]. Initial conditions influence the outcome, and changes in initial conditions are measurable as indicators of the behavior of fluctuations. Boltzmann entropy production is necessarily sensitive to exponentially changing initial conditions in tangent space [2.39]. However, slight perturbations in initial conditions will yield measurement of the degree and characterization of symmetry behavior at any point in the dynamical system.

      Lyapunov exponents comprise a multidimensional spectrum indicative of the rate of (exponential) divergence of multiple symmetry states with respect to direction of changes from one symmetry state to the next. Lyapunov exponents give bounds on information production of a dynamical system. They measure local behavior of the collective properties of a dynamical system yielding a global characterization of the system [2.8, 2.133]. A Lyapunov function close to equilibrium ensures that entropy tends to move toward a stable maximum in a dynamical system [2.29].

Lyapunov exponents are invariant to coordinate transformations and are ordered from largest to smallest [2.29]. They are a measure of the behavior of vectors in a tangent space. As such, these exponents are a measure of stability or instability at each symmetry state. For Lyapunov exponents that are positive, the dynamical system is chaotically unstable [2.29, 2.118], while negative Lyapunov exponents are indicators of a stable dynamical system [2.29]. A Lyapunov exponent with a value of zero may indicate a dissipative, regular [2.35], weakly chaotic [2.73] or intermittent system [2.74], depending on the particulars defining the initial system.

      By quantifying the relation between symmetry states and stability via Lyapunov exponents, instability is characterizable as a specific behavior, either deterministically chaotic or not. However, the picture may be more complicated. Slight changes in initial conditions can precipitate chaotic behavior resulting in instability [2.29] even though the initial conditions may be random. Chaotic systems may appear to be random when considering fewer than all possible states in the dynamical system [2.97]. High-dimensional systems may appear to be ordered despite non-convergent random behavior [2.35]. Different kinds of randomness as “noise” may appear to be chaotic in a given dynamical system [2.118]. Care must be taken to discern the behavior of a dynamical system in terms of randomness or chaoticity.

      2.2.8 Randomness and Instability

      Lyapunov exponents enable the assessment of randomness