Function, differentiable

A final comment on the interpretation of stochastic simulations We are so accustomed to writing continuous functions—differential and integrated rate equations, commonly called deterministic rate equations—that our first impulse on viewing these stochastic calculations is to interpret them as approximations to the familiar continuous functions. However, we have got this the wrong way around. On a molecular level, events are discrete, not continuous. The continuous functions work so well for us only because we do experiments on veiy large numbers of molecules (typically 10 -10 ). If we could experiment with very much smaller numbers of molecules, we would find that it is the continuous functions that are approximations to the stochastic results. Gillespie has developed the stochastic theory of chemical kinetics without dependence on the deterministic rate equations. 

L. H. Erbe et ai. Oscillation Theory for Functional Differential Equations (1995)... 

Semino CE, Merok JR, Crane GG et al (2003) Functional differentiation of hepatocyte-like spheroid structures from putative liver progenitor cells in three- dimensional peptide scaffolds. Differentiation 71 262-270... 

Constant, S., Pfeiffer, C., Woodard, A., Pasqualini, T. and Bottomly, K. (1995) Extent of T cell receptor ligation can determine the functional differentiation of naive CD4+ T cells. Journal of Experimental Medicine 182,1591-1596. 

In this equation, H, the Hamiltonian operator, is defined by H = — (h2/8mir2)V2 + V, where h is Planck s constant (6.6 10 34 Joules), m is the particle s mass, V2 is the sum of the partial second derivative with x,y, and z, and V is the potential energy of the system. As such, the Hamiltonian operator is the sum of the kinetic energy operator and the potential energy operator. (Recall that an operator is a mathematical expression which manipulates the function that follows it in a certain way. For example, the operator d/dx placed before a function differentiates that function with respect to x.) E represents the total energy of the system and is a number, not an operator. It contains all the information within the limits of the Heisenberg uncertainty principle, which states that the exact position and velocity of a microscopic particle cannot be determined simultaneously. Therefore, the information provided by Tint) is in terms of probability I/2 () is the probability of finding the particle between x and x + dx, at time t. 

The conformation and orientation of adsorbed proteins has been examined with monoclonal antibodies that recognize a specific site in a protein of interest. Keselowsky et al. examined the conformation of Fn adsorbed to SAMs that carried methyl, hydroxyl, carboxyl, and amine groups [79]. They used monoclonal antibodies that recognized the central cell-binding domain of Fn near the RGD motif. Different SAM functionalities differentially modulated the binding affinities of the monoclonal antibodies (OH > COOH = NH2 > CH3). The strength of cell adhesion to these... 

By functional differentiation, Equation 4.22 leads us to the Euler-Lagrange deterministic equation for the electron density, viz.,... 

Curtis, W. K., R. O. Fox, and K. Halasi (1992). Numerical stability analysis of a class of functional differential equations. SIAM Journal of Applied Mathematics 52,... 

Seifert, R., Hoer, A., Schwaner, I., Buschauer, A. (1992). Histamine increases cytosolic Ca2+ in HL-60 promyelocytes predominantly via H2 receptors with an unique agonist/ antagonist profile and induces functional differentiation. Am. Soc. Pharmacol. Exp. Therapeut. 42,235-41. 

Morphologically, this is demonstrated by the development of a multinuclear syncytium with apical microvilli [49], while functional differentiation is associated with induction and synthesis of hormones, such as human chorionic gonadotropin (hCG) and human placental lactogen (hPL) [50, 51]. Cultures of primary cytotrophoblasts have been used to study functional expression of P-gp, amino acid uptake, and hormonal stimulation of amino acid uptake and... 

Treating as an independent variable, the chain rule for functional differentiation gives... 

The operators W, A, occurring above, should be taken in the second-quantization form, free of explicit dependence on particle number, and Tr means the trace in Fock space (see e.g. [10] for details). Problems of existence and functional differentiability of generalized functionals F [n] and r [n] are discussed in [28] the functional F [n] is denoted there as Fi,[n] or Ffrac[n] or FfraoM (depending on the scope of 3), similarly for F [n]. Note that DMs can be viewed as the coordinate representation of the density operators. 

SAXS measurements with CBH II indicated a very similar tertiary structure for both CBH I and CBH II, in spite of a different domain arrangement (to be published). Discrete differences in the tail parts could, however, be noticed. The maximum diameter of CBH II (21.5 nm) was higher than in CBH I this might be due to duplication of the glycosylated part in the former case. Thus, the functional differentiation of these cellulases can be reflected by structural differences. 

During the development of an organism, many cells enter a state of terminal differentiation, in which they perform a specialized function. Differentiated cells originate from dividing stem cells and in the process, partially or completely lose the abUity to divide. They are then no longer able to receive and act upon mitogenic signals. 

Nature is economical in her means. She uses many of the same chemicals to accomplish her nervous purposes within the brain that she has already used to the same ends throughout the body. The good news is that once you have worked out the biochemistry and pharmacology of a neuromodulator in the body, you can apply a lot of what you know to its action in the brain. The bad news is that every time you target, for example, the acetylcholine system of the brain, you also hit the body. That means that the heart, the bowel, the salivary glands, and all the rest of the organs innervated by the autonomic nervous system are influenced. What is worse, the target sites within the brain may not only be as spatially dispersed as in the periphery, but may also be as functionally differentiated ... 

The student of thermodynamics must learn to cope with the functional, differential, and derivative relationships in (1.2)—(1.7) from a variety of formulaic, graphical, and experimental aspects. Let us briefly discuss each in turn. 

The generating term J fj(r)p(r) is diagrammatically represented by a wiggly arrow (cf. Fig. 5.18) inserted into polymer lines. We then without any change can take ewer the discussion given above for Z[p7 to find that In Z[p.p, a] is given by the set of all connected diagrams with any number of (a - p) -insertions. Functional differentiation reduces a (u p) - insertion to a density insertion. The result, for I tC then is found from Eq. (A 5.8). 

Tlie density correlations can be derived by functional differentiation, as indicated in Sect. A 5.1. 

Hence the flow of each chapter of this book will lead from a description of specific chemical/biological processes and systems to the identification of the main state variables and processes occurring within the boundaries of the system, as well as the interaction between the system and its surrounding environment. The necessary system processes and interactions are then expressed mathematically in terms of state variables and parameters in the form of equations. These equations may most simply be algebraic or transcendental, or they may involve functional, differential, or matrix equations in finitely many variables. 

Yoshida K, Iwasaka R, Kaneko T, Sato S, Tabata S, Sakuta M. 2008. Functional differentiation of Lotus japonicus TT2s, R2R3-MYB transcription factors comprising a mutigene family. Plant Cell Physiol 49 157-169. 

Ichikawa T., Shiota T., Shimizu I. and Kataoka H. (1996b) Functional differentiation of neurosecretory cells with immunoreactive diapause hormone and pheromone biosynthesis activating neuropeptide of the moth, Bombyx mori. Zool. Sci. 13, 21-25. 

Sinton CM, Fallon SL. Electrophysiological evidence for a functional differentiation between subtypes of the 5-HTj receptor. Eur J Pharmacol 1988 157 173-181. 

---