scholarly journals Exact Solution to the Extended Zwanzig Model for Quasi-Sigmoidal Chemically Induced Denaturation Profiles: Specific Heat and Configurational Entropy

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
G. E. Aguilar-Pineda ◽  
L. Olivares-Quiroz
Keyword(s):  
Exact Solution ◽  
Analytical Solution ◽  
Specific Heat ◽  
Configurational Entropy ◽  
Naturally Occurring ◽  
Thermodynamic Potentials ◽  
Chemically Induced ◽  
Denaturation Profile ◽  
Average Internal Energy ◽  
Proteins And Peptides

Temperature and chemically induced denaturation comprise two of the most characteristic mechanisms to achieve the passage from the native state N to any of the unstructured states Dj in the denatured ensemble in proteins and peptides. In this work we present a full analytical solution for the configurational partition function 𝒵qs of a homopolymer chain poly-X in the extended Zwanzig model (EZM) for a quasisigmoidal denaturation profile. This solution is built up from an EZM exact solution in the case where the fraction α of native contacts follows exact linear dependence on denaturant’s concentration ζ; thus an analytical solution for 𝒵L in the case of an exact linear denaturation profile is also provided. A recently established connection between the number ν of potential nonnative conformations per residue and temperature-independent helical propensity ω complements the model in order to identify specific proteinogenic poly-X chains, where X represents any of the twenty naturally occurring aminoacid residues. From 𝒵qs, equilibrium thermodynamic potentials like entropy 𝒮 and average internal energy 〈E〉 and thermodynamic susceptibilities like specific heat C𝓋 are calculated for poly-valine (poly-V) and poly-alanine (poly-A) chains. The influence of the rate at which native contacts denature as function of ζ on thermodynamic stability is also discussed.

scholarly journals Translation of a Coated Rigid Spherical Inclusion in an Elastic Matrix: Exact Solution, and Implications for Mechanobiology

2019 ◽  
Vol 86 (5) ◽  
Author(s):  
Xin Chen ◽  
Moxiao Li ◽  
Shaobao Liu ◽  
Fusheng Liu ◽  
Guy M. Genin ◽  
...  
Keyword(s):  
Mechanical Properties ◽  
Exact Solution ◽  
Analytical Solution ◽  
Spherical Inclusion ◽  
Matrix Material ◽  
Elastic Matrix ◽  
Protein Ligands ◽  
First Case ◽  
The Matrix ◽  
Matrix Modulus

The displacement of relatively rigid beads within a relatively compliant, elastic matrix can be used to measure the mechanical properties of the matrix. For example, in mechanobiological studies, magnetic or reflective beads can be displaced with a known external force to estimate the matrix modulus. Although such beads are generally rigid compared to the matrix, the material surrounding the beads typically differs from the matrix in one or two ways. The first case, as is common in mechanobiological experimentation, is the situation in which the bead must be coated with materials such as protein ligands that enable adhesion to the matrix. These layers typically differ in stiffness relative to the matrix material. The second case, common for uncoated beads, is the situation in which the beads disrupt the structure of the hydrogel or polymer, leading to a region of enhanced or reduced stiffness in the neighborhood of the bead. To address both cases, we developed the first analytical solution of the problem of translation of a coated, rigid spherical inclusion displaced within an isotropic elastic matrix by a remotely applied force. The solution is applicable to cases of arbitrary coating stiffness and size of the coating. We conclude by discussing applications of the solution to mechanobiology.

On metastable approximations in co-operative assemblies

1956 ◽  
Vol 235 (1201) ◽  
pp. 247-259 ◽  
Keyword(s):  
Exact Solution ◽  
Specific Heat ◽  
Curie Point ◽  
Lattice Structure ◽  
Three Dimensional ◽  
Entropy Change ◽  
Successive Approximations ◽  
Series Expansions ◽  
Temperature Expansion ◽  
Heat Curve

The exact solution of the three-dimensional Ising model of a ferromagnetic presents diffi­culties of a very fundamental nature. It therefore seems that the most reliable information on the behaviour of the model is provided by exact series expansions of the partition func­tion at low and high temperatures. However, the usual low -temperature expansion fails to converge in the neighbourhood of the critical point. By rearranging the terms of the series on the basis of physical considerations, it is possible to obtain a systematic set of successive approximations, each approximation taking exact account of clusters of a given size or less (metastable approximations). By extrapolation accurate estimates can be derived of the Curie point and critical values of the energy and entropy. It is found that there is a marked difference in behaviour between two- and three-dimensional lattices, a far larger proportion of the entropy change taking place in the temperature region below the Curie point in the latter case. The corre­sponding specific heat curves are therefore much closer to those observed experimentally. Finally, a brief discussion is given of the dependence of the specific heat curve on lattice structure.

OBSERVER-BASED SYNCHRONIZATION FOR A CLASS OF FRACTIONAL CHAOTIC SYSTEMS VIA A SCALAR SIGNAL: RESULTS INVOLVING THE EXACT SOLUTION OF THE ERROR DYNAMICS

2011 ◽  
Vol 21 (03) ◽  
pp. 955-962 ◽  
Author(s):  
DONATO CAFAGNA ◽  
GIUSEPPE GRASSI
Keyword(s):  
Exact Solution ◽  
Analytical Solution ◽  
Fractional Order ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Exact Analytical Solution ◽  
Fractional Order Systems ◽  
Fractional Error ◽  
Transmitted Signal ◽  
Error Dynamics

This paper deals with chaos synchronization for a class of fractional-order systems characterized by one nonlinearity. In particular, an observer-based approach is illustrated, which presents two remarkable features: (i) it provides an exact analytical solution of the fractional error dynamics, written in terms of Mittag-Leffler function; (ii) it enables synchronization to be achieved using a scalar transmitted signal. Finally, a synchronization example based on fractional Chua's system is illustrated, with the aim to show the capabilities of the developed approach.

scholarly journals Two-dimensional solutions of Poisson's and Laplace's equations

1919 ◽  
Vol 95 (672) ◽  
pp. 457-475 ◽  
Keyword(s):  
Exact Solution ◽  
Analytical Solution ◽  
Preliminary Investigation ◽  
Fluid Motion ◽  
General Character ◽  
Present Moment ◽  
Two Dimensional ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Analytical Expressions

In the course of a preliminary investigation of fluid motion when viscosity is taken into account, it was found that a solution of Laplace’s equation was obtained in very general form. The solution gives exact analytical expressions where the conditions of the problem admit of exact solution, but the chief claim to notice arises from the extremely general character of the method if graphical and mechanical integration are employed. The problem has been approached from the engineering standpoint, and the absence of any proof of the convergency of the series developed will immediately he evident to the mathematician. The justification by trial in particular cases is all that is attempted at the present moment, but in those cases the result is wholly satisfactory. For the analytical solution of the problems I am indebted to Mr. Arthur Berry who has added a second part to this paper, and has generally acted as helpful critic.

scholarly journals Variational Iteration Method for Analytical Solution of the Lane-Emden Type Equation with Singular Initial and Boundary Conditions

2019 ◽  
pp. 127-142 ◽  
Author(s):  
Muhammad Nadeem ◽  
Hijaz Ahmad
Keyword(s):  
Exact Solution ◽  
Analytical Solution ◽  
Iteration Method ◽  
Initial Value Problems ◽  
Type Equation ◽  
Variational Iteration Method ◽  
Mathematical Physics ◽  
Initial Value ◽  
Emden Equation ◽  
Variational Iteration

In this paper, a well-known equation used in astrophysics and mathematical physics called the Lane-Emden equation is to be solved by a variational iteration method. The main purpose of this approach is to solve the singular initial value problems and also boundary value problem of Lane-Emden type equations. This technique overcomes its singularity at origin rapidly. It gives the approximate and exact solution with easily computable terms. The approach is illustrated with some examples to show its reliability and compactness.

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