Entropic elasticity of polymers and their networks

The elastic energy for many biopolymer systems is comparable to the thermal energy at room temperature. Therefore, biopolymers and their networks are constantly under thermal fluctuations. From the point of view of thermodynamics, this suggests that entropy plays a crucial role in determining the mechanical behaviors of these filamentous biopolymers. One of the main goals of this thesis is to understand how thermal fluctuations affect the mechanical properties and behaviors of filamentous networks, and also how stress affects the thermal fluctuations. Filaments and filamentous networks are viewed as mechanical structures, whose static equilibrium states under the action of loads or kinematic constraints are determined in the first step of the investigation. Typically, a system is discretized and represented by a finite set of kinematic variables that characterizes the configuration space. In the next step, we apply statistical mechanics to study the thermo-mechanical properties of the system. We approximate the local minimum energy well to quadratic order. Such a quadratic approximation for a discrete system gives rise to a stiffness matrix that characterizes the flexibility of the system around the ground state. Using the multidimensional Gaussian integral technique, the partition function is efficiently evaluated, provided that the energy well around the ground state is steep. In this case, the dominant contribution to the partition function is from the states that are close to the equilibrium state, whose energies are well approximated by the quadratic energy expression. All thermodynamic properties of the system can be further evaluated from the partition function. Fluctuation of the system, in particular, scales linearly with the temperature and inversely with the stiffness matrix. Therefore, the stiffness matrix governs the statistical mechanical behavior of the system near its ground state. We also show that a system with constraints on its kinematic variables can be converted into an effective non-constrained system. Using the above theoretical framework, we study the thermo-mechanical properties of filaments and filamentous networks under different loadings and confinement conditions. The filaments need not be homogeneous in the mechanical properties, and they can be subjected to non-uniform distributed loads or non-uniform confinements. Under compression, a filament can buckle. Buckling in a filament network can reduce the stiffness of the structure, which leads to significant thermal fluctuations around the buckling point. Properties of a triangular network under pure expansion, simple shear and uniaxial tension are also investigated in this thesis. As further applications, we discuss the protein forced unfolding problem. We show that different unfolding behaviors of a protein chain can be understood using a system of three equations. We also discuss the internal fluctuations of DNA under confinement and show a length-dependent transition between the de Gennes and Odijk regimes. We also show that entropy plays a role in driving the motion of a piece of DNA along a nonuniform channel. We derive the entropic force on the DNA in this thesis and discuss the coupled migration and deformation of the polymer under non-uniform confinement. Degree Type Dissertation Degree Name Doctor of Philosophy (PhD) This dissertation is available at ScholarlyCommons: http://repository.upenn. edu/edissertations/371 Graduate Group Mechanical Engineering & Applied Mechanics First Advisor Dr. Prashant K. Purohit Subject Categories Applied Mechanics | Atomic, Molecular and Optical Physics | Biological and Chemical Physics | Biomechanical Engineering | Mechanical Engineering | Physics This dissertation is available at ScholarlyCommons: http://repository.upenn. edu/edissertations/371 ENTROPIC ELASTICITY OF POLYMERS AND THEIR NETWORKS