Lecture4 - UC Davis, BME BIM 162 Quantitative Concepts in...

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Unformatted text preview: UC Davis, BME BIM 162 Quantitative Concepts in Biomolecular Engineering Lecture 04 Interactions between biomolecules and their aggregates wikipedia.com Follow-up Things worth remembering (among others) Simple applications of Boltzmann distribution Relationship between probability density and distribution (i.e., histogram) of a measured random quantity Born energy and partitioning of ions between different media Calculation of the net interaction energy by addition of energies of pairwise interactions Follow-up Spring constant k kBT var x “Blur model”: knaïve model cP 1 1 exp 2 2 cPte Example Bistable potential 1 Example Bistable potential Electrostatic interactions between biomolecules – Recall: Coulomb energy: E r Ion-dipole interaction: 1 q1q2 40 r e 1.602 1019 C … elementary charge 0 8.854 1012 C2 J -1m -1 … permittivity of free space relative permittivity, … relative permittivity, dielectric constant z1,2 … ionic valencies E r 1 q2 cos 40 r 2 q1 … dipole moment Interactions between biomolecules: Electrostatics E r Ion-dipole interaction: 1 q1q2 q1q2 1 q2 1 1 40 r r 40 r r q1 … dipole moment 30 Example: H 2O 1.85 3.336 10 Cm 1D ("Debye") E r 1 q2 cos 40 r 2 How much more likely is the case =0 compared to =? 1 q2 4 r 2 cos 0 0 exp kBT 1 q2 4 r 2 cos 0 exp kBT Interactions between biomolecules Ion rotating-dipole interaction interaction For fixed dipole (previous slide): E r 1 q2 cos 40 r 2 How is this energy affected by rotations of the dipole about ? What is the likelihood of finding the dipole at a given angle ? Given the probability p(x) of some event x, how do we calculate the average (expectation value) of any function f(x)? What is the effective (average) energy of interaction between a charge (ion) is the effective (average) energy of interaction between charge (ion) and a freely rotating dipole? Ion rotating-dipole interaction For fixed dipole (previous slide): fixed dipole (previous slide): E r, 1 q2 cos g r cos 40 2 r g r Effective (average over all possible angles) energy of interaction: (taking into account the Boltzmann weight for each configuration with given ) into account the Boltzmann weight for each configuration with given E if g r kBT E 1 Eeff 11 r 3kBT 40 2 q2 2 Always attractive! r4 (This is the average interaction energy, not the (Helmholtz) free energy, which is 1/2Eeff here.) Interactions between biomolecules Interaction between two dipoles between two dipoles For fixed dipoles: For fixed dipoles: E r 1 1 2 40 r 3 sin 1 sin 2 cos 1 2 2cos 1 cos 2 Example: coplanar dipoles 1 = 2 (HW) Interactions between biomolecules Interaction between two dipoles between two dipoles Free energy for freely rotating dipoles: 11 E r 3kBT 40 2 1 2 2 Always attractive! r6 (“Keesom” free energy) Interactions between biomolecules ind = × Electric field Interactions involving non-polar molecules “induced dipoles” involving non molecules dipoles” di di permanent dipole induced dipole charge (ion) induced dipole dipole fixed E r 1 r6 dipole freely rotating E r 1 r6 (“Debye free energy) r4 Always attractive! (Israelachvili) E r 1 Interactions between biomolecules Interactions between two non-polar molecules between two non molecules “Dispersion forces”: caused by “dancing charges” – instantaneous charge distributions as well as their fluctuations generate electromagnetic fields that induce dipoles, which in turn create fields … always present long-range complicated, not necessarily following a simple power law, usually attractive (but not necessarily) E r 1 r6 (“London dispersion” free energy) Interactions between (bio)molecules (bio)molecules Summary Which interactions are attractive? In medium other than vacuum, th 2. divide by or “van der Waals” interactions Add “hardcore repulsion” energy: 1 EHCR r n r “Lennard-Jones” potential (empirical) (Israelachvili) van der Waals interactions between “aggregates” pairwise addition (disregards cooperativity) (Glaser) (Israelachvili) Interactions between (bio)molecules Covalent bonds: What’s worth knowing about them? They’re strong. Backbones of macromolecules strong. of macromolecules Unless sterically hindered, single covalent bonds allow rotations about their axis. Flexibility, e.g., folding of proteins Origin: Shared electrons Description: Quantum mechanics, Molecular orbital theory Other important insight from quantum mechanics: • Heisenberg uncertainty principle uncertainty principle • Pauli exclusion principle • Spatial “probability distribution” of electrons given by the square of the wave function that solves the the square of the wave function that solves the Schrödinger equation http://www.chem.ufl.edu/~chm2040/ Notes/Chapter_11/covalent.html Interactions between biomolecules X H X Hydrogen bond bond most common: O, N, F always involves a hydrogen atom exhibits features of dipole-dipole interactions but also partially covalent character typically stronger than van-der-Waals interaction but weaker than covalent bond • strength can vary from 5 to 40 kJ/mol, depending on temperature, pressure, bond angle, dielectric constant, …) • typical length: 0.18 nm length: nm examples from previous lectures: Repulsion electrostatic: between like charges “steric” hardcore repulsion defines size of molecules: “packing radius”, “crystal radius”, “van der Waals radius”, “hardsphere radius”, “bare ion radius” Other “size” measures exist, for example: “hydration radius” (includes shell of oriented water molecules) “Stokes radius” (defined by electrophoretic electrophoretic mobility) (Israelachvili) BIM 162 Homework 02 1. 2. Due date: Thursday, 01/20/11 Compare the energy of a typical covalent single bond with the energy of electrostatic attraction in vacuum between a Na+ and a Cl- ion at contact (where r = 0.276nm). 1 1 2 Derive the interaction energy E r sin 1 sin 2 2cos 1 cos 2 between 40 r 3 two coplanar dipoles. di (See sketch for notation. Make sure you give all important steps of the derivation.) Note: This example is more complex than the one shown in class. The following strategy should lead to the right answer. Use a rectangular (Cartesian) coordinate system as shown below and express the position of all charges in terms of given quantities, e.g., (x-2,y-2) … position of charge labeled “-q2” where x2 r 2 cos 2 2 y2 2 sin 2 2 Recall that the distance x12 between two points (x1,y1) and (x2,y2) is given by: x122 = (x2 - x1)2 + (y2 - y1)2 When calculating the Taylor expansion of terms of the form 1 1 (where 1 ), keep also the second-order terms. ...
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This note was uploaded on 07/12/2011 for the course BIM 162 taught by Professor Heinrich during the Spring '11 term at UC Davis.

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