August 13, 2020 Draft Physical Principles in Biology Biology 3550/3551 Fall 2020 Chapter 1: The Scale of Things; Units and Dimensions David P. Goldenberg University of Utah [email protected] © 2020 David P. Goldenberg ii Contents 1 The Scale of Things: Units and Dimensions1 1.1 Measurements as comparisons..........................1 1.2 Units versus dimensions and a brief history of the metric system.......5 1.2.1 Early metric systems...........................6 1.2.2 Establishment of the Modern Metric System, the Syst`emeInterna- tional d'unit`es(SI) and Further Revisions...............8 1.2.3 The base dimensions of the SI and their current definitions...... 11 1.2.4 Other Units................................ 15 1.3 Using units in calculations............................ 15 1.4 Units of Concentration.............................. 18 1.4.1 Different ways of expressing concentration............... 18 1.4.2 Units of atomic and molecular mass................... 20 1.4.3 Special units of concentration for hydrogen and hydroxide ions.... 21 1.5 Further reading.................................. 23 2 Probability 25 2.1 An example of a random process: Brownian Motion............. 25 A mathematical description - random walks.................. 26 2.2 Introduction to probability theory........................ 27 Some introductory comments.......................... 27 A coin toss..................................... 29 A bit of mathematical formalism........................ 29 Adding and multiplying probabilities...................... 32 A final comment about independent events and the law of large numbers.. 35 2.3 Plinko probabilities: 6 rows........................... 35 Formulation of the problem........................... 35 Outcomes..................................... 36 Events....................................... 37 2.4 Plinko probabilities: The general case for n rows................ 40 Another way to count the paths to bucket 2 in a 6-row plinko........ 40 Labeled beans in a cup.............................. 41 The factorial function, permutations and combinations............ 42 iii CONTENTS Back to the plinko................................ 44 2.5 Biased plinkos................................... 46 2.6 Binomial coefficients, Pascal's triangle and the binomial distribution function 48 Binomial coefficients in algebra......................... 48 Pascal's triangle.................................. 49 The binomial probability distribution function................. 51 2.7 Random variables, expected value, variance and standard deviation..... 51 Playing for money................................. 51 Random variables................................. 52 Expected value, or mean............................. 54 The variance and standard deviation...................... 57 2.8 Continuous probability distribution functions.................. 59 The spinner.................................... 59 Expected value and variance for continuous random variables......... 62 Some other random variables from the spinner................. 64 2.9 The Gaussian, or normal, probability distribution function.......... 68 The general form of the Gaussian function................... 68 Approximation of the binomial distribution by the Gaussian distribution.. 70 2.10 Simulating randomness with a computer: (Pseudo) random numbers............................ 72 3 Random Walks 77 3.1 Random walks in one dimension......................... 77 The final position of the walker......................... 77 Other averages: The mean-square and root-mean-square........... 80 The mean-square and RMS end-to-end distance of a one-dimensional random walk.................................... 81 3.2 Random walks in two dimensions........................ 85 The random walks along the x- and y-axes................... 85 The end-to-end distance............................. 89 3.3 Three-dimensional Random Walks........................ 90 3.4 Computer Simulations of Random Walks.................... 93 Simulating large samples of random walks................... 97 4 Diffusion 101 4.1 Flux: Fick's First Law.............................. 101 The derivation................................... 101 The distribution of molecules diffusing from a single position......... 104 biological example................................ 106 4.2 Fick's second law................................. 108 The derivation................................... 108 4.3 Diffusion from a Sharp Boundary........................ 110 A solution to the diffusion equation...................... 110 Graphical representations of the solution.................... 113 4.4 Estimating a Diffusion Constant from a Simple Experiment.......... 115 iv CONTENTS 4.5 Molecular Motion and Kinetic Energy...................... 116 Kinetic energy................................... 116 Thermal energy.................................. 117 Steps in the random walk............................ 118 The relationship between molecular size and diffusion coefficient....... 120 4.6 A Plant Faces Diffusion.............................. 123 A plant's demand for CO2 ............................ 123 Leaf structure and stomata............................ 123 Diffusion of CO2 through stomata........................ 124 The big problem: Water diffusion........................ 126 The Crassulacean Acid Metabolism Cycle................... 127 Changes in atmospheric CO2 concentration................... 128 4.7 Bacterial Chemotaxis: Overcoming the Limits of Diffusion.......... 129 Bacteria under the microscope.......................... 132 Chemotaxis: Movement to or from specific chemicals............. 133 The rotary motor................................. 135 The sensory and signaling system........................ 135 5 Thermodynamics 137 5.1 Energy, Work and Heat.............................. 137 Units of energy.................................. 137 An important distinction: Temperature versus heat.............. 138 Some examples based on the expansion and compression of gasses...... 139 The first law of thermodynamics......................... 143 Reversible expansion and compression...................... 144 The maximum work from gas expansion.................... 146 State functions versus path functions...................... 147 5.2 Entropy and the Second Law........................... 148 The classical definition of entropy........................ 148 The statistical definition of entropy....................... 149 Microstates with different probabilities..................... 152 Entropy and information............................. 153 The second law.................................. 154 5.3 Thermodynamics of Chemical Reactions.................... 155 E and ∆E reconsidered............................. 155 Enthalpy (H)................................... 157 ∆G, the change in Gibbs free energy...................... 158 Free energy changes for chemical reactions................... 160 Concentrations and standard states....................... 164 Calculating the entropy change for a bimolecular reaction........... 164 Activity versus concentration.......................... 166 5.4 \Chemical Energy" and Metabolism....................... 167 Glucose oxidation................................. 167 ATP hydrolysis.................................. 169 Enzymatic coupling................................ 171 v CONTENTS 6 Formation of Biomolecular Structures 175 6.1 Water, Ionization and the Hydrophobic Effect................. 175 Hydrogen bonding................................ 175 Ionization..................................... 177 Dynamics of hydrogen ion diffusion....................... 178 The hydrophobic effect.............................. 179 6.2 Lipid Bilayers and Membranes.......................... 183 Amphiphilic molecules, micelles and bilayers.................. 183 Permeability of bilayers.............................. 185 Primitive membranes............................... 190 6.3 Protein Folding and Unfolding.......................... 191 Native and unfolded protein states....................... 192 Entropy of the unfolded state.......................... 193 Protein-stabilizing factors............................ 198 7 Molecular Motors 203 7.1 Some Basic Principles.............................. 203 Steam engines................................... 203 Measuring forces at a molecular scale and stretching a DNA molecule.... 204 A Brownian ratchet and Maxwell's demon................... 208 A hypothetical ATPase ratchet......................... 210 7.2 Adenylate kinase: Coupling a chemical reaction to conformational change.. 212 7.3 Myosin and Muscle Contraction......................... 213 The structure of muscle fibers.......................... 214 The ATPase cross-bridge cycle.......................... 219 Atomic resolution structures of myosin and actin................ 222 Non-muscle myosins............................... 224 vi Chapter 1 The Scale of Things: Units and Dimensions Aside from establishing links among the various sciences, a goal of this class is to help strengthen some of the skills that are required in all of the sciences, especially quantitative skills. Working with dimensions and units is one of the most important of these skills. Historically, one thing that has tended to distinguish biology from the physical sciences is the extent to which mathematics is used. This distinction is diminishing, but there is certainly a strong tradition in biology that is very descriptive. The greatest of all biologists was (arguably) Charles Darwin, who used little or no