BIOLOGICAL PHYSICS MAJOR OPTION QUESTION SHEET 1 - will be covered in 1st of 3 supervisions

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1 1 BIOLOGICAL PHYSICS MAJOR OPTION QUESTION SHEET 1 - will be covered in 1st of 3 supervisions MODULE A Lecture 1: Intro and concepts A1 Intro (a) Roughly 2-3 kg of bacteria are in your large intestine. Estimate the number of bacteria that this corresponds to. Estimate the number of human cells in your body, and compare the two figures. (b) Look up estimates of protein density in a cell. Estimate cell volumes for bacteria, yeast and a typical mammalian cell. Work out rough total number of proteins in these cells. (c) Look up rough fractions of protein, RNA, lipids and other classes of molecules in the dry mass of cells. What is a typical water content in a cell? Explore, and keep in mind, the online resource: A2 Growth laws - a numerical exercise on growth dynamics (adapted from BBSRC Sysmic course.) A modified growth model, with similar behaviour to logistic growth, can be developed from the differential equation: where dn/dt = r(n)n, ( N ) 2 ) r(n) = r 0 (1. K For the following use parameters values r 0 = and K = i) Use Matlab or your favorite package to plot dn/dt. ii) Use this plot to infer how the population will change at different values of N. iii) Describe how this plot differs from the equivalent plot for the logistic growth model. iv) Using Matlab (or otherwise!) find the value of N at which the growth rate is a maximum. (Hint: there are two solutions but we require N > 0. ) v) Find the solution of this differential equation, for example using Matlab s dsolve command. vi) Create two figures overlaying plots showing population growth, for the model introduced here and for Logistic growth. Do this for both N 0 = 100 and N 0 = vii) Describe the differences between the graphs. A3 Cell size control Consider the growth control models discussed in the first lecture as examples of physics based models (in this case models based not on stat mech as many others in the course, but trying to tease out consistent mechanisms from data). The added size to a cell is given by a s b +, where s b is the size at birth, 1 < a < 1 is a parameter that distinguishes timer control (a = 1), from adder (a = 0) and sizer (a = 1). > 0 is a fixed added size. Show that in an organism performing symmetric divisions, starting from a cell of arbitrary size s b, the size of its progeny will eventually converge. Find an expression for how this steady state size depends on and a. Advanced: Make some considerations on the speed of convergence (in generations) for varying a. For thought: What experiments are done to study these control mechanisms? Size control is necessary for all populations of dividing cells; do you think the control mechanism will be universal? What might you imagine as being plausible timer and sizer mechanisms at a molecular level? Lecture 2: Numbers in the cell, and Central Dogma A4 Small numbers - (2.7 from [Phillips et al., 2013]) In considering the cell to cell variability, one aspect is the random partitioning of molecules to daughter cells. Derive the result that if there are N molecules, and this partitioning follows purely a random process, then the variance in the number n 1 of the molecules that go to cell 1 is: n 2 1 n 1 2 = N p q, where p is the probability of the molecule going into daughter cell 1, and q = 1 p is the probability of that molecule going to cell 2. If the molecule of interest are fluorescent, and the mother cell has total fluorescence I tot, show that you can expect the variability in the fluorescence of daughter cells (I 1 and I 2 ) to follow: (I 1 I 2 ) 2 = αi tot, where α is a coefficient relating the number of molecules to the fluorescence as I = αn. For thought: what would you think if there was found to be deviations from this last results in experiments? Can you think of plausible complications from purely the experimental side? Can you think or do a quick literature search for possible biological effects? Numerical investigation Can you modify the countingbydilution.m script to plot the distributions of (I 1 I 2 ) 2 for some values of I tot? Do these resemble more closely Gaussian distributions (normals) or log-normal?

2 2 A5 On timing of DNA replication, speed of forks - (3.3 from [Phillips et al., 2013]). dummy text dummy text The electron microscopy image, and the helpful schematic cartoon (both from [Alberts et al., 1994]) show a snapshot in time of replication forks duplicating the genome of a drosophila fly. (a) Estimate the fraction of the total fly genome captured in the micrograph. (The total fly genome is about nucleotide pairs in size.) (b) Extrapolating from this snapshot, estimate the number of DNA polymerase molecules in a eukaryotic cell like this one from drosophila. (c) There are 8 forks in this micrograph. Estimate the lengths of the strands between the forks 4 and 5 counting from the bottom. If a fork moves at a replication speed of about 40 bp/s how long will it take forks 4 and 5 to collide? (d) Given the mean spacing between the forks seen here, estimate how long it will take to replicate the entire genome. A6 Ribosomes - (adapted from 3.4d from [Phillips et al., 2013]) E. coli is growing in a condition where cells divide every 3000 s. If each cell contains typically ribosomes, and assuming no ribosome degradation, how many RNA polymerase molecules must be synthesising rrna (RNA that codes for ribosomes) at any instant? What percentage of RNA polymerase molecules in E. coli are involved in transcribing rrna genes? Lecture 3: Networks A7 General definitions Let G be a graph with n vertices labelled 1, 2, 3,..., n. The adjacency matrix A(G) of G is the n n matrix in which the entry in row i and column j is the number of edges joining the vertices i and j. (a) Write down the adjacency matrix of each of the following graphs: (b) Draw the graph represented by each of the following adjacency matrices: Let D be a digraph (directed graph) with n vertices labelled 1, 2, 3,..., n. The adjacency matrix A(D) of D is the n n matrix in which the entry in row i and column j is the number of arcs (directed arrows ) from vertex i to vertex j. (c) Write down the adjacency matrix of each of the following digraphs:

3 3 (d) (e) Draw the digraph represented by each of the following adjacency matrices: Calculate the clustering coefficients for the red nodes in the graphs below: Convince yourself that the maximum number of edges between k vertices in an undirected graph with no loops is k(k 1)/2. Think: How is this related to a special property of the adjacency matrix? (f) Classify each of the feedforward motifs below as coherent or incoherent (see glossary in notes handout): A8 Numerical network question. (problems 4 and 5, p.219 of [Sneppen and Zocchi, 2005].) Simulate a network that grows with preferential attachment, say at each time step one node with one link is added with probability R new = 0.1, and if not, then only a link between two existing nodes is added. Let the network reach a steady state, by removing nodes plus all their links with small probability (try ɛ 10 3 or Quantify the connectivity distribution. (Optional: Quantify the degree of correlation of the number of links per node, between nearest neighbour nodes. Is this same or different to that of a random graph?) Then, let n(k, t) be the number of nodes with connectivity k at time t. Can you find the analytical expression for the steady-state distribution of n(k) for different values of R new? (TURN OVER

4 4 A9 Thinking about size of regulatory networks, from [Sneppen and Zocchi, 2005] Regulatory genetic networks are essential for epigenetics and thus for multicellular life, but are not essential for life. In fact, there exist prokaryotes with nearly no genetic regulation. The course handout shows a figure for the fraction of regulators as a function of genome size, for a number of prokaryotic organisms. Understand that: If life was just a bunch of independent switches, i.e. if living cells could be understood as composed of a number of modules (genes regulated together) each, for example, associated with a response to a corresponding external situation, then the fraction of regulators would be independent of the number of genes N. If life was simply hierarchical, with each gene controlling a certain number of downstream genes, then again the number of regulators would grow linearly with N. If life would have been controlled using the maximum capacity for combinatoric control, even fewer transcription factors would be necessary. Show that if the regulation of each gene could include all transcription regulators available, then in principle the total state of all genes could be specified with only log 2 (N) regulators. MODULE B Lecture 4: Statistical Physics of Living Systems (Introduction) B10 F and G in Biological systems, 8.4 from [Biological Physics (Updated 1st Edition, P. Nelson, Freeman Press)] (a) Consider a chemical reaction in which a molecule moves from gas to a water solution. At atmospheric pressure each gas molecule occupies a volume of about 22 L/mole, whereas in solution the volume is closer to the volume occupied by a water molecule, or 1/(55 mole/l). Estimate ( V)p, expressing your answer in units of k B T assuming room temperature. (b) Next Consider a reaction in which two molecules in aqueous solution combine to form one. Compare the expected ( V))p to what you found in (a), and comment on why we usually don t need to distinguish between F and G for such reactions. (From : Nelson, Biological Physics) B11 Michaelis-Menten Kinetics Consider the following chemical reaction : E + S k 1 ES k 2 E + P k 1 where E is enzyme, S is substrate, ES is the enzyme-substrate complex, and P is product. Let s assume that the concentration of substrate is much greater than that of total enzyme ([S]» [E t ]) such that it can be treated as a constant. Derive an expression for the steady-state rate of product formation. Lecture 5: Statistical Physics of Living Systems (Multi-State Systems, Monod-Wyman-Changeux Model) B12 Allosteric transitions Explain what is meant by allostery. Enzymes that undergo allosteric transitions are typically composed of a large number of identical subunits. A complex is made of N identical subunits, each of which can bind to a ligand molecule. The ligand occupancy of the ith subunit is given by σ i : σ i = 0, 1 for vacant and occupied subunits respectively. The Monod-Wyman-Changeux (MWC) model for cooperative binding is one of the classic two-state models for binding. In the all-or-none MWC model, the activity s of the entire complex is either active (s = 1) or inactive (s = 0). For the MWC model, the energy of the complex depends on s and the σ i in the following way: H = (E + ɛσσ i )s + µσσ i. E is the energy difference between active and inactive state in the absence of ligand; each occupied subunit enhances the activity by changing the energy of the active state by ɛ < 0; µ is the free energy for ligand binding in either state, µ depends on the ligand concentration and a dissociation constant, K 0, for the inactive state. All energies are in units of the thermal energy k B T. The parameters used in the orignal MWC model are the equilibrium constant L for the transition from the inactive to the active state in the absence of ligand, and the ratio of the dissociation constants for the inactive and active states, C = K 0 /K 1. Show that these are related to the above energy parameters by: exp( E) = L, exp( ɛ) = C, exp( µ) = [L]/K 0, where [L] is the concentration of ligand. Show that the partition function is Z = [ 1 + exp( µ) ]N + exp( E) [ 1 + exp( ɛ µ) ] N and hence that the average activity of the complex is s = L(1 + C[L]/K 0 ) N (1 + [L]/K 0 ) N + L(1 + C[L]/K 0 ) N Lecture 6: Statistical Physics of Living Systems (Dynamics/Diffusion)

5 5 B13 Diffusion Limited Size Some bacteria can be idealised as spheres of radius R. Suppose that such a bacterium is suspended in a lake and that it is aerobic (it needs oxygen to survive). Oxygen is available in the lake at a concentration c 0 except in the immediate surrounding of the bacterium because every oxygen molecule reaching the bacterium is immediately being consumed. Determine the steady-state concentration profile of oxygen and hence show that the rate at which oxygen is absorbed by the bacterium is : k = 4πDRc 0 where D is the diffusion coefficient of oxygen. Evaluate the absorption rate for D = 0.197µm 2 ms 1, R = 1µm, and c 0 = 200 mmm 3. The absorption rate of oxygen increases linearly with R. We might expect, however, that the consumption of oxygen will increase linearly with the volume of the bacterium. This implies that there is an upper limit to the size of a bacterium! If R were too large, the bacterium would literally suffocate. A convenient measure of an organism s overall metabolic activity A is its rate of O 2 consumption divided by its mass. Find the maximum possible metabolic activity of a bacterium of arbitrary radius. The actual metabolic activity of a bacterium is about A = 10mMkg 1 s 1. What limit do you then get on the size of a bacterium (use a reasonable estimate of their mass density)? Compare your answer to the size of real bacteria. What ways do small organisms have to evade this limit? B14 Diffusion Controlled Reaction Rate Randomly diffusing particles are absorbed on the surface of a sphere of radius a. If the concentration of particles a long way from the sphere is maintained at c 0, show that the steady-state concentration profile varies as ( c(r) = c 0 1 a ), r where r is the radial distance from the centre of the sphere. Hence show that the rate at which particles are absorbed the sphere is k = 4πDac 0, (1) where D is the diffusion coefficient of the particles. The globular proteins A and B have diameters a and b. B is an enzyme that binds to A and converts it to a product A. Suppose that there are NA copies of A and only one copy of B in a cell of volume V. Treating the cytoplasm as a medium of viscosity η in which the proteins can freely diffuse, use the previous result show that the fastest rate at which the product can be formed is k = 2 N A k B T (a + b) 2. (2) 3 ηv ab Evaluate this rate for parameter values typical of a bacterial cell: V = m 3, η = 0.1 kgm 1 s 1, N A = 1,000 and protein sizes (i) a = b = 3nm; (ii) a = 1nm, b = 9nm. References [Alberts et al., 1994] Alberts, B., Bray, D., Lewis, J., Raff, M., Roberts, K., and Watson, J. D. (1994). Molecular Biology of the Cell. Garland Publishing, New York. [Phillips et al., 2013] Phillips, R., Kondev, J., Theriot, J., Garcia, H., and Orme, N. (2013). Physical Biology of the Cell, 2nd Ed. Garland Science, London. [Sneppen and Zocchi, 2005] Sneppen, K. and Zocchi, G. (2005). Physics in Molecular Biology. Cambridge University Press, Cambridge.