Systems Biology 7.81 / 8.591 / 9.531 Problem Set 2 Assigned: 10.03.06 Due in class or in 13-2042 before 3pm Due: 10.19.06 1. Circadian clocks Several protein expression levels in plant and animal cells go through a daily cycle, driven by exposure to sunlight during the day and darkness at night. However, even in complete darkness, these expression levels oscillate with an intrinsic period of about 24 hours. The systems which drive these oscillations are known as circadian clocks. At the heart of most of these systems is a pair of transcriptionally regulated proteins: an activator (X) and an inhibitor (Y). In this problem set, we will see how such a simple system can be made to generate oscillations. We consider two possible system architectures (arrows represent activation, blunt ends represent inhibition): The corresponding dynamical equations are (using x = [X] and y = [Y]) (A) −++=−++=yxAxkvdtdyxyAAkvdtdxyyyxxxγγ211 (B) −++=−+++=yxAxkvdtdyxyAAxAxkvdtdxyyyxxxγγ2112232. (4) a. Identify the parameters corresponding to basal transcription rate, maximal transcription rate, and degradation rate. (4) b. We have assumed the following: for both (A) and (B), the X promoter is inactivated by the binding of a single molecule of Y, and the Y promoter is activated by the binding of a single molecule of X. In addition, for (B), the X promoter is activated by the cooperative binding of two molecules of X. The various fractions that appear in the dynamical equations represent the fractions of promoters that are active under these conditions. Give a biochemical interpretation of each of these fractions. (4) c. Normalization of units. Assume that A2 >> x. Redefine variables and show that by choosing units properly the previous equations can be written in the form (A) −+=−++=yxkvtdydxykvtdxdyyxxx)11(γ (B) −+=−+++=yxkvtdydxyxxkvtdxdyyxxx)111(22γ. Calculate the values of the new (barred) parameters in terms of the old parameters and discuss whether these equations can be further simplified by normalization of units. X Y (A) X Y (B)Systems Biology 7.81 / 8.591 / 9.531 From now on we will work with the simplified equations, dropping the bars on our variable and parameter symbols. These equations are of the general form ==),(),(yxgdtdyyxfdtdx. Let {x0, y0} be a fixed point of the system (f(x0, y0) = 0, g(x0, y0) = 0) and recall that if we define 00,yxygxgyfxfA=δδδδδδδδ fixed points of this two dimensional system are stable if and only if Tr(A) < 0 and Det(A) > 0. (5) d. For systems (A) and (B) separately, assume vx = 0.1, vy = 0.0, kx = 4.0, ky = 2.0 and γx = 10.0 and on a graph of y vs x plot the nullclines f(x, y) = 0 and g(x, y) = 0. You should find that the physical regions gets divided by these lines in several regions; for each one of them draw some arrows indicating the direction of motion and, based on these figures, comment on the stability of the fixed point. (5) e. For system (A) prove that the system will always converge to a stable fixed point. (6) f. For system (B) assume vx = 0.1, vy = 0.0, kx = 4.0, ky = 2.0 and let γx be a variable parameter. Numerically find the fixed point of the system as a function of γx. If this fixed point becomes unstable oscillations will arise. By numerically analyzing Tr(A) and Det(A) as a function of γx write down the conditions on γx under which the system is oscillatory. (6) g. For system (B) solve the dynamical equations of motion numerically using the parameters given in f and two values of γx, one which gives stable oscillations and other for which the system approaches a fixed point. On the same graph plot the nullclines. (6) h. Look up in the literature for two examples of simple genetic oscillators. Are the related network structures similar to system (A) or system (B)? (0) i. CHALLENGE. In the limit γx >> 1 estimate the period of oscillations. 2. The Repressilator Box 1 of the Elowitz and Leibler1 paper propose the accompanying set of normalized differential equations for modeling the “repressilator” system. In these equations mi stands for the normalized concentration of mRNA of the ith species and pi is the normalized concentration of the associated protein. 1 M.B. Elowitz and S. Leibler, A synthetic oscillatory network of transcriptional regulators, Nature 403, p. 335 (2000). +++−=+++−=+++−=000111ααααααntetRcIcInlacItetRtetRncIlacIlacIpmdtdmpmdtdmpmdtdm−−=−−=−−=)()()(cIcIcItetRtetRtetRlacIlacIlacImpdtdpmpdtdpmpdtdpβββSystems Biology 7.81 / 8.591 / 9.531 (5) a. Draw a diagram showing how the different variables affect each other, analogous to the ones shown in Problem 1. Does the lacI protein inhibit or enhances its own production? (5) b. Show that the system has a unique steady state and that in that state all dynamical variables are equal. If we name the equilibrium value a derive a nonlinear equation for it. (7) c. Define a column vector δ that represents deviations from equilibrium and show that it satisfies the equation: terms.nonlinear +=δδAdtd . 1 1 as defined function with the)( and ,010001100 , 100010001I with , Where3333nx f(x) fafCIICIA+=′===−−=αγββγ (6) d. Suppose that you are given an eigenvalue λ of A. Show that it is related to an eigenvalue c of C through the relation ).1)((++=λβλβγc Using the fact that C3 = I3 show that two of the eigenvalues of A satisfy 0)1)((=−++βγλβλ and the other four, .0)1)(()1()(2222=++++++γββγλβλλβλ (6) e. Use the result deduced in d and the Routh-Hurwitz conditions to show that the repressilator is stable if and only if -2 < γ < 1 and ββγγ22)1()2(23 +<+. (6) f. Use the last result to draw a phase diagram in (γ, β) space separating the regions in which the system is oscillatory from those in which it is not. Indicate the behavior in each region. 3. Revisiting the S. cerevisiae GAL system In the GAL system studied in the first problem set transcription and translation of Gal80 was considered to be decoupled from the