1. OR-gate logic. Analyze the C1-FFL with OR-logic at the Z promoter. Are there delays following ON or OFF steps of Sx? What could be the biological use of such a design? 2. A decoration on the FFL. The regulator Y in C1-FFLs in transcription networks is often negatively auto-regulated. How does this affect the dynamics of the circuit, assuming that it has an AND input function at the Z promoter? How does it affect the delay times? The Y regulator in an OR-gate C1-FFL is often positively auto-regulated. How does this affect the dynamics of the circuit? How does it affect the delay times? 3. The diamond. The four-node diamond pattern occurs when X regulates Y and Z, and both Y and Z regulate gene W. (a) How does the mean number of diamonds scale with network size in random ER networks? (b) What are the distinct types of sign combinations of the diamond (where each edge is either activation + or repression -)? How many of these are coherent? (Answer: 10 types, of which 6 are coherent). (c) Consider a diamond with four activation edges. Assign activation thresholds to all edges. Analyze the dynamics of W following a step of Sx, for both AND and OR logic at the W promoter. Are there sign-sensitive delays? 4. Type three. Solve the dynamics of the type-3 coherent FFL with AND-logic at the Z promoter in response to steps of Sx and Sy. Are there delays? What is the steady-state logic carried out by this circuit? Compare to the other coherent FFL types. X Y Z 5. The diamond again. The diamond pattern occurs when X regulates Y1 and Y2, and both Y1 and Y2 regulate gene Z. Analyze the 10 types of diamond structures (where each edge is either activation + or repression -) with respect to their steady-state responses to the inputs Sx, Sy1 and Sy2. Use an AND input function at the Z promoter. Do any diamond types lack responsiveness to any input? To all three inputs? 6. Repressilator. Three repressors are hooked up in a cycle X|Y|Z and Z|X. What are the resulting dynamics? Solve graphically using logic input functions. This circuit was constructed in bacteria using three well studied repressors, one of which was also made to repress the gene for green-fluorescent protein (Elowitz and Leibler, 2000). What would the resulting bacteria look like under a microscope that dynamically records green fluorescence? 7. Robust timing. Consider a SIM controlled by regulator X which activates downstream genes Zi, i= 1 .. n genes with thresholds Ki. At time t=0, X begins to be produced at a constant rate . (a) Are there biological reasons that favor placing the thresholds Ki much smaller than the maximal level of X? Consider the case in which X is an activator that begins to be produced at time t=0, and consider the effects of cell-cell variations in the production rate of X. (b) Would a design in which X is a repressor whose production stops at time t=0 provide more robust temporal programs? Explain. 8. The multi-output OR-FFL. In a multi-output C1-FFL with OR-gate logic at the Z promoters, transcription-factor X begins to be produced at a constant rate at time t=0. At time t=T, the production rate suddenly becomes equal to zero. Calculate the dynamics of the downstream genes Zi. What are the delays between genes? (use logic input functions). Assume KXZ2(K2)>KXZ1(K1) and KYZ2(K2')