lecture_14 - D e v e l o p m e n t o f C o m p l e x i t y...

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Unformatted text preview: D e v e l o p m e n t o f C o m p l e x i t y Catastrophe Theory Consider a potential function V ( x ) = x 3 + ax . When a < there is both a stable minimum (dots) and an unstable maximum in the potential. As a is slowly increased, the equilibrium system moves smoothly to smaller x : x eq = p- a/ 3 . When a = 0 , the stable minimum disappears, and for a infinitesimally positive, the system moves abruptly to a new equilibrium very far removed from the old. Lattimer, AST 248, Lecture 14 p.1/18 Boolean Networks A logical system composed of units or nodes that can be in either on or off states (analogy, lightbulbs). One could have a system of N nodes, each connected to K other nodes. Whether a node is on or off is determined by the state of the conected neighbors and a set of logical rules. For example, suppose K = 2 , so that node A is connected to nodes B and C, and the logical rules are: if B and C are both on or off, then A is turned on. If B is on and C is off or vice versa then A is turned off. Beginning from an initial state, the network takes a step and alters the state of each node, if necessary, to comply with the logical rules. Since some state of the network will eventually repeat an earlier state, the system will eventually enter a repeating cycle. The length of the cycle could be as short as one step ( i.e. the system is fixed) or extremely large. A large network will seem to behave chaotically. In between these extremes, Boolean networks show interesting behavior, such as quickly reverting to cyclical behavior, known as an attractor. For example, if N = 100 , 000 and K = 2 there are about N = 316 different cycles, each about 316 steps long. This is surprisingly orderly behavior: out of the 10 30 , 000 possible states, it will end up in one of these 316 cycles. Thus the vast majority of the systems possible states are never visited after the limit cycle is obtained. As the network goes into a limit cycle, about 70% of the nodes become frozen, with the other 30% randomly varying. When the frozen sea forms marks a phase transition between chaos and order. Lattimer, AST 248, Lecture 14 p.2/18 Stuart Kauffman has applied this model to genes to describe genetic regulatory networks and to the development of complex organisms. He examines how one goes from an undif- ferentiated egg by repeated cell division to differentiated cell types: red blood cells, retinal cones, etc. Genes are the nodes of a Boolean network and cell types are the attractors or limit cycles; cell division represents a timestep. He reasons that cell specialization is obtained for free from the connectivity rules. The number of genes in humans is about 100,00 and the number of cell types is about 256, close to its square root. This scal- ing also seems to work for other organisms. Furthermore, about 70% of the genes are permanently turned on in all cell types....
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lecture_14 - D e v e l o p m e n t o f C o m p l e x i t y...

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