Team:Peking/Modeling/CalculationProcess

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   Calculating Process

Network Enumeration

We use three nodes as a minimal framework: one node that receives input( A in Figure 2 ), a second node that transmits output( C in Figure 2 ), and a third node that can play diverse regulatory roles( B in Figure 2 ). There are 9 direct links among the three nodes and there are altogether 39=19683 three-node topologies. With 3,645 topologies that have no direct or indirect links from the input to the output occluded, there remain a total of 16,038 possible three-node topologies that contain at least one direct or indirect causal link from the input node to the output node. For each topology, we sampled 10,000 sets of network parameters with the method of latin hypercube sampling (LHS, Figure 3). In all, we have analyzed a total of 16,038*10,000 different circuits. This search resulted in an exhaustive circuit function map used to extract core topological motifs essential for IOA.
Figure 2 Three-node network with all of its possible directed links(Ref. 7) There are altogether 9 possible links and the input here is Hg(Ⅱ), and the concentration of C is taken as output.
Figure 3 Latin Hypercube Sampling When sampling a function of N variables, the range of each variable is divided into M equally probable intervals, M sample points are then placed to satisfy the Latin Hypercube requirements. Then each sample is theonly one in each axis-aligned hyperplane containing it.

Equations Set Up

Our model is based on the following statements:
(1) The nodes are restricted to TF nodes so that the links stand for TF-TF interactions via DNA. The expression level is quantified by the equilibrium binding probability P of TF binding on its site and the maximum expression rate constant , and we adopt a constant to modify P to make different TFs equal status. When it comes to several TF factors, we use the multiplication of their or to indicate their interactions.
(2) We take into consideration only the transcription and translation and TF-DNA interactions because other reactions such as signal-transduction activities typically operate much faster and can be considered to be approximately at steady state on the slow timescales of transcription networks. Also the TF activity levels can be considered to be at steady state within the equations that describe network dynamics on the slow timescale of changes in protein levels. So that the equations contain only the accumulation and degradation of the protein products ( here the TFs).
(3) It has been observed that one ordinary gene usually has a nonzero expression level with no TFs on its binding site. We propose that one repressor will lower the initial expression level and one activator will shift it, further on, each TF has its unique contribution to the final expression level, which means for example that the expression level with two repressor and one activator binding on is different from the level under the regulation of two activators and one repressor, but may not necessarily lower than the latter if the only one activator is very strong.
  Consider first the simplest condition under which there is only one link from a node to another. (AC, Figure 4)
It is widely accepted that the possibility of TF binding to the binding site in promoter is
(X*: the effective concentration of one TF; Kd : the dissociation constant )
According to hypothesis (1)&(3), the link from A(node1) to C(node3) can be translated as

( :the basal translation rate factor;:the effective translation rate factor.)

Figure 4 simplest network There is only one link from node A to node C and we don’t regulate the property of the link. Actually, whether it is activation, repression or no regulation is represented by .
As (Figure 5)
Thus (The subscript of X is the node number.) Actually, the first component of the equation is the fundamental expression level of the network in which the is the possibility that the TF is off the DNA target site, and the second component of the equation is the effective expression level in which is the possibility that TF is on its site.
As to general conditions, there are

network topologies’ analysis

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