Overview of Modeling

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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

And in order to make the equations fit hypothesis 3，we deduced the proper range of , that is, no regulation , activation and repression .The choice of other Parameter values and their reference is in Table 1

Network Topologies’ Analysis

Aiming at getting the values of r for each circuit, we need to numerically simulate the ODE equations to get the steady-state concentration of output node C under each input concentration, and then making linear fit of input and output concentrations. As our input concentration range is 10-9~10-5, we select points that have the same logarithmic distance intervals, then simulate output evolution curve to get the steady concentration one point by one. We choose the fourth-order Runge-Kutta method to solve the ODE equations and so as to save calculation time, we adopt Implicit Runge-Kutta algorithm to get the output steady concentration when Input=10-9M and set the very concentration as initial value for the Newton-Raphson method for following different Input concentration. And considering the possibility of bistable network topology, we calculate the two directions (positive sequence and the reverse ) that Input concentration changes to avoid wrongly supposing it as one IOA function circuit.

Identifying Minimal IOA Networks

Here we define Q value as the number of IOA function circuit among 10000 sets of network parameters, and it indicates the robustness of one topology to finish the IOA function—the larger Q value is, the more robust the topology is. We sort in reverse sequence all the network topologies according to their Q value and the x axis is their rank( Figure 6).We can observe that most network topologies have 0 or low Q value while there’s only a small part of the topologies having large Q value.</bn> We firstly analysis the first 160 network topologies ( Q >=705 ) and list in Figure 7 all the simplest topologies that have only 3 or less direct links between the three nodes.( Figure 7) Among the 14 topologies, there are 12 three-link networks and 2 two-node networks so that we can see that the minimal number of links for the topologies to be functional is two but the most usual number is 13. The common features of the networks capable of IOA are either one negative control loop (NCL) or one negative feedback loop (NFL). Here we define NCL as a topology that has one negative control on the input-receiving node (A) from the intermediate node (B) and one positive regulation on the output node (C) from A node.(as the 1st topology in Figure 7 and the 1st one in Figure 8) Similarly we define NFL as a topology that share the positive-regulation link from A to C, while uniquely have one negative feedback from C to A.(as the 2nd topology in Figure 7 and the 2nd one in Figure 8) And the NCL topology seemingly more robust than the NFL topology, as there are 9 topologies out of the 14 simplest networks contain NCL topology compared to 6 of NFL, so that B node appears to be important, which will be discussed in details in the following part.