Overview of Modeling

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Mechanisms of Minimal IOA Networks and Key Parameters Analysis

Aimed at answering the question why the two topologies defined above (Figure 8) is functional in IOA, we unravel their mechanisms using the ODE equations in this part, also getting the parameter restrictions of each topology.

NCL Topology

When the network has built steady state:

Solve the equations:

If there is , then
Because X1 is constant, the coefficient of I is also constant. So there comes the linear correlation between X3(output concentration) and I.
When ,
And the concentration of B node is actually part of the slope coefficient the linear equation, we can imagine that the function of B node is to lower the concentration of A straightly without the interference of other factors as well as control the concentration of A more precisely and more freely to make the parameter restriction easier to achieve and at the same time the output range is not too small. As we know, the stochastic error may make vague the linear relationship when the values of y axis are too near. The lower concentration X2 is, the steeper the line is, and so the bigger the range is, which in biology means that the bioreporter is more sensitive to certain environmental signal. Through modifying the parameters of node B, we can get a proper concentration of A node to achieve a good r. In all, the node B is a proportion node.

NFL Topology

Solve the equations:



Compare X1 here with the concentration of node A in NCL topology, one can easily discover that X1 here is higher and more dull, which makes it more hard to get a balance between satisfying the second parameter restriction and the need for the range to be rational. Then we can understand why NFL topology is less than NCL in high-Q-value topologies.
Aimed at unraveling the relative importance among parameters, we got all functional sets of parameters of NCL&NFL and selected randomly one set, respectively, from the two topologies and processed them with Matlab to compare the differences once the parameters change. The results are in Figure 9. From changes of the two important characters – the output range and r with the change of parameters (Table 2), we got the answer to previous questions.

Figure 9 Analysis for key parameters We got all functional sets of parameters of NCL&NFL and selected randomly one set, respectively, from the two topologies and processed them with Matlab to compare the differences once the parameters change. When analyzing one parameter, we only change this very parameter and keep others the same, and when we change the parameter to a lower level, we get the blue line, when to a higher level, we get the red line and the black line is for the unchanged parameter set. Each line has its Pearson Correlation Coefficient r marked in the figure. The X Axis is the concentration of Hg ion as INPUT whose range is 1 to 10000 nM, and the Y Axis is the concentration of node C with the unit of nM.

Analysis of All Possible Three-Node Networks

The above analyses focused on minimal( less than or equal to 3 links) three-node networks and identified simple topologies that are sufficient for IOA function, also unraveling the mechanism that the topologies work. But whether the topologies are necessary for the IOA function is not understood yet. In other words, are the identified minimal topologies the foundation of all possible networks, or are there more complex higher-order solutions that do not contain these minimal topologies? To answer the questions above, we analysis the first 160 topologies (Q>705) that are well capable of the IOA function.

Figure 10 Distribution of NCL topology parameters which can establish linear response curves
Analysis of these robust topologies shows that they are overrepresented with NCL and NFL. All 160 topologies contain at least one NCL or NFL motif ( or both ). These results indicate that at least one of these motifs is necessary for IOA function.
Supplementarily, the NCL average Q value(AQV)of all 19683 topologies is 17.14 while the NFL AQV is only 9.36, which again indicate that the NCL is more robust than NFL that have drew conclusion in the minimal topology analysis.

Motif Combinations that Improve IOA

To investigate what additional features can improve the functional performance in some more complex and more robust networks than minimal topologies, we clustered the first 160 networks and then cluster them respectively in three categories: NCL, NFL and the combination of the two.

Figure 11 Distribution of NFL topology parameters which can establish linear response curves.
The results clearly indicate that apart from the link from A to C, there should be no positive regulation, and the NFL topology hates the link from A to A, while the combination topologies show no additional tendency.

In all, by exhaustively searching all network and analyzing the results, we draw the conclusion that NCL meets our need in application well, and we get the proper parameter range for practice.

Figure 12 Analysis of the first 160 networksWe count all the NCLs and NFLs and also the IOA networks, and discover that all of the IOA networks can be classified into NCL/NFL/the combination of the two. And there are more IOA functional network featured in NCL than those characterized with NFL.

Figure 13 The clustergrams of the networksWe use the clustergram command in matlab to get the additional features of the functional networks. The nine vertical rectangle bar stand for nine links in Figure 2 which are, respectively, from A to A, from A to B, from A to C, from B to A, from B to B, from B to C, from C to A, from C to B, from C to C. And red stands for activation, green for repression and black for no regulation. The topologies on the right are corresponding minimal topologies that is shown in the clustergrams on the left.

More Advanced Model

As the IOA function still cannot work well to ensure the linear relationship when the input range spans several orders of magnitude, we search for semilog networks further using the same method.

Identifying Minimal Semilog Networks

We again get the Q-Rank Figure ( Figure 1). Topologies that have large Q value are still the minority.
Listed in Figure 2 are all the simplest topologies whose Q value is above 100 that have 4 or less direct links between the three nodes.( Figure 2) There are only one 3-link topology( as the 7th topology in Figure 2) out of all the 7 simplest topologies, and it has exactly the common features of the seven topologies: two positive controls from A node to respectively B node and C node, and one positive control from B to C. We call it the All Activated Network(AAN). The role of different nodes and links will be discussed in the next part.

Figure 2 All Functional Networks The numbers below each network are their ranks. The bigger topology has 2 links while the other 6 networks have 3 links. In each network, the green arc with one short straight line at one end stands for repression from the start node to the end node and the red arc with one arrow at one end stands for activation.

Mechanisms of Minimal Semilog Networks and Key Parameters Analysis

In order to unravel the mechanism that AAN functions, we analysis the topology with the ODE equations, and we also get the key parameters in this part.
When the network reaches its steady state,

Solve the equations,

When, there is
,so:

Parameters Tendency of the IOA Networks

It can be seen from the distribution of K and α values that compared to networks with linear response curve, single parameter shows much less tendency. Instead, to establish a semi-log response curve may require the cooperation of multiple parameter values.