A&S 320: Mathematical Modeling in Biology

Transcription

1 A&S 320: Mathematical Modeling in Biology David Murrugarra Department of Mathematics, University of Kentucky These slides were modified from Matthew Macauley s lecture at Clemson University. Spring 2016 David Murrugarra (University of Kentucky) A&S 320: Reduction of Boolean network models Spring / 8

2 Application: Boolean model of Th-cell differentiation White blood cells or leukocytes are in the immune system and fight diseases and infections. One subtype are the lymphocytes, which includes the natural killer (NK) cells, B cells, and T cells, all which have different cellular functions. The T-cells circulate throughout our bodies in the lymph fluid, looking for cellular abnormalities, infections, and diseases. Helper T-cells (Th-cells) are a certain type of T-cells. They begin as naïve, or Th0 cells, and then differentiate into one of two phenotypes: 1 Type 1 are the Th1 cells which fight intracellular bacteria and protozoa. 2 Type 2 are the Th2 cells which fight extracellular parasites. Malfunctions of immune responses involving Th1 phenotypes can result in autoimmune diseases, whereas malfunctions involving Th2 phenotypes can result in allergic reactions. The biochemical signals that determine Th1 and Th2 differentiation act as a bistable switch, which permits either GATA3 or T-bet to be expressed, but not both. This was modeled using a 23-node Boolean network in Mendoza et. al (2006). David Murrugarra (University of Kentucky) A&S 320: Reduction of Boolean network models Spring / 8

3 Boolean model of Th-cell differentiation (Mendoza, 2006) x 1 = GATA3 f 1 = (x 1 x 21 ) x 22 x 2 = IFN-β f 2 = 0 x 3 = IFN-βR f 3 = x 2 x 4 = IFN-γ f 4 = (x 14 x 16 x 20 x 22 ) x 19 x 5 = IFN-γR f 5 = x 4 x 6 = IL-10 f 6 = x 1 x 7 = IL-10R f 7 = x 6 x 8 = IL-12 f 8 = 0 x 9 = IL-12R f 9 = x 8 x 21 x 10 = IL-18 f 10 = 0 x 11 = IL-18R f 11 = x 10 x 21 x 12 = IL-4 f 12 = x 1 x 18 x 13 = IL-4R f 13 = x 12 x 17 x 14 = IRAK f 14 = x 11 x 15 = JAK1 f 15 = x 5 x 17 x 16 = NFAT f 16 = x 23 x 17 = SOCS1 f 17 = x 18 x 22 x 18 = STAT1 f 18 = x 3 x 15 x 19 = STAT3 f 19 = x 7 x 20 = STAT4 f 20 = x 9 x 1 x 21 = STAT6 f 21 = x 13 x 22 = T-bet f 22 = (x 18 x 22 ) x 1 x 23 = TCR f 23 = 0 David Murrugarra (University of Kentucky) A&S 320: Reduction of Boolean network models Spring / 8

4 Boolean model of Th-cell differentiation Reduced model (by removing, in order, x 23, x 21, x 20,... ): Variable Boolean function Polynomial function x 1 = GATA3 h 1 (x 1, x 22 ) = x 1 x 22 h 1 (x 1, x 22 ) = x 1 x 22 + x 1 x 22 = T-bet h 22 (x 1, x 22 ) = x 1 x 22 h 22 = x 1 x 22 + x 22 There are three fixed points: (0, 0): GATA3 and T-bet are inactive, the signature of Th0 cells. (0, 1): Only T-bet is active, the signature of Th1 cells. (1, 0): Only GATA3 is active, the signature of Th2-cells. David Murrugarra (University of Kentucky) A&S 320: Reduction of Boolean network models Spring / 8

5 AND Boolean models In this section, we study a reduction method tailored specifically to AND Boolean models. AND networks are of the form f = (f 1,..., f n) : {0, 1} n {0, 1} n, where f i = x j1 x j2... x jk. Example The following Boolean network is an AND network. In Boolean form f 1 = x 2 x 3, f 2 = x 1 x 2 x 6, f 3 = x 2, f 4 = x 1 x 5, f 5 = x 5 x 6, f 6 = x 1 x 5 x 6, or in polynomial form f 1 = x 2 x 3, f 2 = x 1 x 2 x 6, f 3 = x 2, f 4 = x 5 x 1, f 5 = x 5 x 6, f 6 = x 1 x 5 x 6. David Murrugarra (University of Kentucky) A&S 320: Reduction of Boolean network models Spring / 8

6 AND Boolean models AND networks are completely determined by the wiring diagram. For example, f 1 = x 2 x 3 will be represented by arrows from x 2 and x 3 to x 1 (and no other arrows to x 1 ). A subgraph G i of a wiring diagram is strongly connected if there is a directed path from any vertex of G i to any other vertex in G i Figure: Wiring diagram (left) and some strongly connected subgraphs (right). A subgraph G i of a wiring diagram is a strongly connected component (scc) if it is strongly connected and is the largest strongly connected subgraph that contains G i, that is, G i cannot be made larger and still be strongly connected. If a scc has only one node and it does not have a self loop, then we call that scc trivial. David Murrugarra (University of Kentucky) A&S 320: Reduction of Boolean network models Spring / 8

7 AND Boolean models The main result in the reduction of AND networks is that strongly connected components can be replaced by a single node with a self loop. Example Consider the Boolean network f = (x 2 x 3, x 1 x 3, x 2, x 1 x 5, x 6, x 3 x 4 ) with wiring diagram shown in Figure 4 (left) Figure: Wiring diagram of AND network from Example 3 (left) with its strongly connected components highlighted in circles and the reduced wiring diagram (right). David Murrugarra (University of Kentucky) A&S 320: Reduction of Boolean network models Spring / 8

8 David Murrugarra (University of Kentucky) A&S 320: Reduction of. Boolean network models Spring / 8 AND Boolean models The main result in the reduction of AND networks is that strongly connected components can be replaced by a single node with a self loop. Example Consider the Boolean network f = (x 2 x 3, x 1 x 3, x 2, x 1 x 5, x 6, x 3 x 4 ) with wiring diagram shown in Figure 4 (left). The reduced network given in Figure 4 (right), which corresponds to the Boolean network h = h(x 1, x 4 ) = (x 1, x 1 x 4 ) and has steady states (x 1, x 4 ) = (0, 0), (1, 0), (1, 1). Then, the steady states of the original AND network are found using the equations x 2 = x 3 = x 1 and x 6 = x 5 = x 4 : x = , , Figure: Wiring diagram of AND network from Example 3 (left) with its strongly connected components highlighted in circles and the reduced wiring diagram (right)