Heterogeneous Boolean Networks with Two Totalistic Rules By: Katie Toh Advisor: Dora Matache Hello, my name is Katie Toh. I am graduate student in the department of Mathematics here at UNO. I have been researching heterogeneous Boolean Networks under my advisor Dora Matache. Heterogenous Boolean networks are used to model and understand numerous biological processes such as gene networks, but very little is know about how heterogenous Boolean networks behave in general so my research asks the questions: can we predicate the dynamics of heterogeneous Boolean network? In this talk I will first go over some background information, then talk about my research, and conclude with its impacts. The first thing we need to know is… University of Nebraska - Omaha Department of Mathematics

What is a Boolean Network? We will do this with a simple example.

What is a Boolean Network? A network made up of: Nodes or Elements A Boolean is a network made up of nodes or elements.

What is a Boolean Network? A network made up of: Nodes or Elements People In our example the nodes will be people.

What is a Boolean Network? A network made up of: Nodes or Elements People That each have a binary state (ON or OFF, 0 or 1, etc) A network made up of: Nodes or Elements People Each node has a binary state such as ON or OFF.

What is a Boolean Network? A network made up of: Nodes or Elements People That each have a binary state (ON or OFF, 0 or 1, etc) Either they like cats OR they like dogs A network made up of: Nodes or Elements People In our example people either like cats or they like dogs, never both.

What is a Boolean Network? A network made up of: Nodes or Elements People That each have a binary state (ON or OFF, 0 or 1, etc) Either they like cats OR they like dogs Are connected to other nodes which make up their neighborhood Each node is connected to other nodes which make up its neighborhood.

What is a Boolean Network? A network made up of: Nodes or Elements People That each have a binary state (ON or OFF, 0 or 1, etc) Either they like cats OR they like dogs Are connected to other nodes which make up their neighborhood Through social media In our example each person’s neighborhood is made up of people they are connected to through social media.

What is a Boolean Network? A network made up of: Nodes or Elements People That each have a binary state (ON or OFF, 0 or 1, etc) Either they like cats OR they like dogs Are connected to other nodes which make up their neighborhood Through social media Update their state based on some rule Each node has an update rule.

What is a Boolean Network? A network made up of: Nodes or Elements People That each have a binary state (ON or OFF, 0 or 1, etc) Either they like cats OR they like dogs Are connected to other nodes which make up their neighborhood Through social media Update their state based on some rule If more people they are connect to like cats, they like cats, if more people they are connect to like dogs, they like dogs In our example, if more people on their social media like cats then the person will like cats. If more people on their social media like dogs then they will like dogs. Since this person has more people that like cats on their social media they will change from liking dogs…

What is a Boolean Network? A network made up of: Nodes or Elements People That each have a binary state (ON or OFF, 0 or 1, etc) Either they like cats OR they like dogs Are connected to other nodes which make up their neighborhood Through social media Update their state based on some rule If more people they are connect to like cats, they like cats, if more people they are connect to like dogs, they like dogs To liking cats.

What is a Boolean Network? A network made up of: Nodes or Elements People That each have a binary state (ON or OFF, 0 or 1, etc) Either they like cats OR they like dogs Are connected to other nodes which make up their neighborhood Through social media Update their state based on some rule If more people they are connect to like cats, they like cats, if more people they are connect to like dogs, they like dogs They update over time The nodes will continue to update over time.

What is a Boolean Network? A network made up of: Nodes or Elements People That each have a binary state (ON or OFF, 0 or 1, etc) Either they like cats OR they like dogs Are connected to other nodes which make up their neighborhood Through social media Update their state based on some rule If more people they are connect to like cats, they like cats, if more people they are connect to like dogs, they like dogs They update over time So if this second person has more people on their social media that now like dogs…

What is a Boolean Network? A network made up of: Nodes or Elements People That each have a binary state (ON or OFF, 0 or 1, etc) Either they like cats OR they like dogs Are connected to other nodes which make up their neighborhood Through social media Update their state based on some rule If more people they are connect to like cats, they like cats, if more people they are connect to like dogs, they like dogs They update over time They will switch to liking dogs.

What is a Boolean Network? A network made up of: Nodes or Elements People That each have a binary state (ON or OFF, 0 or 1, etc) Either they like cats OR they like dogs Are connected to other nodes which make up their neighborhood Through social media Update their state based on some rule If more people they are connect to like cats, they like cats, if more people they are connect to like dogs, they like dogs They update over time Then our first person now has more people on their social media that like dogs so they will switch from liking cats…

What is a Boolean Network? A network made up of: Nodes or Elements People That each have a binary state (ON or OFF, 0 or 1, etc) Either they like cats OR they like dogs Are connected to other nodes which make up their neighborhood Through social media Update their state based on some rule If more people they are connect to like cats, they like cats, if more people they are connect to like dogs, they like dogs They update over time To liking dogs. One of the main things we are interested in when looking at a Boolean network is the long term behavior of the network.

Long Term Behavior of a Boolean Network We will discuss four different long term behaviors.

Long Term Behavior of a Boolean Network Stable Fixed Point: Example: 60% of people like cats and 40% like dogs and it stays that way One type of long term behavior we might see is called a stable fixed point. In our example of cats and dogs this might mean that eventually 60% of people like cats and 40% like dogs and it stays that way. Their neighborhoods don’t change and they don’t change.

Long Term Behavior of a Boolean Network Stable Fixed Point: Example: 60% of people like cats and 40% like dogs and it stays that way Stable Period-2 Orbit: 60% of people like cats and 40% like dogs, then, 70% of people like cats and 30% like dogs, then it repeats Another long term behavior we might see is called a period-2 orbit. An example of this would be if eventually 60% of people like cats and 40% like dogs, then it changes to 70% of people like cats and 30% like dogs, and then it repeats going back and forth between these two states.

Long Term Behavior of a Boolean Network Stable Fixed Point: Example: 60% of people like cats and 40% like dogs and it stays that way Stable Period-2 Orbit: 60% of people like cats and 40% like dogs, then, 70% of people like cats and 30% like dogs, then it repeats Stable Period-3, Period-4, Period-5, etc. orbit: 3, 4, 5, etc. states of the network are repeat over and over A third long term behavior we might see is a larger period orbit, such as a period-3 orbit. In this case the percentage of people that like cats and dogs would go through three different states and then repeat.

Long Term Behavior of a Boolean Network Stable Fixed Point: Example: 60% of people like cats and 40% like dogs and it stays that way Stable Period-2 Orbit: 60% of people like cats and 40% like dogs, then, 70% of people like cats and 30% like dogs, then it repeats Stable Period-3, Period-4, Period-5, etc. orbit: 3, 4, 5, etc. states of the network are repeat over and over Chaos: There is no pattern seen Different percentages of people like cats and dogs and never repeats Lastly we might see chaos. This would mean that the percentage of people that like cats and dogs keeps changing and never repeats or there is no clear pattern after a large number of updates.

Heterogeneous Boolean Networks Boolean networks have been used to model: Gene networks (Shmulevich, Dougherty, & Zhang, 2002) (Schlatter R1, 2009) Cellular networks (Saez-Rodriguez, et al., 2007) (Albert & Othmer, 2003) Gut Bacteria (Steinway, et al., 2015) Despite the simplicity of Boolean networks they have been successfully used to model complicated biological networks such as genetic networks and cellular networks.

Heterogeneous Boolean Networks Boolean networks have been used to model: Gene networks (Shmulevich, Dougherty, & Zhang, 2002) (Schlatter R1, 2009) Cellular networks (Saez-Rodriguez, et al., 2007) (Albert & Othmer, 2003) Gut Bacteria (Steinway, et al., 2015) For example, one research paper used a Boolean network model to looked at how antibiotics effect the interaction of gut bacteria in order to find a way to prevent intestinal infection.

Heterogeneous Boolean Networks Boolean networks have been used to model: Gene networks (Shmulevich, Dougherty, & Zhang, 2002) (Schlatter R1, 2009) Cellular networks (Saez-Rodriguez, et al., 2007) (Albert & Othmer, 2003) Gut Bacteria (Steinway, et al., 2015) Heterogeneous Boolean Network: different nodes use different update rules In many of these applications different nodes use different update rules. We call this a heterogeneous Boolean network.

Heterogeneous Boolean Networks Boolean networks have been used to model: Gene networks (Shmulevich, Dougherty, & Zhang, 2002) (Schlatter R1, 2009) Cellular networks (Saez-Rodriguez, et al., 2007) (Albert & Othmer, 2003) Gut Bacteria (Steinway, et al., 2015) Heterogeneous Boolean Network: different nodes use different update rules Rule 1 Going back to our example about cats and dogs, this might mean that while some people like cats if the majority of people on their social media like cats…

Heterogeneous Boolean Networks Boolean networks have been used to model: Gene networks (Shmulevich, Dougherty, & Zhang, 2002) (Schlatter R1, 2009) Cellular networks (Saez-Rodriguez, et al., 2007) (Albert & Othmer, 2003) Gut Bacteria (Steinway, et al., 2015) Heterogeneous Boolean Network: different nodes use different update rules Rule 1 Rule 2 Others might like cats if at least one person in their social media likes cats.

Heterogeneous Boolean Networks Boolean networks have been used to model: Gene networks (Shmulevich, Dougherty, & Zhang, 2002) (Schlatter R1, 2009) Cellular networks (Saez-Rodriguez, et al., 2007) (Albert & Othmer, 2003) Gut Bacteria (Steinway, et al., 2015) Heterogeneous Boolean Network: different nodes use different update rules Research Question: Is it possible to predict the dynamics of heterogeneous Boolean networks? While heterogeneous networks are being used to analyze these biological networks, very little is know about how these networks behave in general. So in my research I ask the question, could we predict the dynamics of such a network if we know what rules are being used in the network?

Totalistic Rules Does a node turn ON or OFF when: 0 nodes in it neighborhood are ON 1 nodes in its neighborhood are ON 2 nodes in its neighborhood are ON etc. I analyzed heterogeneous networks in which each node uses totalistic rules, which are a special class of Boolean rules. In a totalistic rule a node can turn either ON or OFF when there are zero nodes ON in its neighborhood, ON or OFF when there is one node ON in its neighborhood, ON or OFF when there are two nodes ON in its neighborhood, and so on.

Totalistic Rules Does a node turn ON or OFF when: 0 nodes in it neighborhood are ON 1 nodes in its neighborhood are ON 2 nodes in its neighborhood are ON etc. Example: If 0 OR 2 nodes in its neighborhood are ON Then, the node will turn ON Otherwise, the node will turn OFF For examples, a totalistic rule could be that a node turns ON when there are 0 or 2 nodes ON in its neighborhood. This would be like if a person liked cats if no one else they know likes cats, or if exactly two people they know like cats, and otherwise they like dogs. There can be as many or few of these conditions as you want. (pointing to the 0 or 2 on the power point) In my research its important to note two main categories of totalistic rules.

Totalistic Rules 0 nodes in a neighborhood are ON (0 people on someone’s social media like cat): The two categories will be based on what happens when 0 nodes in a neighborhood are ON.

Totalistic Rules 0 nodes in a neighborhood are ON (0 people on someone’s social media like cat): One category would be all the totalistic rules in which a node turns ON when 0 nodes in its neighborhood are ON. This would be like someone likes cats if 0 people on someone’s social media like cats.

Totalistic Rules 0 nodes in a neighborhood are ON (0 people on someone’s social media like cat): The other category would be all the totalistic rules in which a node turns OFF when 0 nodes in its neighborhood are ON. This would be like if 0 people on someone’s social media like cats they would decide not to like cats. Note that these two categories say nothing about what happens when one, two, three, or more people on someone’s social media like cats, only what happens when no one on someone’s social media like cats. That’s what makes these categories of rules.

Proportion of nodes that are ON I used bifurcation diagrams to determine the long term behavior of these heterogeneous networks. In a bifurcation diagram our vertical axis represents the proportion of nodes that are ON in the network. For example, this would be 20% of the nodes ON (pointing to 0.2 on the vertical axis) and this would be 60% of the nodes ON in the network (pointing to .6 on the vertical axis).

Proportion of nodes that are ON Each of the networks I looked at uses two totalistic rules, out of 52 totalistic rules that I considered. In general we will call the two totalistic rules for a network rule 1 and rule 2. The horizontal axis represents the proportion of nodes that are using each rule. For example, at 0 (pointing to 0 on the horizontal axis) all of the nodes use rule 1, at 0.2 (pointing to it) 20% of the nodes use rule 1 and 80% use rule 2. Now we will use the bifurcation diagram to look at the long term behavior of the network, such as the periodic orbits and chaos I discussed before. All nodes use rules 1 All nodes use rules 2

Proportion of nodes that are ON Here’s part of our bifurcation diagram. Rule 1 is chaotic so when most of the nodes use rule 1, the network is chaotic shown by this block of orange in which the percentage of ON nodes will switch through different values between 0% and 40% and never repeat. All nodes use rules 1 All nodes use rules 2

Proportion of nodes that are ON If more the nodes are using rule 2 we start to see some periodic orbits shown in yellow. Here the percentage of nodes that are ON cycles through 0% ON, 10% ON, 30% ON, 50% ON and then repeats (point to each as I go through them). All nodes use rules 1 All nodes use rules 2

Proportion of nodes that are ON If even more the nodes use rule 2 then we see a period-2 orbit in blue where the percent of nodes ON go back and forth between about 0% and 50% (pointing to them as I say them). All nodes use rules 1 All nodes use rules 2

Proportion of nodes that are ON If more nodes use rule 2 than rule 1 we get back to some chaotic behavior because rule 2 is also chaotic. All nodes use rules 1 All nodes use rules 2

When 0 nodes in a neighborhood are ON: The two rules have the different behavior Rule 1: the node turns ON Rule 2: the node turns OFF Proportion of nodes that are ON This heterogeneous network is made up of two rules that have different behavior when 0 nodes in a neighborhood are ON, one of each of the two categories of rules we just talked about. In fact most of the heterogeneous networks that have one of each type of rule will look like this with a period-2 orbit in the middle. All nodes use rules 1 All nodes use rules 2

Chaos + Chaos ≠ Chaos When 0 nodes in a neighborhood are ON: The two rules have the different behavior Rule 1: the node turns ON Rule 2: the node turns OFF Chaos + Chaos ≠ Chaos Proportion of nodes that are ON One interesting things to note is that if you combine two chaotic rules you may actually get something that is not chaos such as the period 2 orbit seen here. The other interesting thing to note is that this period-2 orbit is almost never seen when both rules have the same behavior for when there are zero nodes ON in a neighborhood. All nodes use rules 1 All nodes use rules 2

When 0 nodes in a neighborhood are ON: The two rules have the same behavior For both rules: the node turns ON OR For both rules: the node turns OFF For example, in this heterogeneous network both rules are from the same category in that both either have a node turn ON or both have a node turn OFF when zero nodes in its neighborhood are ON. Think of both liking cats or both not liking cats when zero people on their social media like cats. In these heterogeneous networks we might see large areas of chaos like the one here.

When 0 nodes in a neighborhood are ON: The two rules have the same behavior For both rules: the node turns ON OR For both rules: the node turns OFF Or, there might be areas of larger periodic orbits such as the period 3 orbit seen here.

When 0 nodes in a neighborhood are ON: The two rules have the same behavior For both rules: the node turns ON OR For both rules: the node turns OFF While there are many examples of these…

When 0 nodes in a neighborhood are ON: The two rules have the same behavior For both rules: the node turns ON OR For both rules: the node turns OFF Almost none have a period-2 orbit as was seen when the two rules have opposite behaviors for what happens when zero nodes in a neighborhood are ON.

Results Rule 1: The node turns ON Rule 2: The node turns OFF Both Rules: The node turns ON This table shows the results for all the heterogeneous networks looked at. Each number along the axis is a different totalistic rule. And each colored square represents the heterogeneous network which is a combination of two of these rules. Example. In blue are all of the heterogeneous networks with period-2 orbits and this is where the rules have different behavior when 0 nodes in the neighborhood are ON. In green are all of the heterogenous networks without a period-2 orbit. These two sections are where the rules have the same behavior when 0 nodes in a neighborhood are ON. Lastly yellow shows all the networks that do not follow the trends seen here, the few that are outliers. Note that the symmetry in the table is due to the fact that it does not matter which of the two rules is rule 1 and which is rule 2. Looking at the clear distinction of the between the areas where we find the green and where we find the blue networks shows how strong the pattern is. Rule 1: The node turns OFF Rule 2: The node turns ON Both Rules: The node turns OFF

Research Contributions Intestinal infections kill good bacteria We use heterogenous Boolean networks to model biological networks like the paper on gut bacteria I described before. In that paper the nodes were the different bacteria and they were either present or not present. Using a heterogenous Boolean model they were able to determine that a certain treatment helped reduce the risk of infection. While we can run these types of models and get results, such as finding a possible way to prevent intestinal inflection, how the different rules in the heterogeneous networks interact, why the network does what it does, is a black box. We just don’t know what makes the network behave one way or another. Will this stop infection? It does stop infection! Boolean Model

Research Contributions It is possible to predict the dynamics of heterogeneous networks. Intestinal infections kill good bacteria My research shows that it is possible to find patterns and predict the dynamics of heterogeneous network. With more research in this area we may one day be able to fully understand this process and know how these networks behave. Will this stop infection? It does stop infection! Boolean Model

Future Research Future Research Questions: What other patterns can be found? Can similar patterns be found with non-totalistic rules Can similar patterns be found when more than two rules are used? In order to do this we need to see what other patterns can be found besides the period-2 orbit found in this research, if similar patterns can be found when the network has more than two rules, and when the network uses non-totalistic rules?

Thank you! Thank you very much for your time! Are there any questions?

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