Yeast pheromone pathway modeling using Petri nets

Problem description

Yeasts are single celled microorganisms in the Fungi kingdom. Saccharomyces cerevisiae a particular species of yeast, has been widely studied in genetics and cell biology. S. cerevisiae has both asexual and sexual reproduction. Sexual reproduction takes place between two haploid cells of opposite types a and α. The process of mating is initiated by secretion of pheromone by one of the cells. Receptors on the opposite cell detect the presence of pheromone and initiates a series of protein-protein interactions within the cell that ultimately might facilitate mating. This series of protein-protein interactions in the cell is known as the yeast pheromone pathway. This pathway is well-studied. We have a working knowledge of how the pathway functions, the different proteins that take part in this pathway and their respective roles. However, several questions still remain unanswered. Our interest lies in one particular question: how does the cell dynamically adapt the pathway to continue mating under severe environmental changes or under mutation (which might result in the loss of functionality of some proteins known to participate in the pheromone pathway).

In our model, the "conditions" mentioned in Question 1 typically refer to the different edge weights between the different components of the Petri net-based pathway model. Different combinations of the values of the edge weights represent different environmental conditions faced by the cell. "Perturbations" mentioned in Question 2 refer to possible methods employed by the cell so that it can mate. We conjecture that one method might be the use of accessory proteins who otherwise are not so prominent in the pheormone pathway. Using appropriate amounts of proteins other than the core pathway component proteins can be a possible compensation method used by the cell to facilitate mating.

We generate a large number of networks and run experiments to identify "conditions" for a positive response. We employ decision trees [1] to analyse the effect of conditions on the pathway. The Petri net-based model gives us a set of conditions that allow us to predict whether the pathway responds positively. It also supports our conjecture about the possible use of other proteins as a compensation process to allow mating by giving positive instances of pheromone response for the networks that simulated the mentioned idea. Finally, we come across several rules or conditions that are highly consistent across all the simulated networks indicating their importance in determining the outcome of the networks.

Petri nets

Petri nets were first proposed by Carl Adam Petri in 1962. Petri nets can be used for describing and modeling dynamic systems that can be characterized as concurrent, asynchronous, distributed, parallel, non-deterministic, and/or stochastic systems. The following is based on the discussion in [2, 3].

A Petri net is a directed weighted bipartite graph with an initial state M 0. The two types of nodes of the bipartite graph are called places and transitions, represented by circles and boxes respectively. There can be arcs from places to transitions as well as from transition to places. The arc weights are positive integers and absence of a weight implies unit weight. A marking is a vector that represents an assignment of a non-negative number of tokens (denoted by dots) in all places in a given Petri net. In a Petri net model of a dynamic system, conditions are represented by places and events by transitions.

Definitions

A Petri net is defined as a 5-tuple π = (P, T, E, W, M₀), where P = {p₁, p₂, .., p _m } denotes a set of places, T = {t₁, t₂, .., t _n } represents a set of transitions, E ⊆ (P × T) ∪ (T × P) defines flow relation in terms of arcs, W : E → {1, 2, 3, ...} is an arc weight function and M 0: P → {0, 1, 2, ...} is the initial marking. It may be noted that the set of places P and the set of transitions T are totally disjoint sets.

Below we define some terminologies related to Petri nets. As stated earlier, a Petri net is a directed graph. A preplace of a transition t, is a place that is adjacent to t. The set of preplaces of t is denoted by pre( t ). Mathematically,

$pre\left(t\right)=\left\{p|\left(p,t\right)\in E\right\}.$

Similarly, a postplace of a transition t, is a place adjacent from t and the set of postplaces of t is denoted by post( t ). Mathematically,

$post\left(t\right)=\left\{p|\left(t,p\right)\in E\right\}.$

The pre-transition and post-transition concepts are defined similarly.

$pre\left(p\right)=\left\{t|\left(t,p\right)\in E\right\}$

and

$post\left(p\right)=\left\{t|\left(p,t\right)\in E\right\}.$

A set of rules defined below control the behavior of a Petri net model for simulating a dynamic system.

1. 1

   Let w(p,t) define the weight of an arc between p and t. We say that a transition t is enabled if each p ∈ pre( t ) has at least w(p,t) tokens.

    
2. 2

   If an event takes place, the corresponding enabled transition will fire otherwise not.

    
3. 3
   Let | p | denote the number of tokens in place p. Let w(t,p) define the weight of an arc between t and p. After a transition t has been fired the tokens will be updated as follows:

   • ∀p ∈ pre(t), |p| = |p| − w(p, t)

   • ∀ṕ ∈ post(t), |ṕ| = |ṕ| + w(t, p)

    

Figure 1 illustrates the workings of a Petri net.

Related work

In this section we survey some of the papers in which a Petri net approach has been used to model biological networks.

Sackmann et al. [4] provide a systemic modeling method of signal transduction pathways in terms of Petri net components. The authors present a process of representing the following three different cases of a signal transduction model.

Case 1: A substance A does not lose its activity by interacting with a second substance B.

Case 2: A substance C triggers several reactions that are independent of each other.

Case 3: A substance changes state from being phosphorylated to being unphos-phorylated and vice versa.

Case 1 indicates phosphorylation reactions between different proteins in a network. Case 2 describes participation of a protein in multiple independent reactions. Both cases are implemented by using read arcs (bidirectional edges between places and transitions) in their Petri net representations. Case 3 indicates the different states of a protein, which is implemented in form of a sub-network. Having described these, the authors propose the following simple steps for representing a signal pathway. First, translate the biological components into logical strucures like conjunction, disjunction, exclusive disjunction and implication. Second, translate the logical structures in corresponding Petri net forms. Finally, assimilate the Petri net components to form a whole network. Our work uses the modeling approach used by this paper [4] and forms the basic structure of our model on the model provided in this paper [4].

Chaouiya [5] provides an overview of the different types of Petri net models available and their uses in modeling different types of biological networks. These include Coloured Petri Net (CPN), Stochastic Petri Net (SPN), Hybrid Petri Nets (HPNs) and Hybrid Function Petri Nets (HFPNs). Hardy and Robillard [6] also discuss the different types of Petri nets extensions used for analysis, modeling and simulation of molecular biology networks. They identify two categories of goals of Petri net biological modeling: qualitative and quantitative analysis. Qualitative analysis is the analysis of the different biological properties while quantitative analysis is the simulation of system dynamics. For quantitative analysis, a Petri net representation with sufficient modeling power should be chosen. For quantitative analysis of a biological system, kinetic parameters like reaction rates and stoichiometric quantities of reactants are necessary. Since no such data are available, we use the basic Petri net structure for our quantitative analysis. In the future, pending availability of data, we plan to upgrade our model to a HFPN or something similar. Monica et al. [7] demonstrate a generalized approach towards modeling and analysis of biological pathways using Petri nets.

Yeast pheromone pathway

In this section, we describe the process of pheromone binding to its receptor on the cell surface and the subsequent effects of that phenomenon on the cell functionality. The summary description below is based on the description from [8, 9]. The yeast mating process is initiated when a yeast cell detects the presence of pheromone secreted by a cell of the opposite sex. There are two cell types in yeast, called a and α that are analogous to egg and sperm cells of animals. The a and α cells can mate to produce an a/α cell. The cell a/α in turn undergoes meiosis to produce the haploid gametes (child cells) a and α cells. The pheromones produced respectively by a and α cells are a-factor and α-factor. An a cell contains the α-factor receptor Ste2 whereas an α cell contains the a-factor receptor Ste3. So a cells can mate with α cells only and vice-versa.

When either Ste2 and Ste3 binds with pheromone, its ability to bind with intracellular G protein complex is compromised. The G protein comprises three subunits known as Gpa1, Ste4 and Ste18. These subunits are commonly referred to as G _α , G _β , and G _γ , respectively. The subunits G _β and G _γ units form a complex G _βγ . If G _α is bound to GDP then G _βγ is bound to G _α . When a pheromone binds to the receptor (Ste2 or Ste3), the receptor interacts with G _α , causing it to replace its GDP with GTP. G _α without its GDP cannot keep the G _βγ complex bound to itself. As a result, the G _βγ complex is liberated and goes on to interact with other proteins. Gradually, hydrolyzation of GTP bound to G _α takes place. G _α then binds back and inhibits the G _βγ complex in absence of pheromone.

The liberated G _βγ complex, activates four protein kinases linked in form of a cascade. Protein Ste5 acts as a scaffold to hold the three other proteins Ste11, Ste7 and Fus3 in place. These three proteins activate each other in series by phosphorylation. So an activated Ste11 phosphorylates Ste7 which becomes active and in turn phosphorylates Fus3. The activated Fus3 then enters the nucleus. The Ste11 at the top of the kinase is activated by a protein Ste20. The protein Ste20 itself becomes activated when it is in the plasma membrane where it is phosphorylated by Cdc42 which is a membrane associated monomeric GTPase.

Activated Fus3 plays an important role in both cell cyle arresting as well as the transcription of genes. Activated Fus3 phosphorylates protein Far1 which blocks the cell cycle in G1 phase, to prepare for mating. Fus3 in the nucleus activates the transciption factor Ste12. Normally, Ste12 is inhibited by proteins Dig1 and Dig2, when pheromone signal is not present. Due to pheromone signalling, activated Fus3 phosphorylates proteins Dig1 and Dig2 which in turn release Ste12. The Ste12 is then free to bind and promote the transcription of a-specific genes (a-sgs) and α-specific genes (α-sgs).

The process of growing projection called a schmoo between cells, is an important feature of mating. The cell surface which faces the highest concentration of pheromone contains the most activated receptors. So the concentration of activated G _βγ is highest here. The G _βγ complex engages proteins for the formation of the shmoo. Far1 engages the proteins Cdc42, Cdc24 and Bem1, to promote schmoo after binding to G _βγ complex. Cdc24 activates Cdc42, which together with Bem1 recruit proteins to promote cell membrane growth such as Bni1 and others. A mating process can succeed or fail. However yeast cells have a mechanism to re-enter the cell cycle using negative feedback loops.

Model

We use Petri nets to model the pheromone response pathway. We represent each protein as a place in the Petri net and each interaction as a transition. Using this representation, the full pathway can be obtained by combining these individual reaction representations. Such a model is already available in the paper by Sackmann et al. [4] We base our model on this avaiable network structure [4] and make several changes to suit our approach. Motivating the first change, we know that the reaction between two or more proteins takes place if the strength of their interaction (kd value) exceeds a certain threshold. A traditional Petri net does not allow one to implement this concept. In our approach we transform the preplaces of all transitions to a single place (colored red in our structure Figure 2), which has inputs from different reactant places. We add a dummy transition to each reactant place. Only for transitions with Ste-type proteins as pre-places are left unchanged. The benefit of having a single pre-place to a transition that originally required several pre-places is that it emphasizes the notion of weighted cumulative concentration of the reactants.

We take these 37 additional proteins and add them to our network structure in the following manner. For each protein i which has j as a neighboring protein, we make i participate in all the reactions in which j is a reactant. In terms of our model, i becomes a preplace to all the post-transitions of j. After adding the additional proteins we add regulatory edges (dashed blue line) in Figure 3 in the network to control the order in which transitions may fire in the network. We define regulatory edges as bidirectional egdes of weight one between a place and a transition which makes sure that the transition cannot fire until that place has at least one token. Bidirectionality ensures that the token content of the place is not affected by the firing of the transition. We illustrate this with the help of Figure 3. The full pathway representation is shown in Figure 2.

Experimental setup

We have developed a Java program that generates instances of the model described in the previous section. Due to the absence of real world data about the kd values for the different reactions in the pathway, we generate all the edge weights in our model randomly. The range of values for the edge weights used in our experiments is between 1 and 100 (extremities included). The places representing the components α-factor, Ste2-receptor, Ste20, Ste5, Fus3, Akr1, Ste11, Ste7, Ste50 and Bem1 were provided with initial concentration values. Let ψ represent the set of these 10 core component proteins. All places representing the additional components were also provided with initial concentration values. Let λ represent the set of all 41 additional protein components in our model. For a given value of concentration of all the proteins in sets ψ and λ, the network is simulated. It is checked whether the transition producing Ste12 has fired or not. If yes, then the pathway has responded successfully and the resultant concentration values of the different proteins are recorded.

Experiments

Interpretation of results

Analysis of experiments