Departamento de Física, Universidade Federal de Santa Maria, Santa Maria (UFSM), RS, Brasil
Universidade Federal do Pampa (UNIPAMPA), São Gabriel, RS, Brasil
Departamento de Eletrônica e Computação, Universidade Federal de Santa Maria, Santa Maria (UFSM), Brasil
Abstract
Background
We introduce a method to analyze the states of regulatory Boolean models that identifies important network states and their biological influence on the global network dynamics. It consists in (1) finding the states of the network that are most frequently visited and (2) the identification of variable and frozen nodes of the network. The method, along with a simulation that includes random features, is applied to the study of stomata closure by abscisic acid (ABA) in
Results
We find that for the case of study, that the dynamics of wild and mutant networks have just two states that are highly visited in their space of states and about a third of all nodes of the wild network are variable while the rest remain frozen in True or False states. This high number of frozen elements explains the low cardinality of the space of states of the wild network. Similar results are observed in the mutant networks. The application of the method allowed us to explain how wild and mutants behave dynamically in the SS and determined an essential feature of the activation of the closure node (representing stomata closure),
Conclusions
For the biological problem analyzed, our method allows determining how wild and mutant networks differ ‘phenotypically’. It shows that the different efficiencies of stomata closure reached among the simulated wild and mutant networks follow from a dynamical behavior of two nodes that are always synchronized. Additionally, we predict that the involvement of the anion efflux at the plasma membrane is crucial for the plant response to ABA.
Availability
The algorithm used in the simulations is available upon request.
Results
In plants transpiration occurs mainly at the leaves while their stomata open for the passage of CO_{2} and O_{2} during photosynthesis. This process is extremely costly for a plant, especially when the water supply is limited. The largest amount of water transpired by a higher plant is lost through the stomata that are controlled by surrounding guard cells that regulate the rate of transpiration. When guard cells become turgid they cause stomata to open allowing water to evaporate. When transpiration exceeds the absorption of water by the roots a loss of turgor occurs and the stomata close balancing the water lost and CO_{2} gain.
Sometimes severe environmental conditions, such as drought, low temperature, heat, high salinity or flooding, have adverse effects on plant growth and development. Plants respond to environmental stresses at cellular and molecular levels, as well as at physiological levels, so as to confer tolerance of the stress and ensure survival. During droughts conditions, the plant synthesizes a hormone called ABA which acts as a signal in a pathway that closes the plant stomata to reduce the water loss by transpiration
We apply our method (see section Methods) to a slightly modified version of the Boolean model of stomata closure regulation by ABA in
Adapted from
Adapted from
In the Figure
We developed a simulation of the wild and 4 different mutant networks studied by the authors
The networks have an initial period that lasts for about 3 or 4 updates where the closure node switches between open (False state) or closed (True state) depending on the initial conditions but it eventually converges to only one of this states (Fig.
Comparison of efficiency among wild and mutant networks of affecting the closure node. Blue circles represent the wild network and yellow, red, green, orange and black represent the ABI1, S1P, pHc, NOS and PA mutants, respectively. Circles represent the percentage of simulations (300 simulations) with the closure node in state True,
Comparison of efficiency among wild and mutant networks of affecting the closure node. Blue circles represent the wild network and yellow, red, green, orange and black represent the ABI1, S1P, pHc, NOS and PA mutants, respectively. Circles represent the percentage of simulations (300 simulations) with the closure node in state True,
The wild network Space of States (SS) Cardinality
During reception, transduction and induction promoted by ABA in Arabidopsis several enzymes, proteins and small molecules are activated or inhibited within the cell. To study the relationship between the elements involved in signaltransduction of ABA, we use an initial random state for the network with the node ABA fixed in the True state meaning that it is being signaled. For illustration of the typical results obtained from the simulation, for a given initial condition and using 100 updates of the network, we find 28 different states in the SS that are shown in Figure
(A) The Space of States Cardinality. Every rectangle represents an element of the SS and the subgroups are formed by the relation of transitivity. For a given initial condition and using 100 updates of the network, we find 28 different states and the different graphs correspond to the number of times and instant of time that the specific state of the network was visited. (B) Histogram of the number of elements in each SS subgroup in Fig.
(A) The Space of States Cardinality. Every rectangle represents an element of the SS and the subgroups are formed by the relation of transitivity. For a given initial condition and using 100 updates of the network, we find 28 different states and the different graphs correspond to the number of times and instant of time that the specific state of the network was visited. (B) Histogram of the number of elements in each SS subgroup in Fig.
Depending on the number of updates and the initial condition, we obtain different cardinalities for the SS. Using 100, 200 and 300 updates and a set of 30 random initial conditions for each we obtain the average and the standard deviation of the cardinality of SS. The values obtained are shown in Table
Updates vs. number of states. Algorithm timings obtained with 32bit version of Mathematica 7.0 running in a personal computer with Intel Xeon X5355 processor and 4Gb of RAM. For the simulations with 100, 200, 300 and 400 updates 30 initial conditions were used except for 500 updates where only initial condition was used given to its very long duration.
Updates
Average number of states
StandardDeviation
Average running time (s)
100
27.9
1.97135
88.3
200
33.5
1.35824
2250
300
34.4333
1.04
17900
400
36
1.01678
68000
500
36

253700
Checking the state of each node for the updates we find that only 16 out of 43 are variable elements, while the remaining 27 are frozen in True or False states. Due to the fast increasing computational time, we had to manage the number of initial conditions and updates used. The variable elements were determined using simulations with 300 different initial conditions and 10 updates, 100 initial conditions with 50 updates and 2 different initial conditions with 800 updates. In each simulation the variable and frozen elements are the same and the results show that the wild network and the mutant networks have different sets of variable elements (see Fig.
The set of 16 variable elements of the wild network (see Fig. 1) arranged in the columns and the result of 10 initial condition are shown in the rows. The numbers in the figure represent the number of times, out of 20 (updates), that a given variable element was found in the True state. Groups of elements with the same number are in the same state. The colors help to distinguish the different groups. The elements in the three last columns (KAP, KEV e Ca^{2+}_{c} ) are not in the same state.
The set of 16 variable elements of the wild network (see Fig. 1) arranged in the columns and the result of 10 initial condition are shown in the rows. The numbers in the figure represent the number of times, out of 20 (updates), that a given variable element was found in the True state. Groups of elements with the same number are in the same state. The colors help to distinguish the different groups. The elements in the three last columns (KAP, KEV e Ca^{2+}_{c} ) are not in the same state.
Considering the results obtained in Fig.
The variable elements of the wild and mutant networks are marked by gray boxes in the table.
The variable elements of the wild and mutant networks are marked by gray boxes in the table.
This is illustrated in Figure
Figure
Analysis of the mutant networks
In their original paper Albert
In addition, we predict the behavior of a new
The behavior of the wild and mutant networks is shown in Figure
Through our simulations we found that it is impossible to have random boolean states for the pair of nodes { AnionEm, closure }. For the mutants PA, S1P and pH_{c} , whose efficiency to set the closure node on state ON is low, this pair alternates between the states {True,True} or {False,False}. For the wild and the mutants NOS and ABI1, whose efficiency to set the closure node on state ON is high, the pair assumes only the states {True,True}. This shows that these two nodes are synchronized in all different networks and we believe that this is an essential component of the dynamics. This observation explains the difference in behavior between the wild and mutant networks and was not observed in the original paper of Albert
The intersection set of variable elements involving the wild and the 5 mutants listed in Figure
{ Ca^{2+}ATPase, Ca^{2+}_{c} , CIS, GCR1, InsP3, KAP, KEV, PLC }
Is clearly seen from Figure
Cardinality of the SS of the networks. The cardinalities were determined by taking the average of 30 simulations during 100 updates with random initial conditions.
Network type
SS Cardinalities
wild
27.9
S1P mutant
26
PA mutant
16
pH_{c} mutant
20
ABI1 mutant
28
NOS mutant
18.5
Conclusions
In this paper we introduced a method of analysis of random boolean models of regulatory networks based on the study of the SS of the network. Its application to a modified version of the model proposed by Albert and coworkers
Methods
Definition: A boolean network consists in a group of
is the state of the
The boolean operations are used as follows: if two or more elements can induce the activation of a node in an independent way, we combine both with the logical function OR, if two or more components cannot induce the activation in an independent way, we associate to both the logical operator AND and finally, the operator NOT will be associated to the inversion of the state of the element.
Our method of analysis consists in finding the states of the network that are most frequently visited and the identification of the variable elements since they show the activated or deactivated network subpathways and their biological influence on the global network behavior. As detailed below.
The algorithm describing the network logics is updated sequentially changing the states of the nodes as a result of their interactions. However, some elements with no prior information about what determine their states are updated with a random boolean state what defines a hybrid synchronous/asynchronous dynamics that is different from that used in reference
the state of the nodes in the update
represents a set of possible states of the network for
To characterize a trajectory we search for equal elements in (1) for each initial condition. The search for these states allows us to establish a relation with the frequency that the state is visited. To find equal states we use the Hamming distance
Another important feature of the dynamics of a boolean network is the general behavior of its nodes. Often, a network breaks apart in two groups of nodes for all initial conditions: in the first group the nodes are always in a frozen state and in the second the nodes are always in a variable state
Authors’ contributions
C. A. Bugs developed the Mathematica code of the simulations, generated the results and wrote the paper. J. C. M. Mombach designed the investigation and wrote the paper. G. R. Librelotto helped with the analysis of the results. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Acknowledgements
We acknowledge partial support from CNPq (500733/20104; 304997/20099) and Fapergs (10/00080). C. A. Bugs thanks the support of Universidade Federal do Pampa.
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