In order to start a new model, choose New project in the File menu. GNA opens a desktop with two components: the project tree and the influence graph window. The project tree lists the variables of the model as well as the initial conditions, atomic propositions and properties, while the Influence graph window summarizes the relations between the variables, thus capturing the regulatory structure of the network. More precisely, an arrow in the influence graph means that the protein at the tail of the arrow influences the synthesis or degradation of the protein at the head of the arrow. In the special case that a protein influences its own synthesis or degradation, the arrow is replaced by an emphasized border. If you start a new model, no variables and initial conditions have been defined yet, and the influence graph is empty. Most of the time, however, you will not build a model from scratch, but start from an earlier model. In that case, you choose Open project... in the File menu to load the earlier model you have decided to use as a template. You may import models from earlier GNA versions (text files with the extension .gna) by choosing Import model... in the File menu. It is also possible to import a model in SBML format, by choosing Import model from SBML... in the same File menu.

The variables in the model represent the concentrations of the proteins encoded by the genes in the network. The variables are created by choosing the option New variable in the Edit menu. You can also press the right button of your mouse when the cursor is on the model item in the Project window. In the remainder of this page, we will omit the shorthand commands for menu options, which are listed in the overview of GNA functions. The creation of a variable is accomplished by specifying its name and leads to the opening of the Variable window. In the case of the example network, three variables are defined: x_a, x_b, and x_c, as shown in Figure 2.2 below. After their creation, variables can be renamed or deleted, by means of the options Rename variable and Delete variable in the Edit menu.

Fig. 2.2.  Project tree and Influence graph window after creation of state variables x_a, x_b, and x_c

Two different types of variables are distinguished: state variables and input variables. While the former vary as a function of the other concentration variables in the model, the latter are determined by factors outside the scope of the system. In addition, GNA assumes that they are constant. In the Variable window, a variable can be defined as an input variable. This hides a number of fields in the Variable window that are irrelevant for input variables, or makes them inaccessible.

The Variable window also allows you to specify a number of parameters. Upon creation of a state or input variable, a zero parameter and a maximum parameter are automatically defined in the parameter tree on the left. The zero and maximum parameters indicate a lower and upper bound for the concentration of the protein, respectively. By selecting these parameters in the parameter tree, the default name assigned by GNA can be changed. For the variable x_c, we define a zero parameter zero_c and a maximum parameter max_c, as shown in Figure 2.3 below.

Fig. 2.3.  Variable window after creation of the parameters for the state variable x_c

The parameter tree may also contain a number of threshold parameters for the variable. A threshold indicates the concentration above or below which a regulatory protein has an influence on the expression of the target gene. As a protein may be involved in several regulatory interactions, it may have several different thresholds The thresholds are created by pressing the New button of the tree menu, which asks the name of the threshold and inserts the newly created threshold into the tree. Once created, a threshold parameter can be deleted or renamed. In the case of x_c, we define thresholds t_c1 and t_c2, arising from the two interactions in which the protein encoded by gene c is involved. Notice that, if two thresholds are equal, they are given the same name.

Finally, if the variable is a state variable, you can specify so-called synthesis and degradation parameters. These indicate the rate of synthesis and degradation, respectively, of the protein whose concentration is represented by the variable. The synthesis and degradation parameters are specified in the parameter tree. The same buttons New and Edit allow the parameters to be created and maintained. Because there must always be at least one degradation parameter, GNA creates one by default. In the example, we introduce two synthesis parameters for x_c, k_c1 and k_c2, representing two possible synthesis rates of protein C, respectively. In addition, we introduce the degradation parameter g_c. The result is shown in Figure 2.3.

Next, still in the Variable window, we add information on the regulatory logic of the synthesis and the degradation of the proteins. That is, we specify state equations, which define the rate of change of the protein concentrations as a balance between the synthesis and the degradation rates of the protein. The state equations are entered in the State equation field, according to the syntax of GNA models specified in the appendix.

The synthesis term of the state equation is a sum of step-function expressions, each weighted by a synthesis parameter. Recall that, in the case of protein C, we defined two synthesis rates, k_c1 and k_c2, respectively. Suppose that the protein is not produced at all, if the concentration of protein C is above its threshold t_c2. That is, for x_c > t_c2, negative autoregulation of the gene sets in. On the other hand, for xc < t_c2, the protein is produced at a rate which is assumed to depend on the concentration of protein A. If the latter concentration is below the threshold t_a2, i.e. x_a < t_a2, then gene c is expressed at a low rate k_c1. However, if x_a > t_a2, then RNA polymerase is stabilized by protein A, and gene c is expressed at a high rate k_c1 + k_c2. The following synthesis term captures this regulatory logic: k_c1 * s-(x_c,t_c2) + k_c2 * s-(x_c,t_c2) * s+(x_a,t_a2), where s-(x_c,t_c2) and s+(x_a,t_a2) are step-function expressions. That is, s-(x_c,t_c2) evaluates to 1, if x_c < t_c2, whereas it evaluates to 0, if x_c > t_c2. The case of s+(x_a,t_a2) goes analogously, while bearing in mind that s+(x_a,t_a2) = 1 – s-(x_a,t_a2).

The degradation term is defined analogously to the synthesis term, except that we demand that an unregulated, spontaneous degradation term is included, in order to ensure that the degradation rate of the protein is always positive. If we only have spontaneous degradation or growth dilution, as in the example of Figure 2.4, then the degradation term is simply a degradation parameter multiplied by the state variable (e.g., g_c * x_c). In case the degradation of the protein is also regulated by other proteins, one or more step-function expressions, each weighted by a degradation parameter, are added to the degradation parameter accounting for spontaneous degradation. The resulting sum is multiplied by the state variable. For example, above a threshold concentration t_b2, protein B might favor degradation of protein C, by binding to it and thus presenting it to the proteolytic machinery. This can be represented by the degradation term (g_c1 + g_c2 * s+(x_b,t_b2)) * x_c, where g_c1 is a low, spontaneous degradation rate, and g_c1 + g_c2 a high, regulated degradation rate.

In general, the step-function expressions in the synthesis and the degradation terms may take many forms, depending on the number of proteins involved and the regulatory logic they implement. Further examples of step-function expressions, sometimes quite complex, can be found in the model of the initiation of sporulation in B. subtilis and the model of the carbon starvation response in E. coli.

The state equation is a differential equation whose righthand-side is the difference of the synthesis and degradation terms. In the example, the state equation for protein C is d/dt x_c = k_c1 * s-(x_c,t_c2) + k_c2 * s-(x_c,t_c2) * s+(x_a,t_a2) – g_c * x_c, as shown in Figure 2.4. Once entered in the State equation field, the syntax and semantics of the state equation can be checked by clicking the Check equation button. It is then verified that the equation satisfies the syntax of GNA models and that it is consistent with the parameter definitions in the other Variable windows.

Fig. 2.4.  Variable window after definition of the state equation for the variable x_c

The simulation method does not require precise numerical values for the threshold, synthesis, and degradation parameters. Instead, inequality constraints on the parameter values are specified to predict the qualitative dynamics of the regulatory system (see the section on running simulations). More precisely, GNA demands the user to order the so-called threshold parameters and focal parameters. We briefly explain the biological meaning of these parameters and their ordering by means of the example.

Protein C has two thresholds, t_c1 and t_c2. Assume that the protein influences the expression of gene b at lower concentrations than it influences the expression of its own gene. This gives rise to the inequality t_c1 < t_c2. By default, zero_c < t_c1 and t_c2 < max_c. The thresholds divide the phase space into regions in which the step functions evaluate to either 0 or 1, so that the state equations there reduce to linear and uncoupled differential equations. Inside a region, each protein concentration monotonically converges towards a so-called focal parameter, given by a quotient of synthesis and degradation parameters of the protein. The state equation above, for instance, implies that in a region in which x_a lies below t_a2, and x_c below t_c2, x_c converges towards the focal parameter k_c2/g_c. The behavior of the system on the threshold planes separating the regions can be determined from the behavior in the adjacent regions, as explained in the publications on the qualitative simulation method.

The possible focal parameters in the different regions of the phase space can be ordered with respect to the threshold parameters. Intuitively speaking, this defines the strength of gene expression in the regions of the phase space in a qualitative way, on the scale of ordered threshold concentrations. In the case of variable x_c, the possible focal parameters are 0, k_c1/g_c, and (k_c1 + k_c2)/g_c. At the highest expression level, corresponding to the focal parameter (k_c1 + k_c2)/g_c, protein C should be able to repress its own synthesis. This implies that t_c2 < (k_c1 + k_c2)/g_c < max_c. In addition, we set t_c1 < k_c1/g_c < t_c2.

The ordering of the threshold and focal parameters, and thus the specification of the inequality constraints, is achieved in the Parameter ordering field. It contains an axis on which the previously defined threshold, zero, and maximum parameters, as well as the focal parameters automatically generated from the state equation are positioned. The elements of the list can be ordered by means of the mouse, by dragging them up and down the axis. Elements can also be moved out of the ordered list, by dropping them left from the zero parameter. Notice that this results in a total order of the threshold and focal parameters, contrary to what was the case in previous versions of GNA (in which focal parameters did not need to be totally ordered with respect to each order). As a consequence, when models prepared with previous versions of GNA (version 6.0 or below) are loaded in GNA 7, the program issues a warning that the parameter order is not total and appends the unordered focal parameters at the left of the zero parameter. Figure 2.5 shows the ordered threshold and focal parameters for the variable x_c.

Fig. 2.5.  Variable window after definition of the inequality constraints for the variable x_c

After you have specified the inequality constraints, the GNA model of the genetic regulatory network is complete. You can save it by means of Save project or Save project as... in the File menu. The project is saved in an XML file with the extension .xml or .gnaml. You may also decide to export the model to a file with the .gna extension (former GNA files) by the means of Export model... in the File menu, or even export it to SBML using Export to SBML....

Examples of complete model files can be consulted in the samples directory of the GNA distribution.