Challenges in Integrating Genetic Control in Plant and Crop Models

Abstract

Predicting genotype-to-phenotype relationships under contrasting environments is a great challenge for plant biology and breeding. Classical crop models have been developed to predict crop yield or product quality under fluctuating environments but they are usually calibrated for a single genotype, restricting the validity range of the model itself. To overcome this limitation, genetic control has to be integrated into crop models and genotype × environment interactions have to be made explicit.

The aim of this chapter is to provide an overview of a panel of approaches that have been developed to integrate genetic control in plant and crop models. The fundamentals of quantitative genetics of complex traits are first introduced, with special attention to methods for quantitative trait loci (QTL) cartography and QTL genetic modelling. The integration of genetic control within ecophysiological models is then discussed. Several methods are reviewed, ranging from classical statistical approaches, relying on specific model parameters reflecting gene or QTL effects, to recent multi-scale models, explicitly integrating molecular networks. This chapter proposes a review of a few techniques from systems biology that can be used to describe the behaviour of cellular networks, in a simplified way. Coupling different organizational scales is finally discussed and a few examples of multi-scale plant models are presented.

Appendices

Appendix 1: Qualitative Modelling

Qualitative approaches aim to characterize key properties of the dynamics of the system as the existence of steady states, limit cycles (oscillation) but also specific dynamical patterns, like for instance the subsequent activation of two network components. Due to their qualitative nature, these approaches are generally non-deterministic: the result of a qualitative simulation is a state transition graph containing all possible dynamical pathways that are coherent with the initial model definition. In this perspective, model checking tools provide an useful complement, allowing the automated verification of dynamical properties of the system, regardless of the specific trajectory (Monteiro et al. 2007).

In the following we present the basics of three common qualitative models; the interested reader can find relevant references for more information.

1.1.1 Logical and Boolean Model

Variables (usually genes or signaling molecules) are represented as discrete entities that can assume a set of integer values (0 and 1 in the Boolean case, 0,1,2, … in logical models) describing which variable is present and, in the case of logical models, at which concentration (state of the system). The temporal evolution of the system, i.e. the transition between system states, is determined by the interactions among variables. A steady state is reached when the output state is equal to the input one. A combination of logical functions (AND, OR, NOT) or, alternatively, a table of truth can be used to define the interaction rules. In the case of logical models, the interaction between two variables can be submitted to an additional constraint on the concentration of the regulatory variables. So for instance, in the example in Fig. 1.1 gene C is activated by protein B if and only if the latter is present at high concentration (variable level 2). The dynamics of Boolean and logical models (but not their steady states) depends on the specific updating scheme employed. Two main kinds of update are possible, either synchronous or asynchronous. In the synchronous scheme, all nodes are updated simultaneously according to their values at previous time, whereas in the asynchronous update a single node, selected at random, is changed at time, according to the current state of the network. For more information on logical models and their application, see de Jong (2002), Fauré et al. (2006) and Morris et al. (2010).

Fig. 1.1

Example of qualitative modelling of a gene regulatory network using a Boolean, logical and piece-wise linear formalism. (a): example of gene regulatory network. (b): Boolean equations and corresponding table of truth, for network in panel (a). Boolean operators OR, AND and NOT are used to describe the logic of gene interactions. The steady states of the system are indicated with a star. (c): Graph representation of the network in A, where the edges are labelled to express the rank number of the interaction, and corresponding table of truth in the logical formalism. (d): Piece-wise linear equations corresponding to the network in panel A. k_A, k_B and k_C are synthesis rates, γ_A, γ_B and γ_C are degradation rates and θ_A, θ_B, and θ_C are threshold concentrations for proteins A, B and C, respectively

1.1.2 Piece-wise Linear (PL) Model

PL model, introduced by Glass and Kauffman (1973), lies in between ODE and logical models (Davidich and Bornholdt 2008): a continuous step function describes the activation (inhibition) of a gene, whenever the concentration of a regulatory variable x_j is above given threshold ϴ_j (Fig. 1.2).

Fig. 1.2

Piece-wise linear approximation of a sigmoid function into a step function

$$ {s}^{+}\left(xj,\theta j\right)=\Big\{\begin{array}{cc}1& xj>\theta j\\ {}0& xj<\theta j\end{array} $$

$$ {s}^{-}\left(xj,\theta j\right)=1-{s}^{+}\left(xj,\theta j\right) $$

Threshold concentration thus defines a rectangular partition of the phase space, such that in every region not located on a threshold, the PL model reduces to a linear system of differential equations. Moreover, in every such region the derivatives (trends) of the concentration variables have a determinate sign, which is shown to be invariant for rather weak constraints (inequalities) on the parameter values. In a qualitative analysis, each of these regions corresponds to a qualitative state of the network, analogous to the n-tuple of zero and ones of Boolean model s.

PL models are often used to describe gene regulatory networks, based on the observation that gene expression rates are often a sigmoid function of the transcription factor concentration. Step-functions are thus seen as an approximation of this sigmoidal behaviour. For more information on PL formalism and its application to biological systems see Baldazzi et al. (2011) and de Jong et al. (2004).

1.1.3 Petri Net Model

Petri nets contain two kinds of nodes: places and transitions. Places (graphically described as circle) represent the resources or variables of the system (e.g., metabolites in a metabolic network) whereas transitions (rectangles) correspond to events (e.g., reactions) that can change the state of the variables. At any time, each place contains a zero or positive number of tokens (small black dots): when the number of tokens is sufficient, the transition is enabled and the reaction can take place. The firing of an enabled transition results in the consumption of tokens in the input place and the creation of a given number of tokens in the output place, according to edge weight (Fig. 1.3). Due to their structure, Petri nets are usually employed to describe signalling and metabolic networks. Indeed, following a metabolic analogy, the number of tokens in a given place can be associated with metabolite concentrations whereas edge weights correspond to reaction stoichiometry. For a more complete review we refer to Chaouiya (2007).

Fig. 1.3

Firing of a Petri net. At time t, places A and B both contain one token (small black dot), enough to allow the reaction A + B– > 2C to take place, as indicated by the edge weights (small numbers on the arrows). At time t + 1, tokens in A and B have been consumed whereas two tokens have been created in place C

Appendix 2: Constraint-based Models

1.2.1 Metabolic Pathway Analysis

The idea is to study the solution space (flux vector v) of the following system at steady-state:

$$ \begin{array}{l}N\cdot v=0\hfill \\ {}v\in C\hfill \end{array} $$

where N is the stoichiometric matrix and C is the set of constraints. In the simplest version C are thermodynamics constraints, defining irreversible reactions (i.e., vi > 0 for some flux i), but it can also include flux capacity constraints (i.e., v_min < vi < v_max) or experimental data (measured flux values, i.e., vi = mi). In this framework, omics data can be used directly to reduce the search area (in blue in Fig. 1.4).

Fig. 1.4

Schematic representation of Constraint-Based Analysis of metabolic network (inspired by Orth et al. (2010)). In absence of constraints, the flux distribution of a metabolic network can be everywhere in the solution space. Stoichiometry, thermodynamics and limitations to fluxes capacity act as constraints that reduce the allowable space of solutions (blue surface). An optimization principle allows selecting one optimal solution, at one edge of the solution space

Within constraints C, any admissible flux distribution is represented as a nonnegative combination of a valid set of pathways that is (a) unique and (b) minimal, for each network topology. In particular each pathway is “non decomposable”, i.e., it consists of the minimum number of reactions needed to exist as a functional unit. Two alternative definitions of these pathways exist, elementary modes (Stelling et al. 2002) and extreme pathways (Schuster et al. 2000), that can provide slightly different information (see Papin et al. (2004) for a comparison between the two).

1.2.2 Flux Balance Analysis and Related Techniques

Within the solution space defined by the Metabolic Pathway Analysis, Flux Balance Analysis aims at identifying the most likely flux distribution as the ones that maximize a given objective Z, i.e.:

$$ \begin{array}{l}N\cdot v=0\hfill \\ {}v\in C\Big|Z(v) \max \hfill \end{array} $$

Several choices are possible regarding the objective function Z and the formulation of an appropriate one is still subject of research, especially in the case of higher organisms (Schuetz et al. 2007). Among the most popular cost functions in literature are the biomass or ATP productions, growth rate or the production of specific metabolites.

Whatever the objective function, a common problem with FBA is the possible existence of multiple optimal solutions, i.e. several flux distributions with same cost score. In this case the prediction is not unique, and some conclusions may change depending on the selected solution. However, alternative solutions may have a biological meaning: biological systems show a high degree of redundancy that is often associated with a certain functional robustness. Mahadevan and Schilling (2003) explicitly use the existence of multiple optima to investigate redundancies in the network and to derive a flux range in which the optimality is guaranteed. In the case of engineered organisms (e.g., mutants or knock-outs), MOMA technique predicts the expected flux distribution by assuming that it is the closest to the wild type one (principle of minimal modification), irrespectively of its optimality (Segrè et al. 2002).