Comparison of gene regulatory networks via steadystate trajectories.  
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PMID: 18309365 Owner: NLM Status: PubMednotMEDLINE 
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The modeling of genetic regulatory networks is becoming increasingly widespread in the study of biological systems. In the abstract, one would prefer quantitatively comprehensive models, such as a differentialequation model, to coarse models; however, in practice, detailed models require more accurate measurements for inference and more computational power to analyze than coarsescale models. It is crucial to address the issue of model complexity in the framework of a basic scientific paradigm: the model should be of minimal complexity to provide the necessary predictive power. Addressing this issue requires a metric by which to compare networks. This paper proposes the use of a classical measure of difference between amplitude distributions for periodic signals to compare two networks according to the differences of their trajectories in the steady state. The metric is applicable to networks with both continuous and discrete values for both time and state, and it possesses the critical property that it allows the comparison of networks of different natures. We demonstrate application of the metric by comparing a continuousvalued reference network against simplified versions obtained via quantization. 
Authors:

Marcel Brun; Seungchan Kim; Woonjung Choi; Edward R Dougherty 
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Type: Journal Article 
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Title: EURASIP journal on bioinformatics & systems biology Volume:  ISSN: 16874145 ISO Abbreviation: EURASIP J Bioinform Syst Biol Publication Date: 2007 
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Created Date: 20080229 Completed Date: 20100628 Revised Date: 20120426 
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Nlm Unique ID: 101263720 Medline TA: EURASIP J Bioinform Syst Biol Country: United States 
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Languages: eng Pagination: 82702 Citation Subset:  
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Computational Biology Division, Translational Genomics Research Institute, Phoenix, AZ 85004, USA. 
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Journal Information Journal ID (nlmta): EURASIP J Bioinform Syst Biol Journal ID (publisherid): BSB ISSN: 16874145 ISSN: 16874153 Publisher: Hindawi Publishing Corporation 
Article Information Download PDF Copyright ? 2007 Marcel Brun et al. openaccess: This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Received Day: 31 Month: 7 Year: 2006 Accepted Day: 24 Month: 2 Year: 2007 Print publication date: Year: 2007 Electronic publication date: Day: 23 Month: 5 Year: 2007 Volume: 2007Elocation ID: 82702 ID: 1950252 PubMed Id: 18309365 DOI: 10.1155/2007/82702 
Comparison of Gene Regulatory Networks via SteadyState Trajectories  
Marcel Brun*^{1}  
Seungchan Kim^{1, 2}a2  
Woonjung Choi^{3}  
Edward R. Dougherty^{1, 4, 5}a4a5  
^{1}Computational Biology Division, Translational Genomics Research Institute, Phoenix, AZ 85004, USA 

^{2}School of Computing and Informatics, Ira A. Fulton School of Engineering, Arizona State University, Tempe, AZ 85287, USA 

^{3}Department of Mathematics and Statistics, College of Liberal Arts and Sciences, Arizona State University, Tempe, AZ 85287, USA 

^{4}Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX 77843, USA 

^{5}Cancer Genomics Laboratory, Department of Pathology, University of Texas M.D. Anderson Cancer Center, Houston, TX 77030, USA 

Correspondence: *Marcel Brun: marcelbrun@yahoo.com [other] Recommended by Ahmed H. Tewfik 
The modeling of genetic regulatory networks (GRNs) is becoming increasingly widespread for gaining insight into the underlying processes of living systems. The computational biology literature abounds in various network modeling approaches, all of which have particular goals, along with their strengths and weaknesses [^{1}, ^{2}]. They may be deterministic or stochastic. Network models have been studied to gain insight into various cellular properties, such as cellular state dynamics and transcriptional regulation [^{3}?^{8}], and to derive intervention strategies based on statespace dynamics [^{9}, ^{10}].
Complexity is a critical issue in the synthesis, analysis, and application of GRNs. In principle, one would prefer the construction and analysis of a quantitatively comprehensive model such as a differential equationbased model to a coarsely quantized discrete model; however, in practice, the situation does not always suffice to support such a model. Quantitatively detailed (finescale) models require significantly more complex mathematics and computational power for analysis and more accurate measurements for inference than coarsescale models. The network complexity issue has similarities with the issue of classifier complexity [^{11}]. One must decide whether to use a finescale or coarsescale model [^{12}]. The issue should be addressed in the framework of the standard engineering paradigm: the model should be of minimal complexity to solve the problem at hand.
To quantify network approximation and reduction, one would like a metric to compare networks. For instance, it may be beneficial for computational or inferential purposes to approximate a system by a discrete model instead of a continuous model. The goodness of the approximation is measured by a metric and the precise formulation of the properties will depend on the chosen metric.
Comparison of GRN models needs to be based on salient aspects of the models. One study used the L_{1} norm between the steadystate distributions of different networks in the context of the reduction of probabilistic Boolean networks [^{13}]. Another study compared networks based on their topologies, that is, connectivity graphs [^{14}]. This method suffers from the fact that networks with the same topology may possess very different dynamic behaviors. A third study involved a comprehensive comparison of continuous models based on their inferential power, prediction power, robustness, and consistency in the framework of simulations, where a network is used to generate gene expression data, which is then used to reconstruct the network [^{15}]. A key drawback of most approaches is that the comparison is applicable only to networks with similar representations; it is difficult to compare networks of different natures, for instance, a differentialequation model to a Boolean model. A salient property of the metric proposed in this study is that it can compare networks of different natures in both value and time.
We propose a metric to compare deterministic GRNs via their steadystate behaviors. This is a reasonable approach because in the absence of external intervention, a cell operates mainly in its steady state, which characterizes its phenotype, that is, cell cycle, disease, cell differentiation, and so forth. [^{16}?^{19}]. A cell's phenotypic status is maintained through a variety of regulatory mechanisms. Disruption of this tight steadystate regulation may lead to an abnormal cellular status, for example, cancer. Studying steadystate behavior of a cellular system and its disruption can provide significant insight into cellular regulatory mechanisms underlying disease development.
We first introduce a metric to compare GRNs based on their steadystate behaviors, discuss its characteristics, and treat the empirical estimation of the metric. Then we provide a detailed application to quantization utilizing the mathematical framework of reference and projected networks. We close with some remarks on the efficacy of the proposed metric.
In this section, we construct the distance metric between networks using a bottomup approach. Following a description of how trajectories are decomposed into their transient and steadystate parts, we define a metric between two periodic or constant functions and then extend this definition to a more general family of functions that can be decomposed between transient and steadystate parts.
Given the understanding that biological networks exhibit steadystate behavior, we confine ourselves to networks exhibiting steadystate behavior. Moreover, since a cell uses nutrients such as amino acids and nucleotides in cytoplasm to synthesize various molecular components, that is, RNAs and proteins [^{18}], and since there are only limited supplies of nutrients available, the amount of molecules present in a cell is bounded. Thus, the existence of steadystate behavior implies that each individual gene trajectory can be modeled as a bounded function f (t) that can be decomposed into a transient trajectory plus a steadystate trajectory:
(1)
f(t)=ftran(t)+fss(t), 
The limit condition on the transient part of the trajectory indicates that for large values of t, the trajectory is very close to its steadystate part. This can be expressed in the following manner: for any ? > 0, there exists a time t_{ss} such that  f (t) ? f_{ss} (t)  < ? for t > t_{ss}. This property is useful to identify f_{ss} (t) from simulated data by finding an instant t_{ss} such that f (t) is almost periodical or constant for t > t_{ss}.
A deterministic gene regulatory network, whether it is represented by a set of differential equations or state transition equations, produces different dynamic behaviors, depending on the starting point. If ? is a network with N genes and x_{0} is an initial state, then its trajectory,
(2)
f(?,x0)(t)={f(?,x0)(1)(t),?,f(?,x0)(N)(t)}, 
f(?,x0)(i)(t) 
The decomposition of (1) applies to f_{(?, x0)} (t) via its application to the individual trajectories
f(?,x0)(i)(t) 
Different metrics have been proposed to compare two realvalued trajectories f (t) and g (t), including the correlation
?f,g? 
Let f_{ss} (t) and g_{ss} (t) be two measurable functions that are either periodical or constant, representing the steadystate parts of two functions, f (t) and g (t), respectively. Our goal is to define a metric (distance) between them by using the amplitude cumulative distribution (ACD), which measures the probability density of a function [^{20}].
If f_{ss} (t) is periodic with period t_{p} > 0, its cumulative density function F (x) over ? is defined by
(3)
F(x)=?(?(x)tp), 
(4)
?(x)={ts?t<tefss(t)?x}, 
If f_{ss} is constant, given by f_{ss} (t) = a for any t, then we define F (x) as a unit step function located at x = a. Figure 1 shows an example of some periodical functions and their amplitude cumulative distributions.
Given two steadystate trajectories, f_{ss} (t) and g_{ss} (t), and their respective amplitude cumulative distributions, F (x) and G (x), we define the distance between f_{ss} and g_{ss} as the distance between the distributions
(5)
dss(fss,gss)=?F?G? 
??? 
(6)
dL?(f,g)=sup?0?x??F(x)?G(x), 
(7)
dL1(f,g)=?0?x<?F(x)?G(x)dx. 
The L_{1} norm is well suited to the steadystate behavior because in the case of constant functions f (t) = a and g (t) = b, their distributions are unit steps functions at x = a and x = b, respectively, so that
dL1 
dL1 
Once a distance between their steadystate trajectories is defined, we can extend this distance to two trajectories f (t) and g (t) by
(8)
dtr(f,g)=dss(fss,gss), 
The next step is to define the distance between two multivariate trajectories f (t) and g (t) by
(9)
dtr(f,g)=1N?i=1Ndtr(f(i),g(i)), 
(10)
dtr(f,h)?dtr(f,g)+dtr(g,h) 
The last step is to define the metric between two networks as the expected distance between the trajectories over all possible initial states. For networks ?_{1} and ?_{2}, we define
(11)
d(?1,?2)=ES[dtr(f(?1,x0),f(?2,x0))], 
The use of a metric, in particular, the triangle inequality, is essential for the problem of estimating complex networks by using simpler models. This is akin to the pattern recognition problem of estimating a complex classifier via a constrained classifier to mitigate the data requirement. In this situation, there is a complex model that represents a broad family of networks and a simpler model that represents a smaller class of networks. Given a reference network from the complex model and a sampled trajectory from it, we want to estimate the optimal constrained network. We can identify the optimal constrained network, that is, projected network, as the one that best approximates the complex one, and the goal of the inference process should be to obtain a network close to the optimal constrained network. Let ? be a reference network (e.g., a continuousvalued ODEbased network), let P (?) be the optimal constrained network (e.g., a discretevalued network), and let
?? 
(12)
d(??,?)?d(??,P(?))+d(P(?),?), 

is the overall distance and quantifies the approximation of the reference network by the estimated optimal constrained network;d(??,?) 
is the estimation distance for the constrained network and quantifies the inference of the optimal constrained network;d(??,p(?))  d (P (?), ?) is the projection distance and quantifies how well the optimal constrained network approximates the reference network.
This structure is analogous to the classical constrained regression problem, where constraints are used to facilitate better inference via reduction of the estimation error (so long as this reduction exceeds the projection error) [^{11}]. In the case of networks, the constraint problem becomes one of finding a projection mapping for models representing biological processes for which the loss defined by d (P (?), ?) may be maintained within manageable bounds so that with good inference techniques, the estimation error defined by
d(??,P(?)) 
The amplitude cumulative distribution of a trajectory can be estimated by simulating the trajectory and then estimating the ACD from the trajectory. Assuming that the steadystate trajectory f_{ss} (t) is periodic with period t_{p}, we can analyze f_{ss} (t) between two points, t_{s} and t_{e} = t_{s} + t_{p}. For a continuous function f_{ss} (t), we assume that any amplitude value x is visited only a finite number of times by f_{ss} (t) in a period t_{s} ? t < t_{e}. In accordance with (3), we define the cumulative distribution
(13)
F(x)=?({ts?t?tefss(t)?x})tp. 
(14)
Sx={ts?t?tefss(t)=x}?{ts,te}. 
n=Sx 
(15)
mi=fss(ti+ti+12). 
Let I_{x} be a set of the indices of points t_{i} such that the function f (t) is below x in the interval [t_{i}, t_{i + 1}],
(16)
Ix={0?i?n?2mi?x}. 
(17)
F(x)=?i?Ix(ti+1?ti)tp. 
(18)
F?(x)={1?i?mai?x}m 
In the case of computing the distance between two functions f (t) and g (t), where the only information available consists of two samples, {a_{1}, ? , a_{m}} and {b_{1}, ? , b_{r}}, for f and g, respectively, both cumulative distributions
F?(x) 
G?(x) 
(19)
S={a1,?,am}?{b1,?,br}. 
(20)
d?L?(f,g)=max?0?i?kF?(si)?G?(si) 
(21)
d?L1(f,g)=?0?i?k?1(si+1?si)F?(si)?G?(si). 
To illustrate application of the network metric, we will analyze how different degrees of quantization affect model accuracy. Quantization is an important issue in network modeling because it is imperative to balance the desire for fine description against the need for reduced complexity for both inference and computation. Since it is difficult, if not impossible, to directly evaluate the goodness of a model against a real biological system, we will study the problem using a standard engineering approach. First, an in numero reference network model or system is formulated. Then, a second network model with a different level of abstraction is introduced to approximate the reference system. The objective is to investigate how different levels of abstraction, quantization levels in this study, impact the accuracy of the model prediction. The first model is called the reference model. From it, reference networks will be instantiated with appropriate sets of model parameters. The model will be continuousvalued to approximate the reference system at its fullest closeness. The second model is called a projected model, and projected networks will be instantiated from it. This model will be a discretevalued model at a given different level of quantization.
The ability of a projected network, an instance of the projected model, to approximate a reference network, an instance of the reference model, can be evaluated by comparing the trajectories generated from each network with different initial states and computing the distances between the networks as given by (11).
The origin of our reference model is a differentialequation model that quantitatively represents transcription, translation, cisregulation and chemical reactions [^{7}, ^{15}, ^{21}]. Specifically, we consider a differentialequation model that approximates the process of transcription and translation for a set of genes and their associated proteins (as illustrated in Figure 3) [^{7}].The model comprises the following differential equations:
(22)
dpi(t)dt=?iri(t??p,i)??ipi(t),????i??,dri(t)dt=?ici(t??r,i)??iri(t),????i??,ci(t)=?i[pj(t??c,j),j??i],????i??, 
??i 
?,??i:??i?? 
(23)
?i[pj,j??i]=[1??j??i+?(pj,Sij,?ij)]??j??i??(pj,Sij,?ij),?(p,S,?)=1(1+?p)S, 
A discretetime model results from the preceding continuoustime model by discretizing the time t on intervals n?t, and the assumption that the fraction of DNA fragments committed to transcription and concentration of mRNA remains constant in the time interval [t ? ?t, t) [^{7}]. In place of the differential equations for r_{i}, p_{i}, and c_{i}, at time t = n?t, we have the equations
(24)
ri(n)=e??i?tri(n?1)+?is(?i,?t)ci(n?nr,i?1),pi(n)=e??i?tpi(n?1)+?is(?i,?t)ri(n?np,i?1),ci(n)=?i[pj(n?nc,j),j??i],????i??, 
(25)
s(x,y)=1?e?xyx. 
We generate networks using this model and a fixed set ? of parameters. We call these networks reference networks. A reference network is identified by its set ? of parameters,
(26)
?=(?1,?1,?1,?1,?1,?p,1,?r,1,?c,1,?1,?1,?,?N,?N,?N,?N,?N,?p,N,?r,N,?c,N,?N,?N). 
The next step is to reduce the reference network model to a projected network model. This is accomplished by applying constraints in the reference model. The application of constraints modifies the original model, thereby obtaining a simpler one. We focus on quantization of the gene expression levels (which are continuousvalued in the reference model) via uniform quantization, which is defined by a finite or denumerable set ? of intervals,
L1=[0,?x),L2=[?x,2?x),?,Li=[(i?1)?x,i?x),?, 
??:???? 
?(x)=ai 
The equations for r_{i}, p_{i}, and c_{i}(24) are replaced by
(27)
r?i(n)=?(e??i?tr?i(n?1)+?is(?i,?t)c?i(n?nr,i?1)), 
(28)
p?i(n)=?(e??i?tp?i(n?1)+?is(?i,?t)r?i(n?np,i?1)), 
(29)
c?i(n)=?i[p?j(n?nc,j),?j??i],????i??. 
Issues to be investigated include (1) how different quantization techniques (specification of the partition ?) affect the quality of the model; (2) which quantization technique (mapping ?) is the best for the model; and (3) the similarity of the attractors of the dynamical system defined by (27) and (28) to the steady state of the original system, as a function of ?_{x}. We consider the first issue.
To illustrate the proposed metric in the framework of the reference and projected models, we compare two networks based on a hypothetical metabolic pathway. We first briefly describe the hypothetical metabolic pathway with necessary biochemical parameters to set up a reference system. Then, the simulation study shows the impacts of various quantization levels in both time and trajectory based on the proposed metric.
We consider a gene regulatory network consisting of four genes. A graphical representation of the system is depicted in Figure 4, where ? denotes an activator and ? denotes a repressor. We assume that the GRN regulates a hypothetical pathway, which metabolizes an input substrate to an output product. This is done by means of enzymes whose transcriptional control is regulated by the protein produced from gene 3. Moreover, we assume that the effect of a higher input substrate concentration is to increase the transcription rate ?_{1}, whereas the effect of a lower substrate concentration is to reduce ?_{1}. Unless otherwise specified, the parameters are assumed to be geneindependent. These parameters are summarized in Table 1.
We assume that each cisregulator is controlled by one module with four binding sites, and set S = 4, ? = 10^{8} M^{?1}, ?_{2} = ?_{3} = ?_{4} = 0.05 pMs^{?1}, and ? = 0.05 s^{?1}. The value of the affinity constant ? corresponds to a binding free energy of ?U = ?11.35 kcal/mol at temperature T = 310.15?K (or 37?C). The values of the transcription rates ?_{2}, ?_{3}, and ?_{4} correspond to transcriptional machinery that, on the average, produces one mRNA molecule every 8 seconds. This value turns out to be typical for yeast cells [^{22}]. We also assume that on the average, the volume of each cell in 𝒞 equals 4 pL [^{18}]. The translation rate ? is taken to be 10fold larger than the rate of 0.3/minute for translation initiation observed in vitro using a semipurified rabbit reticulocyte system [^{23}].
The degradation parameters ? and ? are specified by means of the mRNA and protein halflife parameters ? and ?, respectively, which satisfy
(30)
e???=12,??????e???=12. 
(31)
?=ln?2?,???????=ln?2?. 
It is expected that the finer the quantization is (smaller values of ?_{x}), the more similar will be the projected networks to the reference networks. This similarity should be reflected by the trajectories as measured by the proposed metric. A straightforward simulation consists of the design of a reference network, the design of a projected network (for some value of ?_{x}), the generation of several trajectories for both networks from randomly selected starting points, and the computation of the average distance between trajectories, using (9) and (21). Each process is repeated for different time intervals ?t to study how the time intervals used in the simulation affect the analysis.
The first simulation is based on the same 4gene model presented in [^{7}]. We use 6 different quantization levels, ?_{x} = 0, 0.001, 0.01, 0.1, 1, and 10, where ?_{x} = 0 means no quantization, and designates the reference network. For each quantization level ?_{x} and starting point x_{0}, we generate the simulated time series expression and compare it to the timeseries generated with ?_{x} = 0 (the reference network), estimating the proposed metric using (21). The process is repeated using a total of 10 different time intervals, ?_{t} = 1 second, 5 seconds, 10 seconds, 30 seconds, 1 minute, 2 minutes, 5 minutes, 10 minutes, 30 minutes, and 1 hour. The simulation is repeated and the distances are averaged for 30 different starting points x_{0}.
Figures 5 and 6 show the trajectories and empirical cumulative density functions estimated from the simulated system as illustrated in the previous section. Several quantization levels are used in the simulation. The last graph in Figure 5 shows the mRNA concentration for the forth gene, over the 10 000 first seconds (transient) and over the last 10 000 seconds (steadystate). We can see that for quantizations 0 and 0.001, the steadystate solutions are periodic, and for quantizations 0.001 and 0.1, the solutions are constant. This is reflected by the associated plot of F (x) in Figure 6.
Figure 7 shows how strong quantization (high values of ?_{x}) yields high distance, with the distance decreasing again when the time interval (?_{t}) increases. The zaxis in the figure represents the distance
d?L1(f(?x,?t),f(?x=0,?t)) 
In our second simulation, we use a different connectivity (all other kinetic parameters are unchanged), and we again use 10 different time intervals, ?_{t} = 1 second, 5 seconds, 10 seconds, 30 seconds, 1 minute, 2 minutes, 5 minutes, 10 minutes, 30 minutes and 1 hour, and 6 different quantization levels, ?_{x} = 0, 0.001, 0.01, 0.1, 1, and 10. (?_{x} = 0 meaning no quantization). The simulation is repeated and the distances are averaged for 30 different starting points. Analogous to the first simulation, Figure 9 shows how strong quantization (high values of ?_{x}) yields high distance, which decreases when the time interval (?_{t}) increases.
An important observation regarding Figures 8 and 10 is that the error decreases as ?_{t} increases. This is due to the fact that the coarser the amplitude quantization is, the more difficult it is for small time intervals to capture the dynamics of slowly changing sequences.
This study has proposed a metric to quantitatively compare two networks and has demonstrated the utility of the metric via a simulation study involving different quantizations of the reference network. A key property of the proposed metric is that it allows comparison of networks of different natures. It also takes into consideration differences in the steadystate behavior and is invariant under time shifting and scaling. The metric can be used for various purposes besides quantization issues. Possibilities include the generation of a projected network from a reference network by removing proteins from the equations and connectivity reduction by removing edges in the connectivity matrix.
The metric facilitates systematic study of the ability of discrete dynamical models, such as Boolean networks, to approximately represent more complex models, such as differentialequation models. This can be particularly important in the framework of network inference, where the parameters for projected models can be inferred from the reference model, either analytically or via synthetic data generated via simulation of the reference model. Then, given the reference and projected models, the metric can be used to determine the level of abstraction that provides the best inference; given the amount of observations available, this approach corresponds to classificationrule constraint for classifier inference in pattern recognition.
t:  Time 
?:  Network 
x_{0}:  Starting Point 
f (t), g (t), h (t):  Trajectories 
f_{ss}, g_{ss}:  SteadyState trajectories 
f_{?,x0} (t):  Trajectory 
f_{tran}:  Transient part of the trajectory 
f_{ss}:  Steadystate part of the trajectory 
F (x), G (x), H (x):  Cumulative distribution functions 
d_{tr}(?,?):  Distance between two trajectories 
d_{ss}(?,?):  Distance between two periodic or constant trajectories 
? (A):  Lebesgue measure of set A 
f (t):  Multivariate trajectory 
We would like to thank the National Science Foundation (CCF0514644) and the National Cancer Institute (R01 CA104620) for sponsoring in part this research.
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