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IEEE/ACM Trans Comput Biol Bioinform. Author manuscript; available in PMC 2012 September 1.

Published in final edited form as:

PMCID: PMC3211072

NIHMSID: NIHMS330289

Department of Chemistry and Biochemistry, 4254 Urey Hall, UCSD, 9500 Gilman Drive, La Jolla, CA 92093-0359

The publisher's final edited version of this article is available at IEEE/ACM Trans Comput Biol Bioinform

See other articles in PMC that cite the published article.

Deciphering the biological networks underlying complex phenotypic traits, e.g., human disease is undoubtedly crucial to understand the underlying molecular mechanisms and to develop effective therapeutics. Due to the network complexity and the relatively small number of available experiments, data-driven modeling is a great challenge for deducing the functions of genes/ proteins in the network and in phenotype formation. We propose a novel knowledge-driven systems biology method that utilizes qualitative knowledge to construct a Dynamic Bayesian network (DBN) to represent the biological network underlying a specific phenotype. Edges in this network depict physical interactions between genes and/or proteins. A qualitative knowledge model first translates typical molecular interactions into constraints when resolving the DBN structure and parameters. Therefore, the uncertainty of the network is restricted to a subset of models which are consistent with the qualitative knowledge. All models satisfying the constraints are considered as candidates for the underlying network. These consistent models are used to perform quantitative inference. By in silico inference, we can predict phenotypic traits upon genetic interventions and perturbing in the network. We applied our method to analyze the puzzling mechanism of breast cancer cell proliferation network and we accurately predicted cancer cell growth rate upon manipulating (anti)cancerous marker genes/proteins.

The topology and the dynamic realization of genetic networks often play a dominant role in phenotype formation. In order to understand the cause of a disease and/or develop effective therapeutics, it is important to understand the function and regulation of the underlying biological network. In recent years, studies on this problem have been focused on the data-based (reverse engineering) approaches, i.e., modeling a biological network from the experimental data and *prior knowledge* by machine learning algorithms, such as learning a genetic regulatory network (GRN) from microarray data using a Bayesian network.

Friedman et al. were the first to use Bayesian networks in identifying a regulatory network structure [17]. In this work, a best Bayesian model structure is learned from gene expression data by maximizing its posterior probability based on the data. The ability of the model to reproduce certain known regulatory interactions is validated against real experiments and the BN model can also predict new regulatory relationships. In general, Bayesian network inference uses two kinds of cost functions, i.e., BIC and BDe score, to learn the BN structure. Following this line, plenty of works have been proposed to learn the genetic regulatory networks and protein-protein interaction networks by analyzing various data resources, such as gene expression data, ChIP-chip data, and protein expression data, etc. [15], [42], [44]. In addition, time-dependent gene activities and their relationships can be inferred from microarray time series data using dynamic Bayesian networks [29], [18]. Other works have recruited data integration schemes to combine different kinds of data together with the prior knowledge into the learning task [22], [36], [50], [51]. Moreover, some works have been proposed to deal with reconstructing the genetic regulatory network with hidden factors and missing observations [2], [14]. All of the above methods and applications inevitably encounter a similar problem, i.e., there are not enough data samples for learning the network structure with given dimensionalities.

In reality, due to the relatively small amount of experimental data available compared to the size of the genetic regulatory network, the learned network often contains a small number of reliable (confident) edges. In addition, the conventional Bayesian network cannot capture cyclic structures in real biological systems, which often results in inaccuracy and/or error. Algorithms of learning cyclic structures from microarray data with dynamic Bayesian network have been proposed [35]. However, these algorithms often need a large amount of data in time series, which is not necessarily available. Moreover, biological networks consist of various interactions, such as protein-protein and protein-DNA interactions. Due to the variation of the techniques used to generate these data, discrepancies between experiments and various types of data often make the data-driven approach difficult.

However, there are plenty of qualitative statements in the literature. For example, TGF*β* stimulates tumor invasion and metastasis. This statement indicates a direct functional relationship, stimulation, between a cytokine, TGF*β* and phenotypes, tumor invasion and metastasis. Such a qualitative statement lacks quantitative information, e.g., how strongly does TGF*β* regulate those phenotypes and its reliability is dependent on the biological experiments that supported it. Nevertheless, the statement is a concrete conclusion supported by various evidences obtained from different experimental measurements including microarray, ELISA, and northern blot experiments. A qualitative statement is often a summary of the most prominent and consistent observations across multiple studies and it should be thusly treated as the most confident information in modeling the underlying biological network. Other links which are less reliable than qualitative statements emphasized in the literatures may be erroneously captured (false positives) in the learned Bayesian model using the data-based reverse engineering approach.

Consequently, it is very important in systems biology to develop methods based on highly confident qualitative statements in the literature (no quantitative experimental data are involved) to establish a genetic network for a specific phenotype (e.g., cancer). In such networks, vertices indicate cellular molecules at multiple levels, such as proteins and RNA molecules. Direct edges from any node(s) to other node(s) in the network represent direct functional regulations from the parental node(s) to the child node(s). Given this genetic network, it is imperative to parameterize its structure. We can thusly use it to interrogate new genetic programs and discover new knowledge about this network and its associated biological phenotypes.

Unfortunately, a major hurdle in developing this knowledge-based approach is the lack of quantitative parameterization information (in qualitative statements) that is crucial for performing quantitative inference. Thus, the problem boils down to constructing parameters from the qualitative statements and encoding this parameter and structure information into a mathematical model for quantitative manipulations. We proposed in this paper, a knowledge-based predictive framework for modeling the recurrent genetic networks based on dynamic Bayesian networks given qualitative knowledge and our model can then perform quantitative inference.

(Dynamic) Bayesian networks (DBNs) are a popular class of graphical probabilistic models which are motivated by Bayes’ theorem [1]. A DBN represents a joint probability distribution over a set of variables. Once known, this joint distribution can be used to calculate the probabilities of any configuration of the variables. In Bayesian probabilistic inference, the conditional probabilities for the values of a set of unconstrained variables are calculated given fixed values of another set of variables, called observations or evidence. Bayesian models have been widely used for efficient probabilistic inference and reasoning [32], [37]. Numerous algorithms for learning the Bayesian network structure and parameters from data have been proposed [23], [24], [16]. However, as we have discussed above, although the maximum a posteriori approximation, i.e., the selection of a single Bayesian network model from the data by learning, is useful for the case of large data sets, independence assumptions among the network variables often make this single model vulnerable to overfitting. In realistic problems, the data basis is often very sparse and hardly sufficient to select one adequate model, i.e., there is considerable model uncertainty. Selecting a single Bayesian model can then lead to strongly biased inference results.

Besides Bayesian networks, other state-of-the-art statistical and deterministic methods have been proposed to infer the genetic regulatory network from the data. These methods can analyze the full range of the behaviors and dynamics of a system under different conditions. (Probabilistic) Boolean networks were initially used to analyze the network stability in the yeast transcriptional regulatory network [27] and to study the dynamics of cell cycle regulation in yeast [33]. Boolean networks can provide important insights in terms of the existence and nature of network steady states and robustness. However, a Boolean network is largely limited by its level of modeling details and computational expense to analyze the dynamics of large networks, as the number of global states is exponential in the number of entities [26]. Petri net is used to analyze the transition sequence of a network from a global state to another. Moreover, Petri net is used to analyze the dynamics of a regulatory network and large-scale metabolic networks [7], [31], [39], [30]. Modulo network module is introduced to infer the regulation logic of gene modules given gene expression data. A regulation logic is represented by a decision tree, in which a path from the root to a leaf is determined by the up or downregulation of regulatory modules, and a leaf determines the expression level of the corresponding genes. Module networks were tested with experimental data and correctly predicted some regulatory modules [43]. Other successful model can predict the genetic regulatory network based on mutual information [34].

As discussed above, a quantitative data set is a sole resource for all these conventional methods. Therefore, these methods’ performance are inevitably limited to the availability and quality of the data. In particular, the performance of these methods will be severely undermined in any of the following cases: 1) the data contain few samples (comparing to number of predictors/features/random variables of the system); 2) the data are contaminated by relatively high-level noise; 3) the data contain no functional measurements. In our method, we try to model the genetic regulatory network structure and parameters and to predict the system behavior based on solely priori qualitative statements. On the contrary, a qualitative knowledge about a physical interaction is usually evaluated by a combination of direct binding and functional regulation experiments. The qualitative knowledge thusly provide a high-confident landscape of the network structure. The major advantage of our proposed method is that we avoid the usage of noisy data yet to construct a confident network structure.

In this paper, we recruit a qualitative knowledge model [5] to map major types of genetic interactions, i.e., 1) transcription factor-DNA regulations and 2) protein-protein interactions, to set a group of constraints over the structure and parameter space of the dynamic Bayesian network. In particular, the qualitative properties of the statements are dealt with by transforming the fuzziness of these statements into a set of prior joint probability distributions over the nodes in the dynamic Bayesian network. The genetic networks are restricted to a subset of models that are consistent with a body of qualitative knowledge. All dynamic Bayesian models satisfying the constraints over the joint probability space are considered as a candidate for the underlying biological network. In this way, we take model uncertainty into account instead of basing our prediction on a single “best” model. With full Bayesian approach, i.e., model averaging, this class of consistent models is used to perform quantitative inference which can be approximated by Monte Carlo methods. This knowledge-based quantitative Bayesian network modeling algorithm preserves the actual network topology derived from knowledge and is able to capture both “correlation” (joint probability) and “causal/influence” (conditional probability) relations in the Bayesian network. When we combine qualitative statements from various studies, statements targeting the same genetic interaction may be inconsistent. In this case, they can be integrated into a unified representation by calculating a priori distribution over the statements [5].

In summary, our method demonstrates that we can achieve good predictions on the biological network behaviors given qualitative statements without any quantitative data. In Section 2, we present the quantitative inference methods with a dynamic Bayesian model based on a set of qualitative statements. In Section 3, we apply our framework to model the cell proliferation network in normal and cancerous breast cells and also predict cell growth given regulatory interventions to the network. Conclusions are made in Section 4.

In this section, we suggest a way to use qualitative relational statements for inference in the Bayesian framework. We proceed from the general equation for Bayesian inference based on data and knowledge, followed by a detailed recipe to transform knowledge, represented by a set of qualitative statements, into an a priori distribution for models.

A Bayesian model *m* represents the joint probability distribution of a set of variables X = X_{1}; X_{2}; … ; X_{D} [24]. The model is defined by a graph structure *s*, which defines the structures of the conditional independence between variables, and a parameter vector *θ*, the components of which define the entries of the corresponding joint probability tables (CPTs). Hence, a Bayesian network can be written as *m* = {*s, θ*}. If we believe that one single model *m* reflects the true underlying distribution, we can perform inference based on this model. Given some observations or “evidence” *E*, reflected by fixed measured values of a subset of variables, X* _{e}* =

$$\begin{array}{l}Pr(X{X}_{e})=\frac{{\sum}_{h}P(X={X}_{q},{X}_{e},{X}_{h})}{{\sum}_{h}qP({X}_{q},{X}_{e},{X}_{h})}& =Pr(XE,m),\end{array}$$

(1)

which can be efficiently evaluated with known methods [35].

The full Bayesian framework does not attempt to approximate one true underlying distribution. Instead, all available information is used in an optimal way to perform inference, without taking one single model for granted. To formalize this statement for our purposes, let us classify the set of available information into an available set of data, *D*, and a body of nonnumeric knowledge, Ω. The a posteriori distribution of models *m* is then given by

$$Pr(mD,\mathrm{\Omega})=\frac{Pr(Dm)\phantom{\rule{0.16667em}{0ex}}Pr(m\mathrm{\Omega})Pr(D,\mathrm{\Omega}).}{}$$

(2)

The first term in the numerator of (2) is the likelihood of the data given the model, which is not directly affected by nonnumeric knowledge Ω, the second term denotes the model prior, whose task is to reflect the background knowledge. We obtain

$$Pr(mD,\mathrm{\Omega})=\frac{1}{Z}Pr(Dm)Pr(m\mathrm{\Omega}),$$

(3)

where *Z* is a normalization factor which will be omitted from the equations for simplicity. The first term contains the constraints of the model space by the data, and the second term the constraints imposed by the background knowledge. In the full Bayesian approach, we can perform inference by model averaging. Now, given some observation or evidence E, the (averaged) conditional distribution of the remaining variable X is performed by integrating over the models:

$$\begin{array}{l}Pr(XE,D,\mathrm{\Omega})={\int}_{m}Pr(XE,m)Pr(mD,\mathrm{\Omega})dm={\int}_{m}Pr(XE,m)Pr(Dm)Pr(m\mathrm{\Omega})dm.\end{array}$$

(4)

In this paper, we consider the extreme case of no available quantitative data, *D* = Even in this case, it is still possible to perform proper Bayesian inference,

$$Pr(XE,\mathrm{\Omega})={\int}_{m}Pr(XE,m)Pr(m\mathrm{\Omega})dm.$$

(5)

Now the inference is based on the general background information contained in Ω alone, and the specific information provided by the measurements *E*. This is reflected by the fact that inference results are conditioned on both quantities in (5).

In order to determine *Pr*(*m*|Ω), we need a formalism to translate a body of qualitative knowledge into an a priori distribution over Bayesian models. For this, we adopt the following notation for a Bayesian model class. A Bayesian model is determined by a graph structure *s* and by the parameter vector *θ* needed to specify the conditional probability distributions given that structure. We refer to *θ* as one specific joint configuration. A Bayesian model class is then given by 1) a discrete set of model structures = {*s*_{1},*s*_{2}, …, *s _{K}*}, and 2) for each structure

$$\begin{array}{l}Pr(m\mathrm{\Omega})=Pr({s}_{k},\theta \mathrm{\Omega})=\frac{Pr(\theta {s}_{k},\mathrm{\Omega})Pr({s}_{k}\mathrm{\Omega}){\sum}_{a=1}^{K}{\int}_{{\mathrm{\Theta}}_{a}}Pr(\theta {s}_{a},\mathrm{\Omega})d\theta Pr({s}_{a}\mathrm{\Omega}).}{}\end{array}$$

(6)

According to (6), in order to construct a prior distribution of the models, we use each statement to constrain the model space to the subspace which is consistent with that statement. In other words, if a statement describes a relationship between two variables, only structures *s _{k}* which contain the corresponding edge are assigned a nonzero probability

$$Pr(XE,\mathrm{\Omega})=\sum _{k=1}^{K}{\int}_{{\mathrm{\Theta}}_{k}}Pr(XE,{s}_{k},\theta )Pr({s}_{k},\theta \mathrm{\Omega})d\theta .$$

(7)

As we utilize the nonnumeric knowledge in terms of qualitative statements about a relationship between biological molecules in a cell, we hypothesize that Ω contains a list of qualitative statements on cellular molecular interactions. In this form, the information can be used in a convenient way to determine the model prior, (6): 1) Each entity which is referenced in at least one statement throughout the list is assigned to one variable *X _{i}*. 2) Each relationship between a pair of variables constrains the likelihood of an edge between these variables being present. 3) The quality of that statement (e.g., “activates” and “inactivates”) affects the distribution over the local conditional probability distributions (CPDs) entries given the local structures. The joint probability

In this section, we give the detail receipt of probabilistic inference in DBN with only qualitative statements. Please note that, from now on, *m* stands for a Dynamic Bayesian network [35] in stead of a static Bayesian network (BN). A dynamic Bayesian network [11] is a way to extend Bayes nets to model probability distributions over semiinfinite collections of random variables. We only consider discrete-time processes, and we increase the index t by one every time. The variable probability is exclusively dependent on its last step parents state which makes a DBN Markovian process. The conventional Bayesian networks is acyclic graph. Thus, we use DBNs to model the recurrent molecule interactions in cells. An example of DBN is shown in Fig. 1a and it can be unrolled into a series of 2-Time-Slice Bayesian Networks (2TBNs) over time as shown in Fig. 1b. Each 2TBN is a conventional Bayesian model with time-invariant interslice structure *s* and joint probability distributions *θ* over the nodes at time *t* and (*t –* 1), i.e., *P*(_{t}, _{t}_{–1}). If time starts from t = 0, the joint probability of a DBN over time *T* is

$$P({\overline{X}}_{0},\dots ,{\overline{X}}_{T})=P({\overline{X}}_{0})\underset{P({\overline{X}}_{t}{\overline{X}}_{t-1}).}{\overset{}{t=1T}}$$

(8)

For *t*th 2TBN, the joint probability of at time *t*, i.e., *P*(* _{t}*) can be written as

$$P({\overline{X}}_{t})=P({X}_{1,t},\dots ,{X}_{N,t})={\int}_{{\overline{X}}_{t-1}}P({\overline{X}}_{t}{\overline{X}}_{t-1})P({\overline{X}}_{t-1}),$$

(9)

where *X _{n,t}* denotes the

$$\begin{array}{l}P({X}_{n,t})={\int}_{\pi ({X}_{n})}P({X}_{n}\pi ({X}_{n}))P(\pi ({X}_{n}))d\pi ({X}_{n})& =\sum _{j=1}^{J}{\theta}_{j}{P}_{j}^{t-1}(\pi ({X}_{n})),\end{array}$$

(10)

where *θ _{j}* denotes the

$$\begin{array}{l}P({X}_{n}E,\mathrm{\Omega})=\sum _{k=1}^{K}{\int}_{{\mathrm{\Theta}}_{k}}P({X}_{n}E,{s}_{k},\theta )P({s}_{k},\theta \mathrm{\Omega})d\theta \approx \frac{1}{K}\sum _{{m}_{k}}P({X}_{n}E,{m}_{k}).\end{array}$$

(11)

Only the structure which is consistent with the hypotheses is assigned with nonzero probability *P*(*s _{k}*|Ω). Likewise, only parameter values on that structure, which are consistent with the contents of the hypotheses, are assigned a nonzero probability

$$P({X}_{n,t}E,{s}_{k},0)=\sum _{j=1}^{J}{\theta}_{k,j}{P}_{j}^{t-1}(\pi ({X}_{n}),E),$$

(12)

where *θ _{k;j}* represents the

$$P({X}_{n,t}E,\mathrm{\Omega})=\sum _{k=1}^{K}{\int}_{\mathrm{\Theta}}\sum _{j=1}^{J}{\theta}_{k,j}{P}_{j}^{t-1}(\pi ({X}_{n}),E)P(s,{\theta}_{k,j}\mathrm{\Omega})d\mathrm{\Theta}.$$

(13)

The inference in (13) can be calculated for each model *m _{k}* = (

In this paper, we recruit the qualitative knowledge model in [6] to translate the “causal”(causal-like) qualitative statements into constraints over conditional probability distribution. In addition, we define here a novel form of inequality to translate the “correlate”(correlate-like) qualitative statements into constraints over joint probability distribution. The joint probability distribution of a DBN and BN (*θ* in (6) and (7)) can be decomposed into a product of local conditional probabilities of each child node given its parental nodes in the network, i.e., *Pr*(*X _{n}, π*(

If a child node B has a neighbor node(s) A and these nodes impose isolated bidirectional influences from one to the other, then qualitative influence between this node and its neighbor node(s) is referred to as *Correlate influence*. Correlate influence can be further classified into positive correlation and negative correlation.

If presence/absence of one node A or B renders the presence/absence of neighbor node(s) B or A more likely than only one of them is present and the other is absent, then these nodes is said to have a *positive correlate influence* on each other. This can be represented by the inequality

$$Pr(A,B),Pr(\overline{A},\overline{B})\ge Pr(\overline{A},B),Pr(A,\overline{B}).$$

(14)

If presence/absence of one node A or B renders the presence/absence of neighbor node(s) B or A less likely than only one of them is present and the other is absent, then these nodes is said to have a *negative correlate influence* on each other. This can be represented by the inequality

$$Pr(A,B),Pr(\overline{A},\overline{B})\le Pr(\overline{A},B),Pr(A,\overline{B}).$$

(15)

In summary, we demonstrate our algorithm of utilizing solely qualitative knowledge to generate quantitative probabilistic inference with Bayesian networks in Fig. 2.

In the following paragraphs, we will demonstrate how to make use of the qualitative knowledge model to construct the parameter distributions and network structures from qualitative statements. In the first example, if two entities A and B are positively correlated to each other, inequality constraints in (14) can be imposed over the joint probability distribution of A and B. Monte Carlo sampling can be used to draw the valid joint probability samples which are consistent with the constraints as shown in Fig. 3a. In the second example, entity A activates B (single positive influence in [5], *P*(*B*|*A*) ≥ *P*(*B*|*Ā*)), then the valid conditional probability samples based on these “causal” statements is shown in Fig. 3b.

We apply our method to investigate the underpinnings of the mammary epithelial carcinoma cell proliferation and to predict the breast cancerous cell growth upon genetic interventions.

As the first step of our experiment, we model the core genetic network of breast cancer cell proliferation program by collecting qualitative knowledge on the direct regulations. In our model, each node represents either a gene or a protein and each edge describes either a physical transcriptional binding or a protein-protein interaction. In addition, an artificial node is added to denote the phenotype, i.e., cell proliferation. Several cell cycle regulating proteins in the signaling pathway(s) are determined to be the direct cause of cell proliferation in breast cancer. Therefore, we could link these proteins to this phenotype.

Second, we transform this network into a mathematically manipulatable representation with a directed cyclic structure. For this purpose, we utilize dynamic Bayesian model which encodes these physical interactions into a (cyclic) directed graph with the CPD.

Last, we perform two sets of in silico intervention simulation in various human breast cancer as well as normal mammalian cell lines. Specifically, we both knockdown and overexpress two cell cycle regulators: cyclinD1 and TGF*β*. Upon interference, our method (in Section 2) quantitatively predicts the level of cell proliferation in these breast cancer cell lines. In the first prediction, we evaluate the cell growth rate in MCF-7 breast cancer cell in the case that cyclinD1 is interfered. In the second simulation, we predict the cell proliferation rate in three distinct human mammary cell lines: 1) human normal mammary cell line MCF-10A; 2) human breast cancer cell line: MDA-MB-231; and 3) RAS/ErbB2-transfected breast carcinoma cell line: MCF-10A(Ras/ErbB2).

Deregulated cell cycle activity is one of the main cause for cancer development, such as breast carcinomas. The cell cycle is controlled by a set of regulators which forms a complex system including the cyclins, cyclin-dependent kinases (CDKs), and CDK inhibitors. The progression through the cell cycle requires sequential activation and inactivation of these modulators. The complex signaling networks and pathways mediated by cytokines and hormones can influence cell proliferation in positive or negative ways and activates a cascade of intracellular biochemical events and is ultimately responsible for the biological phenotypic observations.

Among these pathways, signaling from transforming growth factor *β* (TGF*β*) plays key roles in regulation of a wide variety of biological end points from early embryonic patterning to the control of cell differentiation and growth in adult cells. TGF*β* can activate antiproliferation gene responses in G1 phase and impede the completion of the ongoing cell cycle. TGF*β* responses in human epithelial cell lines from skin, lung, and mammary gland originals have revealed a shared proliferation program. This program consists of cyclins and cyclin-dependent kinases. Meanwhile, the cell proliferation is realized by the sequential activation of CDK inhibitors and repression of the growth promoting transcription factor c-MYC.

Besides the cell growth regulating signals mediated by cytokine TGF*β*, Ras genes are the most common targets for mutations in human breast cancer. Ras protein is activated in response to a very wide spectrum of extracellular stimuli, such as cytokines, growth factors, hormones, neurotransmitters, and extracellular matrix components. These factor stimulation leads to a rapid and transient increase of active Ras-GTP [20]. Ras signal transduction involves the passage of information along a chain of proteins which exert control over a host of signal transduction pathways. The best characterized Ras signal transduction pathway is Ras-MAPK cascade. Ras activates the Raf-1 kinase, which, in turn, activates the MEK-ERK kinase cascade. ERK phosphorylates transcription factors in cytosol and nucleus which leads to activation of the genes involved in cell proliferation.

In this study, we include the compact representation of TGF*β* and Ras signaling pathways in the cell proliferation network (Fig. 4). The signal transduction along these two pathways are extracted from a collection of qualitative knowledge in the publications (see Appendix). TGF*β* represses the growth promoter c-MYC and activates the cell growth inhibitor p15 of the INK4-family and p21, p27 of the Cip1/Waf1/Kip1-2-family. Ras influences the transcription of cyclin and CDK genes through the Raf/MEK/ERK cascade. It is also known that many intermediate factors in the cytosol, e.g., smad proteins and other coexpressors, e.g., E2F4/5, p38, p53, p107, p300, ID1, ID2, and Miz-1 coregulates the cell proliferation [48] which can be integrated into our later studies consistently [4]. The loopy circuit in the network serves to providing tight and robust control to this program.

The cell proliferation network of mammalian epithelial cell can be modeled by a DBN as shown in Fig. 4a. The DBN can be unrolled over the time into a series of 2TBNs as shown in Fig. 4b. The parameters are described by the joint probability distribution (Fig. 5). In [6], a qualitative knowledge model is proposed to define a set of inequality constraints over the conditional probability space. In this study, we project this set of conditional-space constraints onto the joint probability space where we transform the conditional probability into a set of joint probabilities.

The parameters in the dynamic Bayesian network of mammary cell proliferation program are denoted by joint probability tables of child nodes and their parents. The joint condition probability is listed in the rightmost column of each table. The secondary **...**

The conditional probability distribution (*a*_{0}*; a*_{1}) transformed from *α* can be modeled by *Single Negative Influence*, i.e., *a*_{0} ≥ *a*_{1}, where
${a}_{0}={\scriptstyle \frac{{\alpha}_{0}}{{\alpha}_{0}+{\alpha}_{2}}}$ and
${a}_{1}={\scriptstyle \frac{{\alpha}_{1}}{{\alpha}_{1}+{\alpha}_{3}}}$ Similarly, the conditional probability distribution transformed from *σ* can be modeled by by *Single Positive Influence* and the conditional probability distribution (b,r) transformed from (*β, γ*) can be described by *Mixed Joint Influence*, i.e.,

$$\begin{array}{cccc}{b}_{0}\ge {b}_{1}& {b}_{2}\ge {b}_{3}& {b}_{2}\ge {b}_{0}& {b}_{3}\ge {b}_{1}\\ {r}_{0}\ge {r}_{1}& {r}_{2}\ge {r}_{3}& {r}_{2}\ge {r}_{0}& {r}_{3}\ge {r}_{1},\end{array}$$

(16)

where the condition probability entries in (16) can be described by joint probability table. For example,
${b}_{0}=Pr(p15\overline{\mathit{TGF}\beta},\overline{\mathit{cMYC}})={\scriptstyle \frac{{\beta}_{1}}{{\beta}_{0}+{\beta}_{1}}}$. The conditional probability distribution transformed from (*ρ*) can be modeled by *Plain Synergy with Positive Individual Influence*, i.e.,

$${p}_{7}\ge \left\{\begin{array}{c}{p}_{3}\\ {p}_{5}\\ {p}_{6}\end{array}\right\}\ge \left\{\begin{array}{c}{p}_{1}\\ {p}_{2}\\ {p}_{4}\end{array}\right\}\ge {p}_{0},$$

(17)

where *p*_{0} is the probability of breast cancer cell growth given its parents being underexpressed, thus,
${p}_{0}={\scriptstyle \frac{{\rho}_{1}}{{\rho}_{1}+{\rho}_{0}}}$. λ can be defined similarly. The parameters (*θ*, *Ø, η*) can be defined by a set of constraints hierarchically. First, the conditional probabilities transformed from these parameters can be modeled by *Mixed Joint Influence* since there are multiple input signals to activate and to repress the complexes *CyclinD-CDK46*, *CyclinE-CDK2,* and *p27-CyclinD-CDK46* from their parents. Therefore, the parameters can be classified according to the number of repressors being overexpressed. Second, these probabilities in each class can be further defined by *Plain Synergy with Positive Individual Influence* based on the number of parental activators being overexpressed. For example, the conditional probability (g) based on the parameter *3* can be first classified into four classes of parameters based on the configuration of p21 and p27, i.e., *G*_{0} = {*g*_{0}, *g*_{4}, *g*_{8}, *g*_{12}}, *G*_{1;1} = {*g*_{1}, *g*_{5}, *g*_{9}, *g*_{13}}, *G*_{1;2} = {*g*_{2}, *g*_{6}, *g*_{10}, *g*_{14}}, and *G*_{2} = {*g*_{3}, *g*_{7}, *g*_{11}, *g*_{15}}. If we assume the inhibitive effects of p21 and p27 on *CyclinD-CDK46* are symmetric (In general, it is possible to model the unsymmetrical effects given specific knowledge), we could merge the parameters of *G*_{1;1} and *G*_{1;2} into one class, *G*_{1}. With the same configuration of the parents, the parameters across the classes can be constrained as

$$\begin{array}{cccc}{g}_{0}\ge {g}_{1,2}& {g}_{4}\ge {g}_{5,6}& {g}_{8}\ge {g}_{9,10}& {g}_{12}\ge {g}_{13,14}\\ {g}_{1,2}\ge {g}_{3}& {g}_{5,6}\ge {g}_{4}& {g}_{9,10}\ge {g}_{11}& {g}_{13,14}\ge {g}_{15}.\end{array}$$

(18)

Second, within each class, the parameters can be further classified by the number of activators being overexpressed as

$$\begin{array}{ccc}{g}_{4,8}\ge {g}_{0}& {g}_{5,6,9,10}\ge {g}_{1,2}& {g}_{7,11}\ge {g}_{3}\\ {g}_{12}\ge {g}_{4,8}& {g}_{13,14}\ge {g}_{5,6,9,10}& {g}_{15}\ge {g}_{7,11},\end{array}$$

(19)

where the condition probability is defined by the network parameters *η*. Similarly, *Ø* and *θ* can be modeled as (18) and (19).

Besides this basic knowledge, further constraints can be added to regulate the sensitivity and specificity in parameter space. In Fig. 4a, p21 and p15’s activities are completely blocked by c-MYC (see the Appendix). We can confine those corresponding conditional parameters close to zero, i.e., ${\scriptstyle \frac{{\beta}_{6}}{{\beta}_{6}+{\beta}_{2}}}=0,{\scriptstyle \frac{{\beta}_{7}}{{\beta}_{7}+{\beta}_{3}}}=0,{\scriptstyle \frac{{\beta}_{6}}{{\beta}_{6}+{\beta}_{2}}}=0$ and ${\scriptstyle \frac{{\beta}_{7}}{{\beta}_{7}+{\beta}_{3}}}=0$.

The inference on cell growth is computed for each possible model with parameters Π = {*α, σ, γ, ρ, ø, β, η, θ, λ, τ, μ*} by (13). However, since the parameter space is rather high-dimensional, we can use Monte Carlo method to approximate the integration. For each parameter in Π, its distribution in the parameter space is defined by a set of constraints as in (16) to (19). By using Monte Carlo Accept-Reject method, we simulate K = 500,000 CPT samples and together with the structure *s* in Fig. 4a define a consistent model class, * =* {*m _{k}*(

After we build up the DBN model of cell proliferation network in mammalian epithelial cells, we can interrogate this network with quantitative in silico interventional simulation and gain insight on the functional mechanism of cell proliferation program in breast cancer cells.

In [13], the relationship of cyclinD1 and breast cancer cell progression is investigated. Transcription factors regulating the cyclinD1 gene(CCND1) are silenced by transfecting the MCF-7 breast cancer cells with RNA interfering (RNAi) vectors. The RNAi against two transcription factors, FoxA1 and NFIC, significantly increases and reduces the mRNA and protein level of cyclinD1 in the cell nucleus. In addition, the effects of estrogen (estradiol) on CCND1’s expression level is explored in combination with the RNAi against CCND1’s transcription factors. In the experiment, real-time PCR assays determined the mRNA level of CCND1 in MCF-7 under each case of the RNAi interference with or without estradiol. The cell proliferation efficiency is evaluated as percentage change of cell number relative to the control. The control is indicated by RNA interference against luciferase gene in MCF-7 cell.

Therefore, in the first prediction, we like to predict the breast cancer lineage MCF-7’s cell proliferation efficiency in each case of RNAi interference to CCND1’s transcription factors. To this end, we first determine the expression levels of several key cell progression regulators, i.e., TGF*β*, Ras protein, cyclinD1, cyclinE, and CDKs in MCF-7 cells. It is known that TGF*β* is a cell growth inhibitor. Ras, cyclinD,E, and CDKs are cell growth promoters. For example, a breast cell line bearing high levels of cyclinD,E will show proactive cell proliferation than a cell with low levels of these regulators. These regulators do not have any parent node(s) in the cell proliferation network (Fig. 4a), therefore, their levels eventually determine the cell state. To define the cell state, we collect the level of these regulators in MCF-7 cells from a set of publications (please see legend of Table 3). In Table 3, the expression level of each regulators is indicated by a set of six symbols. We transform this discrete set of symbols into probability value by equally dividing [0, 1] into six intervals. We set the initial probability of these regulators in MCF-7 cell according to the transformed probability value which is indicated by the value in bracket in Table 3. In [13], cyclinD1 is interfered to five different levels by RNAi silencing experiments. The ratio between cyclinD1’s level and the control level is evaluated. We transfer these ratio values into probability and then we clamp the level of cyclinD1 to these probabilities as evidence (E) in our interfering prediction (13). The ratio is transformed to the fold change which is calculated as *F* = log_{2} *R* and this log-value indicates the expression level of cyclinD1 in each case. The probability of cyclinD1 equals to 1 if the fold change is greater than 2 and 0 if the fold change is less than −1. Given these initial settings, we can predict the probability of cell proliferation in each interference experiment by (13). The experimental cell proliferation efficiency is measured as the relative cell number changes to the control. Thus, we calculate the predicted cell growth efficiency by referring the change in cell proliferation probability to the probability in the control case. The probability change is assessed between the interfered case and the control case. The ratio, fold change, initial probability of cyclinD1, cell proliferation probability, and cell proliferation efficiency are summarized in Table 2.

Protein Expression Levels of the Key Cell Cycle Regulators in Mammalian Normal Epithelial and Cancerous Cells

The prediction and experiment observations in each interference study is shown in Fig. 6a. Since the TGF*β* level is unknown in MCF-7, we simulated three cases where the TGF*β* level varies from low (0.3) to high (0.8). The correlation between prediction and experimental observations is shown in Fig. 6b. The Pearson correlation coefficient equals to 0.9681. We can see that the relative cell proliferation efficiency in case of cyclinD1 overexpression and knockdown is relatively independent on the level of TGF*β*, i.e., TGF*β* in the MCF-7 cell state, exhibits weak inhibitions on the cell growth. CyclinD1’s changes dominate the TGF*β*’s changes. This observation may be partially due to that cyclinD1 is a cell growth promoter in nucleus and is located at the very downstream of signaling pathways.

In [8], it has been identified that the loss of TGF*β* growth inhibition often occurs without a loss of receptors and/or smad cofactors in breast cancer. Instead, the repression of a key cell growth promotion component c-MYC is selectively lost. In breast cancer cell line, MDA-MB-231 and MCF-10A(Ras/ErbB2), this repressive response of c-MYC to the cytokine TGF*β* is lost [8]. However, the normal breast cell line, MCF-10A reserves this repression. In addition, in MCF-10A(Ras/ErbB2) cell, Ras protein is overexpressed. In the experiment, the concentration of TGF*β* is manipulated to different levels from 0 to 100 (*μ*m) in these cell cultures and cell proliferation efficiency is measured.

In the second prediction, we predict the inhibition of cell growth by TGF*β* in these three breast cancer cell lines. Similar to the first experiment, we first determine the probability of the key cell progression regulators in MCF-10A, MCF-10A(Ras/ErbB2), and MDA-MB-231 cells according to Table 3. The level of these regulators defines the distinct cell context of these three cell lines. MCF-10A is a normal human mammary epithelial cell line. MDA-MB-231 and MCF-10A(Ras/ErbB2) are the human immortal cell lines showing aggressive cell progression. We set the initial probability of these regulators in these cell lines according to the transformed probability value which is indicated by the value in bracket in Table 3.

The level of TGF*β* is similarly discretized into probability with log-transformation as cyclinD1. In addition, the TGF*β*’s repression on c-MYC is lost in MCF-10A(Ras/ErbB2) and MDA-MB-231 cell lines. Thus, the level of c-MYC in these two cell lines are set to 1 according to the experiment [8]. The experimental cell proliferation efficiency is measured as the ratio between cell number given specific TGF*β*’s concentration and the control. The control case of each cell line is indicated by TGF*β* knockout, i.e., TGF*β* = 0 [8]. Thus, we calculate the predicted cell growth efficiency by referring the cell proliferation probability to the probability in the control case. The concentration, fold change, initial probability of TGF*β*, predicted cell proliferation probability, and cell proliferation efficiency are summarized in Table 1.

The Concentration (C), Fold Change (F), Probability of TGF*β*(PT), Probability of Cell Growth (CG), Predicted Cell Growth Efficiency (Eff), and Experimental Cell Growth Efficiency (Exp) is Listed

The prediction on the cell growth efficiency in each cell line is depicted by solid line in Fig. 6c. For comparison, the experimental observation is adopted from [8] and shown as dashed line in Fig. 6c. The correlation between simulation results and experimental observation is shown in Fig. 6d and the pearson correlation equals to 0.978. We observe that the cell proliferation ratio in the two breast cancer cell lines, MCF-10A(Ras/ErbB2) and MDA-MB-231 is much higher than that of the human normal mammary epithelial cell line MCF-10A at any TGF*β*’s level. This may be due to the difference in cell state determined by the key cell cycle regulators. In addition, depending on different cell state, TGF*β* inhibits the cell proliferation distinctly. In the normal epithelial cell line MCF-10A, TGF*β* exhibits prominent inhibition effect on cell proliferation while in the two breast carcinoma cell lines TGF*β* exerts relative faint inhibitions on cell growth.

Several state-of-the-art statistical and deterministic methods have been used to infer the genetic regulatory network from the data. These methods include but are not limited to (Probabilistic) Boolean networks, Petri nets, Modulo networks and mutual-information-based model, and Bayesian networks. Besides their mathematical differences, they encounter a common problem: learning performance and generalization accuracy of these methods are subject to the availability and the quality of the observation data. The performance of these methods will be severely undermined (with slight variance) in any of the following cases: 1) the data contain few samples (comparing to number of predictors/ features/random variables of the system); 2) the data are contaminated by a relatively high-level noise; 3) the data contain no functional measurements.

Comparatively, we propose in this paper, a knowledge-driven systems biology approach using dynamic Bayesian network based solely on knowledge. In this approach, the model structure and parameters are exclusively determined by qualitative knowledge. Given a collection of physical interactions between genes and proteins without parameter details, we can construct not only the structure of the genetic regulatory network, but also automatically recover the parameter distribution to parameterize the network. Consequently, quantitative generalization and prediction can be generated by a model averaging scheme upon genetic interventions. In this study, the prediction of breast cancer phenotypes closely matches the experimental observations.

There are several significant advantages in our proposed knowledge-based framework over the state-of-the-art data-based methods. First, knowledge describing the same biological interactions is often redundant across literatures and knowledge contains qualitative information other than quantitative measurements. These properties of the knowledge help rule out incorrect and noisy information. Second, knowledge can usually identify direct binding and functional regulation through in vitro and in vivo small-scale biochemistry experiments, such as chromo immunoprecipitation, vector reporter, etc. However, the large-scale high-throughput data either cannot provide information on in-/ direct interactions, e.g., mRNA microarray data, or cannot provide functional regulatory information, such as ChIP-chip or ChIP-seq data. Not to mention that in any data format, the information is buried under considerable noise. Third, the constructed network complexity and system dimensionality is solely subject to the amount of knowledge available. The direct benefit is that we need not be concerned about the insufficient statistics in any given set of knowledge since the system dimensionality is limited by the information in the knowledge set. Thus, we do not have the “dimension curse” and “overfitting” problem. Finally, our method has a good scaling ability. By compiling pieces of knowledge together, we can eventually construct a network with an infinite number of variables. Because each piece of knowledge is describing a local structure of a whole network, i.e., substructure of the network, we can obtain the complete network by assembling these substructures together. In addition, the parameters of the whole network are decomposable into a product of local parameters, namely, parameters can be determined locally at each nodes given their parents. No heuristic search for model structures is invoked and the problem remains solvable in linear time.

Like any method, there are inherent limitations with our proposed knowledge-based method. These limitations largely stem from the nature of qualitative knowledge. First, knowledge can be inconsistent, i.e., there can be contradicting information regarding to the same genetic regulation(s). In this case, the entropy of the qualitative knowledge and the dynamic Bayesian model uncertainty is increased. We sort to model the distribution of the inconsistent information and to integrate the knowledge entropy into our proposed method. Following this line, a solution is proposed in [5]. Second, knowledge is usually incomplete. Though it is important to interrogate the system behavior by assembling existing knowledge into a whole network, the existing knowledge does not disclose any new information regarding to the interactions. Knowledge is useful to produce a high-confident network, instead, data-based methods do a good job at discovering new links between molecules but the outcome is noisy and edges are low confident (as discussed in the last paragraph). Therefore, it is natural to combine the two strengths into one framework to recover new links with high confident as well as to generate accurate predictions on the system behavior. Third, we can improve our modeling complexity to cope with more forms of knowledge. Currently, we use basic forms of linear inequality constraints to model “activating” and “repressing” genetic regulations. We can further expand our inequality to incorporate ratios, difference, and boundary features of the parameters. We can even employ nonlinear inequality to model the relations between the parameters. This new set of features is very useful in modeling qualitative knowledge such as a transcription factor A increases the expression level of gene B several times.

One possible reason for the achieved prediction power by our method is that the complex of genetic regulatory network structure with feed-forward and feed-backward regulations ensured the parametric robustness, i.e., the regulatory network behavior becomes robust against parameter configurations under certain structure complexes. It seems that nature tends to preserve certain redundant functional feed-forward and feed-backward links in the biological network so that its functions become robust over insignificant disruptions to the network kinetics. We observed feedback links and feed-forward links in our breast cancer cell growth network, in addition, this hypothesis has been well observed and confirmed in other studies in various organisms [33], [19]. On the other hand, for a simpler network, the network behavior is more sensitive to parameter configurations of the network. Our method may become less predictive than the complex network.

In summary, we believe that our method provides a promising pathway for understanding the underpinnings of a genetic regulatory network. With this method, it is possible to integrate currently available knowledge on genetic regulations and signaling pathways in a disease to a large network and to decipher the biological mechanisms by producing quantitative predictions on interventions based on the network. Thus, it may be especially interesting to drug target discovery research.

This work is partially supported by NIH (R01GM072856 to W.W.).

**Rui Chang** received the PhD degree in computer science and informatics from the Technical University of Munich, Germany, in 2008. He is currently a postdoctoral scholar at the University of California, San Diego. His research focuses on developing and applying statistical algorithms to human genomics. From 2005 to 2007, he worked as a PhD research scientist at Siemens Corporate Technology in Munich, Germany. During this time, he worked on developing novel statistical models and on modeling signaling pathways in breast cancer. He has several approved patents in Germany and USA. He currently works on the development and application of several novel statistical methods in modeling the human embryonic stem cell. He has first-authored several peer-reviewed journal publications and proceeding publications. He has been invited to author a bookchapter and give presentations at several international conferences. He is a member of the IEEE.

**Robert Shoemaker** received the BS degree in biochemistry and the BA degree in German literature from the University California at San Diego, and the PhD degree from the University of California, San Diego. His research interest includes the human epigenome with an emphasis on DNA methylation. Currently, he is exploring DNA methylations influence on cell line phenotypes.

**Wei Wang** is an associate professor in the Department of Chemistry and Biochemistry at the University of California, San Diego (UCSD). His research is focused on understanding the principles governing the topology and dynamics of biological networks. He is particularly interested in how biological networks regulate phenotype formation in response to environmental or developmental signals from both genetic and epigenetic perspectives. His interdisciplinary research integrates statistics, machine learning, chemistry, and physics to uncover biological mechanisms.

A set of qualitative statements, = {Ω* _{i}*|

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