Patent Application: US-71633003-A

Abstract:
embodiments of this invention include application of new inferential methods to analysis of complex biological information , including gene networks . in some embodiments , time course data obtained simultaneously for a number of genes in an organism . new methods include modifications of bayesian inferential methods and application of those methods to determining cause and effect relationships between expressed genes , and in some embodiments , for determining upstream effectors of regulated genes . additional modifications of bayesian methods include use of time course data to infer causal relationships between expressed genes . other embodiments include the use of bootstrapping methods and determination of edge effects to more accurately provide network information between expressed genes . information about gene networks can be stored in a memory device and can be transmitted to an output device , or can be transmitted to remote location .

Description:
in general , a dynamic bayesian network model can be obtained using any suitable method for determining gene expression . in certain embodiments , microarray experiments are desirable because a large number of genes can be studied from a single sample applied to the array , making relative differences in gene expression easy to determine . it maybe desirable to improve accuracy of microarray methods by subtracting background signals from the signal reflecting true gene expression and / or correcting for inherent differences in labels used to measure gene expression ( e . g ., cy3 / cy5 ) using a bayesian network framework , we consider a gene as a random variable and decompose the joint probability into the product of conditional probabilities . for example , if we have a series of observations of the random vector , we can denote the probability of obtaining a given observation can depend upon the conditional probability densities . in certain embodiments , one can use nonparametric regression models for capturing the relationships between the variables . a variety of graphic tools can be used to elucidate the relationships . for example , polynomials , fourier series , regression spline gases , b - spline bases , wavelet bases and the like can be used for defining a graph of gene relationships . certain methods to elucidate network relationships are disclosed in u . s . patent application ser . no . 10 / 259 , 723 , herein incorporated fully by reference . one difficulty in selecting a proper graph is to properly evaluate variance and noise in the system . in some embodiments of this invention , networks can be constructed using bayesian estimation with nonparametric regression using data from time series studies . in many gene networks , an intervention leads to alteration in expression of certain genes before alterations in other genes are observed . one may infer that expression of certain genes after an intervention that occur later in time , may be causally related to genes whose expression is early . time series information is useful to define “ early ” or genes and “ late ” or gene . it is unlikely that an alteration in expression of a late gene could be a cause of an alteration of expression of an early gene , whose expression is altered sooner in time than that of a late gene . although this presumption may not apply in all cases , it is more probable that early genes are more likely “ upstream ” in a network than are late genes , which are more likely to be “ downstream ” genes . therefore , time relations of gene expression can be useful to modify bayesian estimation and nonparametric regression to provide a more reliable network solution . in aspects of this invention , we extend the bayesian network and nonparametric regression model to a dynamic bayesian network model , which can be used to construct cyclic relationships when one has time series gene expression data . information on time delay between changes in gene expression can be included in a model easily , and the model can extract even nonlinear relations among genes easily . in certain embodiments , for constructing a gene network with cyclic regulatory components , an ordinal differential equation model ( chen et al . [ 5 ]; de hoon et al . [ 8 ] can be used . however , this model is based on a linear system and maybe unsuitable for capturing complex phenomena . we have derived a new criterion for choosing an optimal network from bayesian statistical point of view [ 2 ]. the criterion can optimize network structure , which gives the best representation of the gene interactions described by the data with noise . the new criterion is herein termed bnrc dynamic . bnrc dynamic can be evaluated using a first - order markov relation as illustrated in fig1 . in such a relationship , an upstream gene x 1 is depicted as having an effect ( right arrow ) on one or more downstream genes x 2 , which has an effect on x 3 ( not shown ), and so on , until an effect on x n is observed . in situations in which x 1 has no “ upstream ” gene of its own , x 1 is termed a “ parent ” gene within the network . genes under influence of a parent gene are termed “ target ” genes . note that the use of “ target gene ” in this context is not to be confused with a gene that is a target for intervention , such as by a potential drug . in fact , parent genes may be targets for therapeutic intervention . under this scheme , an effect on x n cannot be observed until effects on x 1 , x 2 , etc . have been elicited . note that fig1 illustrates a “ series ” cause / effect relationship , without parallel or feedback systems are present , whereas in many genetic systems , there are series effects , and “ parallel ” effects , in which two or more genes can either be affected by an upstream gene , and / or can themselves affect a downstream gene . moreover , circular effects (“ feedback ”) can be present , in which a gene x a can affect another gene x b , which can affect x c , which itself can affect x a ( or x b ). moreover , such feedback maybe either positive , in which x c stimulates x a or “ negative ” in which x c inhibits x a . further complexities can arise in situations in which both series , parallel , positive feedback and negative feedback relationships are present . in general , relationships between time points may be arbitrary , but in some cases it can be advantageous to use pre - selected time points based on knowledge of the biological effects of the genes and their expression dynamics under study . under first order conditions , a joint probability can be decomposed as shown in equation ( 1 ) in example 1 below . a conditional probability can then be decomposed into the product of conditional probabilities using equation ( 2 ) in example 1 . equations ( 1 ) and ( 2 ) can hold and the density function can be used instead of a probability measure . therefore , the dynamic bayesian network can be represented , for example , using densities described in example 1 to arrive at the local network structure of a gene and its parent genes according to equation ( 3 ) in example 1 . a dynamic bayesian model with nonparametric regression can be applied , for example , as described in example 2 . once experimental data is collected , a the solution to the network can be considered to be a statistical model selection problem . in certain embodiments , we can solve this problem using bayesian approach and derive a criterion for evaluating the goodness of the dynamic bayesian network and nonparametric regression methods . assuming a prior distribution , marginal likelihood and posterior probability can be determined according to equation ( 4 ) in example 2 . subsequent construction of a genetic network involves computation of a high dimensional integral as depicted in equation ( 4 ). in some embodiments , a laplace method for integrals , for example , can be used to approximate the integral . therefore , the criterion bnrc dynamic as shown in equation ( 5 ) in example 2 can be solved . to apply bnrc dynamic to an experimental system , cdna microarray data , for example , can be obtained experimentally at a number of time points after affecting the genetic system . to smooth curves , we can use spline functions , for example b - splines as depicted in example 3 . bnrc dynamic can be decomposed according to equation ( 6 ) in example 3 . optimal network relationships are obtained when bnrc dynamic is minimized . using dynamic bayesian network models with nonparametric regression and the criterion bnrc dynamic , we can formulate a network learning process . however , determining which genes are parent genes and which are target genes can be time consuming when all possible gene combinations and relationships are considered . to reduce the number of analyses needed , we can select candidate parent genes . subsequently a greedy hill - climbing algorithm can be used . bnrc dynamic is calculated and then an addition parent gene is either added or deleted , and bnrc dynamic is re - evaluated according to step 2 in example 3 . the process is repeated until an appropriate convergence is found . then , the order of computation is permutated and bnrc dynamic is reevaluated . the optimal network give the smallest bnrc dynamic . a specific illustration of the above methods are shown in example 4 in fig2 a and 2 b . the efficiencies of the methods are shown through analysis of gene expression data from saccharomyces cerevisiae . fig2 a depicts a group of s . cerevisiae genes involved in regulation of cell cycle . the genes are depicted as grouped based in the overall metabolic pathways involved and focus on the cyclin - dependent protein kinase gene ( ybr160w ). note that the parent / target gene network relationships are unknown based on fig2 a . in contrast , using methods of this invention , network relationships of those genes can be evaluated and are depicted in fig2 b . another example is depicted in fig3 a - 3 c . fig3 a depicts genes involved in metabolic pathways . fig3 a shows no gene network relationships . fig3 b depicts a network solution obtained using bayesian network analysis with nonparametric regression , but without consideration of bnrc dynamic . fig3 c depicts a network solution obtained by minimizing bnrc dynamic . note that in fig3 b , the network relationships are simpler , and compared to those depicted in fig3 b , there are many fewer false positive relationships (“ x ”). boundaries between groups of genes in a network can be determined using methods known in the art , for example , bootstrap methods . such methods include determining the intensity of an edge ( 1 ) providing a bootstrap gene expression matrix by randomly sampling a number of times , with replacement , from the original gene library expression data ; ( 2 ) estimating the genetic network for genes and gene j ; ( 3 ) repeating steps ( 1 ) and ( 2 ) t times , thereby producing t genetic networks ; and ( 4 ) calculating the bootstrap edge intensity between gene i and gene j as ( t 1 + t 2 )/ t . advantages of the new methods compared with other network estimation methods such as bayesian and boolean networks include : ( 1 ) time information can be incorporated easily ; ( 2 ) microarray data can be analyzed as continuous data without extra data pre - treatments such as discretization ; and ( 3 ) fewer false positive relationships are found . even nonlinear relations can be detected and modeled by embodiments of this invention . methods of this invention are useful for analyzing genetic networks and for development of new pharmaceuticals which target particularly genes that control genetic expression of important genes . thus , methods of this invention can decrease the time needed to identify drug targets and therefore can decrease the time needed to develop new treatments . other aspects of methods of this invention are described in the examples below . the examples presented below represent specific embodiments of this invention . other aspects of the invention can be developed by persons of ordinary skill in the art without undue experimentation . all such embodiments are considered part of this invention . suppose that we have an n × p microarray gene expression data matrix x , where n and p are the numbers of microarrays and genes , respectively . usually , the number of genes p is much larger than the number of microarrays , n . in the estimation of a gene network based on the bayesian network , a gene is considered as a random variable . when we model a gene network by using statistical models described by the density or probability function , the statistical model should include p random variables . however , we have only n samples and n is usually much smaller than p . in such case , the inference of the model is quite difficult or impossible , because the model has many parameters and the number of samples is not enough for estimating the parameters . the bayesian network model has been advocated in such modeling . in the context of the dynamic bayesian network , we consider the time series data and the ith column vector x i of x corresponds to the states of p genes at time i . as for the time dependency , we consider the first order markov relation described in fig1 . under this condition , the joint probability can be decomposed as follows : p ( x 11 , . . . , x np )= p ( x 1 ) p ( x 2 | x 1 )× . . . × p ( x 1 x n - 1 ), ( 1 ) where x i =( x i1 , . . . , x ip ) is a random variable vector of p genes at time i . the conditional probability p ( x i | x i - 1 ) can also be decomposed into the product of conditional probabilities of the form p ( x i | x i - 1 )= p ( x i1 | p i - 1 , 1 )× . . . × p ( x ip | p i - 1 , p ), ( 2 ) where p i - 1 , j is the state vector of the parent genes of jth gene at time i − 1 . the equations ( 1 ) and ( 2 ) hold when we use the density function instead of the probability measure hence , the dynamic bayesian network can then be represented by using densities as follows : f ⁡ ( x 11 , … , x np ) = f 1 ⁡ ( x 1 ) ⁢ f 2 ⁡ ( x 2 | x 1 ) × ⋯ × f n ⁡ ( x n | x n - 1 ) = f 1 ⁡ ( x 1 ) ⁢ ∏ i = 2 n ⁢ g 1 ⁡ ( x i1 | p i - 1 , 1 ) × ⋯ × g p ⁡ ( x ip | p i - 1 , p ) = f 1 ⁡ ( x 1 ) ⁢ ∏ j = 1 p ⁢ { ∏ i = 2 n ⁢ g j ⁡ ( x ij | p i - 1 , j ) } . f i ( x i | x i - 1 )= g 1 ( x il | p i - 1 , 1 )× . . . × g p ( x ip | p i - 1 , p ), where p i - 1 , j =( p i - 1 , 1 ( j ) , . . . , p i - 1 , qj ( j ) ) is a q j - dimensional observation vector of parent genes . for modeling the relationship between x ij and p i - 1 , j , we use the nonparametric additive regression model as follows : x ij = m j1 ( p i = 1 , 1 ( j ) )+ . . . + m jqj ( p i = 1 , qj ( j ) )+ ε ij , where ε ij depends independently and normally on mean 0 and variance σ j 2 . here , m jk (•) is a smooth function from r to r and can be expresse ( d by using the linear combination of basis functions m jk ⁡ ( p i - 1 , k ( j ) ) = ∑ m = 1 m jk ⁢ γ mk ( j ) ⁢ b mk ( j ) ⁡ ( p i - 1 , k ( j ) ) , k = 1 , … ⁢ , q j , where γ 1k ( j ) , . . . , γ m jk k ( j ) are unknown coefficient parameters and { b 1k ( j ) (•), . . . , b m jk k (•)} is the prescribed set of basis functions . then we define a dynamic bayesian network and nonparametric regression model of the form f ⁡ ( x 11 , … ⁢ , x np ; θ g ) = ⁢ f 1 ⁡ ( x 1 ) ⁢ ∏ j = 1 p ⁢ [ ∏ i = 2 n ⁢ 1 2 ⁢ ⁢ π ⁢ ⁢ σ j 2 ⁢ exp ⁢ { - ( x ij - μ ⁡ ( p i - 1 , j ) ) 2 2 ⁢ ⁢ σ j 2 } ] , where μ ( p i − 1 , j )= m j1 ( p i − 1 , 1 ( j ) )+ . . . + m jq j ( p i − 1 , jq j ( j ) ). when jth gene has no parent genes , μ ( p i − 1 , j ) is resulted in the constant μ j . we assume f 1 ( x 1 )= g 1 ( x 11 )× . . . × g 1 ( x 1p ) and the joint density f ( x 11 , . . . , x npi ; θ g ) can then be rewritten as f ⁡ ( x 11 , … ⁢ , x np ; θ g ) = ⁢ ∏ j = 1 p ⁢ [ g 1 ⁡ ( x 1 ⁢ j ) ⁢ ∏ i = 2 n ⁢ 1 2 ⁢ ⁢ π ⁢ ⁢ σ j 2 ⁢ exp ⁢ { - ( x ij - μ ⁡ ( p i - 1 , j ) ) 2 2 ⁢ ⁢ σ j 2 } ] = ⁢ ∏ j = 1 p ⁢ ∏ i = 1 n ⁢ g j ⁡ ( x ij | p i - 1 , j ; θ j ) , ( 3 ) where p 0j = θ . thus , g j ( x ij | p i − 1 , j ; θ j ) represents the local structure of jth gene and its parent genes . the dynamic bayesian network and nonparametric regression model introduced in the previous section can be constructed when we fic the network structure and estimated by a suitable procedure . however , the gene network is generally unknown and we should estimate an optimal network based on the data . this problem can be viewed as a statistical model selection problem ( see e . g ., akaike [ 1 ]; konishi and kitagawa [ 17 ]; burnham and anderson [ 4 ]; konishi [ 16 ]). we solve this problem from the bayesian statistical approach and derive a criterion for evaluating the goodness of the dynamic bayesian network and nonparametric regression model . let π ( θ g | λ ) be a prior distribution on the parameter θ g in the dynamic bayesian network and nonparametric regression model and let log π ( θ g | λ )= o ( n ). the marginal likelihood can be represented as ∫ f ( x 11 , . . . , x np ; θ g ) π ( θ g | λ ) dθ g . thus , when the data is given , the posterior probability of the network g is π post ⁡ ( g | x ) = π prior ⁡ ( g ) ∫ f ⁡ ( x 11 , … ⁢ , x np ; θ g ) ⁢ π ⁡ ( θ g | λ ) ⁢ ⅆ θ g σ g { π prior ⁡ ( g ) ⁢ ∫ f ⁡ ( x 11 , … ⁢ , x np ; θ g ) ⁢ π ⁡ ( θ g | λ ) ⁢ ⅆ θ g } , ( 4 ) where πprior ( g ) is the prior probability of the network g . the denominator of ( 4 ) does not relate to model evaluation . therefore , the evaluation of the network depends on the magnitude of numerator . hence , we can choose an optimal network as the maximizer of π prior ( g )∫ f ( x 11 , . . . , x npi ; θ g ) π ( θ g | λ ) dθ g . it is clear that the essential point for constructing a network selection criterion is how to compute the high dimensional integral . imoto et al . [ 14 , 15 ] used the laplace approximation for integrals ( see also tinerey and kadane [ 21 ]; davison [ 6 ]) and we can apply this technique to the dynamic bayesian network model and nonparametric regression directly . hence , we have a criterion , named bnrc dynamic of the form bnrc dynamic ⁡ ( g ) = ⁢ - 2 ⁢ log ⁢ { π prior ⁡ ( g ) ⁢ ∫ f ⁡ ( x 11 , … ⁢ , x np ; θ g ) ⁢ π ⁢ ( θ g | λ ) ⁢ ⅆ θ g } ≈ ⁢ - 2 ⁢ log ⁢ ⁢ π prior ⁡ ( g ) - r ⁢ ⁢ log ⁡ ( 2 ⁢ ⁢ π / n ) + ⁢ log ⁢  j λ ⁡ ( θ ^ g )  - 2 ⁢ nl λ ⁡ ( θ ^ g | x n ) , ( 5 ) l λ ( θ g 51 x n )= log f ( x 11 , . . . , x npi ; θ g )/ n + log π ( θ g | λ )/ n , j λ ( θ g )=∂ 2 { l λ ( θ g | x n )}/∂ θ g θ t g and θ g is the mode of l λ ( θ g | x n ). the optimal graph is chosen such that the criterion bnrc dynamic ( 5 ) is minimal . in this section , we show a concrete strategy for estimating a gene network from cdna microarray time series gene expression data . we use the basis function approach for constructing the smooth function m jk (•) described in section 2 . in this paper we use b - splines ( de boor [ 7 ]) as the basis functions . de boor &# 39 ; s algorithm ( de boor [ 7 ], chapter 10 , p . 130 ( 3 )) is a useful method for computing b - splines of any degree . we use 20 b - splines with equidistance knots ( see also , dierckx [ 10 ]; eiler and marx [ 11 ] for the details of b - spline ). for the prior distribution on the parameter θ g , suppose that the parameter vectors θ j are independent one another , the prior distribution can then be decomposed as π ( θ g | λ )= π j = 1 p π j ( θ j | λ j ). suppose that the prior distribution π j ( θ j | λ j ) is factorized as π j ( θ j | λ j )= π l & lt ; 1 qj π jk ( γ jk | λ jk ), where λ jk are hyper parameters . we use a singular m jk variate normal distribution as the prior distribution on γ jk , π jk ⁡ ( γ jk | λ jk ) = ( 2 ⁢ ⁢ π n ⁢ ⁢ λ jk ) - ( m jk - 2 ) / 2 ⁢  k jk  + 1 / 2 ⁢ exp ⁡ ( - n ⁢ ⁢ λ jk 2 ⁢ γ jk t ⁢ k jk ⁢ γ jk ) , where k jk is an m jk × m jk symmetric positive semidefinite matrix satisfying γ jk t ⁢ k jk ⁢ γ jk = ∑ α = 3 m jk ⁢ ( γ α ⁢ ⁢ k ( j ) - 2 ⁢ ⁢ γ α - 1 , k ( j ) + γ α - 2 , k ( j ) ) 2 . this setting of the prior distribution on θ g is the same as imoto et al . [ 14 , 15 ] nd the details are in those papers . by using the prior distributions in section 4 . 2 , the bnrc dynamic can be decomposed as follows : bnrc dynamic = ∑ j = 1 p ⁢ bnrc dynamic ( j ) , ( 6 ) where bnrc dynamic ( j ) is a local criterion score of jth gene and is defined by bnrc dynamic ( j ) = ⁢ - 2 ⁢ log ⁢ { ∫ π prior ⁡ ( l j ) ⁢ ∏ i = 1 n ⁢ g j ⁡ ( x ij | p i - 1 , j ; θ j ) ⁢ π j ⁢ ( θ j | λ j ) ⁢ ⅆ θ j } ≈ ⁢ - 2 ⁢ log ⁢ ⁢ π prior ⁡ ( l j ) - r j ⁢ log ⁡ ( 2 ⁢ ⁢ π / n ) + log ⁢  j λ j ( j ) ⁡ ( θ ^ j )  - ⁢ 2 ⁢ nl λ j ( j ) ⁡ ( θ ^ j | x ) , l λ j ( j ) ⁡ ( θ ^ j | x ) = ∑ i = 1 n ⁢ log ⁢ ⁢ g j ⁡ ( x ij | p i - 1 , j ; θ j ) / n + log ⁢ ⁢ π ⁡ ( θ j | λ j ) / n , j λ j ( j ) ({ circumflex over ( θ )} j )=−∂ 2 { l λ j ( j ) ({ circumflex over ( θ )} j | x )}/∂ θ j ∂ θ j t and { circumflex over ( θ )} j is the mode of l λ j ( j ) ( θ j | x ). here π prior ( l j ) are prior probabilities satisfying ∑ j = 1 p ⁢ log ⁢ ⁢ π prior ⁡ ( l j ) = log ⁢ ⁢ π prior ⁡ ( g ) . we set the prior probability of local structure π prior ( l j ) as π prior ( l j )= exp {−( the number of parent genes of j th gene )}. by using the dynamic bayesian network and nonparametric regression model together with the proposed criterion , bnrcd1 / namic , we can formulate the network learning process as follows : it is clear from ( 3 ) and ( 6 ) that the optimization of network structure is equivalent to the choices of the parent genes that regulate the target genes . however , it is a time - consuming task when we consider all possible gene combinations as the parent genes . therefore , we cut down the learning space by selecting candidate parent genes . after this step , a greedy hill climbing algorithm is employed for finding better networks . our algorithm can be expressed as follows : we make the p × p matrix whose ( i , j ) th element is the bnrc score of the graph “ gene i → gene j ” and we define the candidate set of parent genes of gene j that gives small bnrc score . we set the number of elements of the candidate set of parent genes 10 . for a greedy hill - climbing algorithm , we start form the empty network and repeat the following steps : step2 - 1 : for genes , implement one from two procedures that add a parent gene , delete a parent gene , which gives smaller bnrc dynamic score . step2 - 2 : repeat step2 - 1 for prescribed computational order of genes until suitable convergence criterion is satisfied . step2 - 9 : permute the computational order for finding better solution and repeat step2 - 1 and 2 - 2 . step2 - 4 : we choose the optimal network that gives the smallest bnrc dyanmic score . we demonstrated one embodiment of this invention through the analysis of the saccharomyces cerevisiae cell cycle gene expression data collected by spellman et al . [ 20 ]. this data contains two short time series ( two time points ; cln3 , clb2 ) and four medium time series ( 18 , 24 , 17 and 14 time points ; alpha , cdc15 , cdc28 and elu ). in the estimation of a gene network , we used four medium time series . for combining four time series , we ignored the first observation of the target gene and last one of parent genes for each time series when we fit the nonparametric regression model . at first , we focused on the cell cycle pathway compiled in kegg database [ 22 ]. the target network is around cdc28 ( ybr160w ; cyclin - dependent protein kinase ). this network contains 45 genes and the pathway registered in kegg is shown in fig2 ( a ) and the estimated network is in fig2 ( b ). the edges in the dotted circles can be considered the correct edges . we thus modeled some correct relations . we denoted the correct estimation by the circle next to edge . the triangle represents the reverse or skip of correct direction . the “ x ” symbols represent incorrect relationships . a second example used to demonstrate our methods is the metabolic pathway reported by derisi et al . [ 9 ]. this network contains 57 genes and the target pathway is shown in fig3 ( a ). we applied a bayesian network and nonparametric regression model [ 14 , 15 ] to this data and the resulting network is depicted in fig3 ( b ). the network of fig3 ( c ) was obtained by the dynamic bayesian network and nonparametric regression model . it is difficult to estimate the metabolic pathway from cdna microarray data . however , our model detected correct relationships between the genes . compared with the bayesian network and nonparametric regression , the number of false positives of this method depicted in fig3 ( c ) was much smaller than those depicted in fig3 ( b ) by the “ x ” symbols . 1 . akaike , j . : information theory and an extension of the maximum likelihood principle . in : petrov , b . n ., csaki , f . 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