## Inferring gene regulatory networks from time-ordered gene expression data of bacillus subtilis using differential equations (2003)

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Venue: | Pac. Symp. Biocomput |

Citations: | 64 - 16 self |

### BibTeX

@INPROCEEDINGS{Hoon03inferringgene,

author = {Michiel De Hoon and Seiya Imoto and Satoru Miyano},

title = {Inferring gene regulatory networks from time-ordered gene expression data of bacillus subtilis using differential equations},

booktitle = {Pac. Symp. Biocomput},

year = {2003},

pages = {17--28}

}

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### Abstract

Abstract. Recently, cDNA microarray experiments have generated large amounts of gene expression data. In time-ordered gene expression data, the expression levels are measured at several points in time following some experimental manipulation. A gene regulatory network can be inferred by describing the gene expression data in terms of a linear system of differential equations. As biologically the gene regulatory network is known to be sparse, we expect most coefficients in such a linear system of differential equations to be zero. In previously proposed methods to infer a linear system of differential equations, some ad hoc assumptions are made to limit the number of nonzero coefficients in the system. Instead, we propose to infer the degree of sparseness of the gene regulatory network from the data, where we determine which coefficients are nonzero by using Akaike’s Information Criterion. 1

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Citation Context ...and take the derivative with respect to Λ. We find a linear equation in Λ: in which the matrices A and B are defined as A ≡ B ≡ n� i=1 n� i=1 ˆΛ = B · A −1 , (13) � (ti − ti−1) 2 · xi−1 · x T � i−1 ; =-=(14)-=- � (ti − ti−1) · � � � T xi − xi−1 · xi−1 . (15) In the absence of errors, the estimated matrix ˆ Λ is equal to the true matrix Λ. We know from biology that the gene regulatory network and therefore Λ... |

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Citation Context ... squared error, as given by Eq. 12, is small. Formally, we would use Akaike’s Information Criterion 15,16 AIC = −2 · � � log-likelihood of the + 2 · estimated model � � number of estimated parameters =-=(16)-=- to decide which matrix elements should be set equal to zero. The AIC avoids overfitting of a model to data by comparing the total error in the estimated model to the number of parameters that was use... |

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Citation Context ... of the measured data x (t). An approximate solution can be found by replacing the differential equation (Eq. 1) by a difference equation: ∆x ∆t = Λ · x , (4)sor x (t + ∆t) − x (t) = ∆t · Λ · x (t) , =-=(5)-=- which is of the form considered by Chen. 13 To be able to statistically determine the sparseness of matrix Λ, we explicitly add an error ε (t), which will invariably be present in the data: x (t + ∆t... |

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Citation Context ...nd a linear equation in Λ: in which the matrices A and B are defined as A ≡ B ≡ n� i=1 n� i=1 ˆΛ = B · A −1 , (13) � (ti − ti−1) 2 · xi−1 · x T � i−1 ; (14) � (ti − ti−1) · � � � T xi − xi−1 · xi−1 . =-=(15)-=- In the absence of errors, the estimated matrix ˆ Λ is equal to the true matrix Λ. We know from biology that the gene regulatory network and therefore Λ is sparse. However, all of the elements in the ... |

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Citation Context ...oretically by Chen. 13 In this model, both the mRNA and the protein concentrations were described by a system of linear differential equations. Such a system can be described as d x (t) = Λ · x (t) , =-=(1)-=- dt in which the vector x (t) contains the mRNA and protein concentrations as a function of time, and the matrix Λ is constant with units of [second] −1 . This equation can be considered as a generali... |

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Citation Context ...r at time ti estimated from the measured data. The maximum likelihood estimate of the variance σ 2 can be found by maximizing the log-likelihood function with respect to σ 2 . This yields ˆσ 2 = 1 nm =-=(9)-=- n� ˆε T i · ˆε i . (10) Substituting this into the log-likelihood function (Eq. 8) yields i=1 L � Λ, σ 2 = ˆσ 2� = − nm 2 ln � 2πˆσ 2� − nm . (11) 2 To find the maximum likelihood estimate ˆ Λ of the... |

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Citation Context ...em of differential equations (Eq. 1), where the vector x (t) contains the expression ratios of the m genes at time t. This system of differential equations can be solved as x (t) = exp � Λt � · x 0 , =-=(2)-=- in which x 0 contains the gene expression ratios at time zero. In this equation, the matrix exponential is defined in terms of a Taylor expansion as 14 exp � A � ≡ ∞� i=0 1 i! Ai . (3) As Eq. 2 depen... |

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Citation Context ...hen. 13 To be able to statistically determine the sparseness of matrix Λ, we explicitly add an error ε (t), which will invariably be present in the data: x (t + ∆t) − x (t) = ∆t · Λ · x (t) + ε (t) . =-=(6)-=- By using this equation, we effectively describe a gene regulatory network in terms of a multidimensional linear Markov model. We assume that the error has a normal distribution independent of time: f... |

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Citation Context ...n information theory and is widely used for statistical model identification, especially for time series model fitting. 17 We use a mask M to set matrix elements of ˆ Λ equal to zero: ˆΛ ′ = M ◦ ˆ Λ, =-=(17)-=- where ◦ denotes the Hadamard (element-wise) product, 14 and the mask M is a matrix whose elements are either one or zero. The corresponding total squared error ˆσ 2 can be found by replacing ˆ Λ by ˆ... |

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Citation Context ...ry network in terms of a multidimensional linear Markov model. We assume that the error has a normal distribution independent of time: f � ε (t) ; σ 2� = � �m 1 √ exp 2πσ2 � − ε (t)T · ε (t) 2σ 2 � , =-=(7)-=- with a standard deviation σ equal for all genes at all times. The log-likelihood function for a series of time-ordered measurements x i at times ti, i ∈ {1, . . . , n} at n time points is then in whi... |

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Citation Context ... (ti − ti−1) xT � � i−1 · Λ · xi−1 , (12)sand take the derivative with respect to Λ. We find a linear equation in Λ: in which the matrices A and B are defined as A ≡ B ≡ n� i=1 n� i=1 ˆΛ = B · A −1 , =-=(13)-=- � (ti − ti−1) 2 · xi−1 · x T � i−1 ; (14) � (ti − ti−1) · � � � T xi − xi−1 · xi−1 . (15) In the absence of errors, the estimated matrix ˆ Λ is equal to the true matrix Λ. We know from biology that t... |

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Citation Context ...he log-likelihood function for a series of time-ordered measurements x i at times ti, i ∈ {1, . . . , n} at n time points is then in which L � Λ, σ 2� = − nm 2 ln � 2πσ 2� − 1 2σ 2 n� ˆε T i · ˆε i , =-=(8)-=- i=1 ˆε i = x i − x i−1 − (ti − ti−1) · Λ · x i−1 is the measurement error at time ti estimated from the measured data. The maximum likelihood estimate of the variance σ 2 can be found by maximizing t... |

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Citation Context ...from the measured data. The maximum likelihood estimate of the variance σ 2 can be found by maximizing the log-likelihood function with respect to σ 2 . This yields ˆσ 2 = 1 nm (9) n� ˆε T i · ˆε i . =-=(10)-=- Substituting this into the log-likelihood function (Eq. 8) yields i=1 L � Λ, σ 2 = ˆσ 2� = − nm 2 ln � 2πˆσ 2� − nm . (11) 2 To find the maximum likelihood estimate ˆ Λ of the matrix Λ, we use Eq. 9 ... |

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Citation Context ... [k]) 2n 2 , (20) i=1 k=1,2 in which xji [k] denotes the data value of measurement k at time point i for gene j. At each time point, we calculate the average log-ratio as ¯xji = 1 2 � k=1,2 xji [k] . =-=(21)-=- Under the null hypothesis, ¯xj· (the average of two gene expression log-ratios at a time point) is a random variable with a normal distribution with zero mean and an estimated standard deviation ˆσ j... |

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Citation Context ...unction with respect to σ 2 . This yields ˆσ 2 = 1 nm (9) n� ˆε T i · ˆε i . (10) Substituting this into the log-likelihood function (Eq. 8) yields i=1 L � Λ, σ 2 = ˆσ 2� = − nm 2 ln � 2πˆσ 2� − nm . =-=(11)-=- 2 To find the maximum likelihood estimate ˆ Λ of the matrix Λ, we use Eq. 9 to write the total squared error ˆσ 2 as ˆσ 2 = 1 nm n� i=1 ��x T i − xT � � � i−1 · xi − xi−1 + (ti − ti−1) 2 xT i−1 · ΛT ... |

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Citation Context ...on levels xi. We then calculate the AIC corresponding to M by substituting the estimated log-likelihood function from Eq. 11 into Eq. 16: AIC = nm ln � 2πˆσ 2� � � �� sum of the mask + nm + 2 · 1 + , =-=(19)-=- elements Mij the estimated parameters being ˆσ 2 and the elements of the matrix ˆ Λ that we allow to be nonzero. From this equation, we see that while the squared error decreases, the AIC may increas... |

28 |
Statistical analysis of a small set of time-ordered gene expression data using linear splines
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Citation Context ... it will be difficult to solve for Λ in terms of the measured data x (t). An approximate solution can be found by replacing the differential equation (Eq. 1) by a difference equation: ∆x ∆t = Λ · x , =-=(4)-=-sor x (t + ∆t) − x (t) = ∆t · Λ · x (t) , (5) which is of the form considered by Chen. 13 To be able to statistically determine the sparseness of matrix Λ, we explicitly add an error ε (t), which will... |

25 |
DNA microarray analysis of cyanobacterial gene expression during acclimation to high light. The Plant Cell 13
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Citation Context ...� Λt � · x 0 , (2) in which x 0 contains the gene expression ratios at time zero. In this equation, the matrix exponential is defined in terms of a Taylor expansion as 14 exp � A � ≡ ∞� i=0 1 i! Ai . =-=(3)-=- As Eq. 2 depends nonlinearly on Λ, it will be difficult to solve for Λ in terms of the measured data x (t). An approximate solution can be found by replacing the differential equation (Eq. 1) by a di... |

22 |
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Citation Context .... We further assume that the log-ratios have a normal distribution with zero mean. The standard deviation is then estimated from all 8 × 2 = 16 measurements: � � � ˆσ j|H0 = � 1 n� � (xji [k]) 2n 2 , =-=(20)-=- i=1 k=1,2 in which xji [k] denotes the data value of measurement k at time point i for gene j. At each time point, we calculate the average log-ratio as ¯xji = 1 2 � k=1,2 xji [k] . (21) Under the nu... |

20 |
S.: XML documentation of biopathways and their simulations
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Citation Context ...on ˆσ j|H0 /√ 2. The joint probability for ¯xj· to be larger in absolute value than the measured values ¯xji is then P = n� Pi = i=1 = n� p (|¯xj·| > |¯xji|) i=1 n� � � |¯xji| 1 − erf ˆσ j|H0 /√ �� , =-=(22)-=- 2 i=1 in which erf is the error function. For a single factor Pi in this product, we would normally choose a significance level α, and reject the null hypothesis if Pi < α. Accordingly, we adopt the ... |

4 | Evolutionary inference of a biological network as differential equations by genetic programming
- Sakamoto, Iba
(Show Context)
Citation Context ...x Λ, we use Eq. 9 to write the total squared error ˆσ 2 as ˆσ 2 = 1 nm n� i=1 ��x T i − xT � � � i−1 · xi − xi−1 + (ti − ti−1) 2 xT i−1 · ΛT · Λ · xi−1 −2 � xT i − (ti − ti−1) xT � � i−1 · Λ · xi−1 , =-=(12)-=-sand take the derivative with respect to Λ. We find a linear equation in Λ: in which the matrices A and B are defined as A ≡ B ≡ n� i=1 n� i=1 ˆΛ = B · A −1 , (13) � (ti − ti−1) 2 · xi−1 · x T � i−1 ;... |