Báo cáo khoa học: Unravelling the functional interaction structure of a cellular network from temporal slope information of experimental data docx

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Báo cáo khoa học: Unravelling the functional interaction structure of a cellular network from temporal slope information of experimental data docx

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Unravelling the functional interaction structure of a cellular network from temporal slope information of experimental data Kwang-Hyun Cho 1,2 , Sung-Young Shin 3 and Sang-Mok Choo 3 1 College of Medicine, Seoul National University, Jongno-gu, Seoul, Korea 2 Korea Bio-MAX Institute, Seoul National University, Gwanak-gu, Seoul, Korea 3 School of Electrical Engineering, University of Ulsan, Ulsan, Korea It is now widely accepted that we need to unravel the functional interaction structure of the underlying cellu- lar network (e.g. signalling cascades or gene networks) in order to get a proper understanding of the biological function of a living system. Despite the constant development of new technology, most of the experi- mental measurements still contain inevitable nonbiolog- ical variations to some extent [1–4] and it is not always easy to get enough replicates for statistical preprocess- ing to eliminate or minimize such nonbiological Keywords cellular networks; functional interaction; structure identification; temporal slope; time-series data Correspondence K H. Cho, Korea Bio-MAX Institute, 3rd Floor, IVI, Seoul National University Research Park, San 4–8, Bongcheon 7-dong, Gwanak-gu, Seoul, 151-818, Republic of Korea Fax: +82 2 887 2692 Tel: +82 2 887 2650 E-mail: ckh-sb@snu.ac.kr (Received 16 April 2005, revised 7 June 2005, accepted 13 June 2005) doi:10.1111/j.1742-4658.2005.04815.x Due to the unavoidable nonbiological variations accompanying many experiments, it is imperative to consider a way of unravelling the functional interaction structure of a cellular network (e.g. signalling cascades or gene networks) by using the qualitative information of time-series experimental data instead of computation through the measured absolute values. In this spirit, we propose a very simple but effective method of identifying the functional interaction structure of a cellular network based on temporal ascending or descending slope information from given time-series measure- ments. From this method, we can gain insight into the acceptable measure- ment error ranges in order to estimate the correct functional interaction structure and we can also find guidance for a new experimental design to complement the insufficient information of a given experimental dataset. We developed experimental sign equations, making use of the temporal slope sign information from time-series experimental data, without a speci- fic assumption on parameter perturbations for each network node. Based on these equations, we further describe the available specific information from each part of experimental data in detail and show the functional interaction structure obtained by integrating such information. In this pro- cedure, we use only simple algebra on sign changes without complicated computations on the measured absolute values of the experimental data. The result is, however, verified through rigorous mathematical definitions and proofs. The present method provides us with information about the acceptable measurement error ranges for correct estimation of the func- tional interaction structure and it further leads to a new experimental design to complement the given experimental data by informing us about additional specific sampling points to be chosen for further required infor- mation. Abbreviations HOG, high osmolarity glycerol response; MAP, mitogen-activated protein; MAPK, MAP kinase; MAPKK, MAPK kinase. 3950 FEBS Journal 272 (2005) 3950–3959 ª 2005 FEBS variations from the experimental data. This makes it difficult to compute any algorithm or to interpret the result based on the measured absolute values. Hence, there is a pressing need to develop a new method by which we can identify the functional interaction structure of a cellular network only through the qualitative information of time-series experimental data. Motivated by this practical need, we investigate in this paper a new identification method based on tem- poral slope changes of the experimental data profiles. There have been diverse approaches to identify (or reverse engineer) the functional interaction structure of a cellular network from given experimental data. These include using differential equations [5]; linear models [6]; linear differential equation models [7]; stochastic models [8]; neural network models [9]; Boolean net- works [10]; Bayesian networks [11]; dynamic Bayesian networks [12,13], etc. However, developing a new method that can unravel the functional interaction structure based only on the qualitative information of time-series experimental data remains a challenging subject. Other recent important developments include the identification methods based on parameter perturbation experiments. In particular, Kholodenko et al. [14] have proposed a general method for identification of a cellu- lar network structure based on stationary experimental data, which is applicable to a network of generalized modules under the assumption that each module con- tains at least one intrinsic parameter that can be directly perturbed without the intervention of other nodes or parameters. Sontag et al. [15] have proposed another complementary method based on time-series measure- ments, which can be useful when stationary data are not available and the strength of self-regulation at each node ⁄ module should be estimated as well. It is, however, only applicable to the case when for each node there are as many parameters as the number of overall network nodes and these parameters do not directly affect the corresponding node. The fundamental concepts of the methods proposed by Kholodenko et al. [14] and Sontag et al. [15] have been expounded (K H. Cho, S M. Choo, P. Wellstead & O. Wolkenhauer, unpub- lished data) and have presented a comprehensive unified framework based on the fact that we need n independent equations to solve n unknowns and these n linearly inde- pendent equations can be obtained by properly chosen n parameter perturbations. All these approaches are based on parameter perturbation experiments which are, however, not always achievable in many practical cases. In this paper, we therefore consider a new identifica- tion method which does not require parameter pertur- bation experiments but utilizes only the qualitative information of time-series experimental data. Specific- ally, we aim at developing an identification method based on the temporal slope changes of experimental data profiles. In other words, we make use of the information only about temporal ascending or des- cending slopes from the given time-series experimental data profiles and do not rely on the measured absolute values at each sampling time point. This implies that we require only an experiment that can guarantee such qualitative information regarding the dynamic pattern change of time-series profiles and thereby we can also get insight into the allowable error ranges in the meas- urements. We can further design a new experiment through the present approach by gathering informa- tion about the required sampling time points to ensure correct dynamic pattern changes. Once we have such time-series experimental data containing (partially) cor- rect dynamic pattern changes then we can infer the (partial) interaction structure of the underlying cellular network by integrating the analysis results on each (partial) time interval of measurement. We note here that only simple algebra on sign changes of the time- series profiles is used in the present method without involving any complicated computations on the meas- ured absolute values of the experimental data. The result is however, verified through rigorous mathemat- ical definitions and proofs. The present method is illus- trated by an artificial example as well as by a simple real example extracted from the HOG (high osmolarity glycerol response) pathway for hyperosmolarity adap- tation in budding yeast (Saccharomyces cerevisiae) and based on the related mRNA expression time-series data from Stanford Microarray Databases. Results Inferring the functional interaction between network nodes from dynamic pattern changes of time-series data Investigations into the conceptual framework of quan- tifying molecular interactions in cellular networks have been getting increasing attention in recent years, e.g. by Brown et al. [17], Bruggeman et al. [18], and Kholodenko et al. [19]. Among them, two recent remark- able developments are that of Kholodenko et al. [14] based on stationary experimental data, and that of Sontag et al. [15] based on time-series experimental data. These two developments have been further extended and unified (K H. Cho, S M. Choo, P. Wellstead & O. Wolkenhauer, unpublished data). All of these methods are however, only applicable to K H. Cho et al. Identification through temporal slope information FEBS Journal 272 (2005) 3950–3959 ª 2005 FEBS 3951 the experimental data obtained by parameter perturba- tions under strict assumptions. As such experiments are not always achievable due to many practical limita- tions, we need to consider a new method that can be applicable to time-series experimental data without parameter perturbations and that can make use only of the qualitative information without relying on meas- ured absolute values. To this end, we first consider the following dynamic equations for a cellular network: dx i ðtÞ dt ¼ f i x 1 ; x 2 ; ÁÁÁ; x n ðÞ; t 2 0; Tð; i ¼ 1; 2; ÁÁÁ; n ð1Þ where a variable x i is the i th network node, denoting the biochemical quantity of an element, and the corres- ponding function f i describes how the rate of change of x i with respect to (w.r.t.) time depends on all the varia- bles of the network. From Eqn 1, we can identify the functional interaction structure of a network if we reveal the sign of f ij xtðÞ½¼ @f i xtðÞ½ @x j ð1 i; j nÞ at some time t under the assumption that the sign of f ij [x(t)] is fixed at all time t (i.e. we assume that the func- tional interaction structure is time-invariant). Specific- ally, we define that a node j affects a node i if and only if f ij „ 0. In particular, if f ij > 0 then we interpret that the node j activates the node i by increasing the net rate of x i and if f ij < 0 then the node j inhibits the node i. We note that the dynamics of the system in Eqn 1 depend on the initial condition and time lapse, and we only assume that f i (x 1 , x 2 , ÁÁÁ, x n )(i ¼ 1, 2, ÁÁÁ,n) is partially differentiable with respect to all its arguments x j ( j ¼ 1, 2, ÁÁÁ, n) (i.e. @f i @x j should exist). Hence, we cannot make use of the information from parameter perturbations represented by either dx i dp or @ 2 x i @t@p [14] and [15], but we can only use the dynamic infor- mation according to the time lapse, e.g. represented by dx i dt , to find the functional interaction structure, i.e. the sign of f ij . As it is difficult to find out the exact value of dx i dt from experimental data, we restrict ourselves to utilizing only the sign of dx i dt from the experimental data to identify the sign of f ij in the present method. For instance, let us consider sample trajectories of Eqn 1 for n ¼ 2 in Fig. 1A. If we focus only on the temporal slope changes, there are time points t a 1 ; t b 1 for which _ x 1 t a 1 ÀÁ > 0 > _ x 1 t b 1 ÀÁ and x j t a 1 ÀÁ < x j t b 1 ÀÁ j ¼ 1; 2ðÞ. If we apply the mean value theorem to _ x 1 ¼ f 1 ðx 1 ; x 2 Þ w.r.t. t a 1 and t b 1 , then we have a theoreti- cal equation: D _ x i t ab i ÀÁ ¼ X n j¼1 f ij ðh j ÞÁDx j t ab i ÀÁ ð2Þ for i ¼ 1 and some h j 2 R n where Dvt ab i ÀÁ  vt b i ÀÁ À vt a i ÀÁ 6¼ 0. From Eqn 2, we know that f 11 ‡ 0, f 12 ‡ 0 are not possible as D _ x 1 t ab 1 ÀÁ < 0; Dx j t ab 1 ÀÁ > 0 j ¼ 1; 2ðÞ, and f 11 £ 0, f 12 ‡ 0 are also impossible as D _ x 1 t cd 1 ÀÁ < 0; Dx 1 t cd 1 ÀÁ < 0; Dx 2 t cd 1 ÀÁ > 0 (Fig. 1). Hence, we can identify the sign of, f 12 , i.e. f 12 < 0, in this way by excluding such impossible combinations from all cases. In other words, we first find the time intervals like Fig. 1A and then identify the impossible combination of signs in those intervals as in Fig. 1B by computing the signs of D _ x 1 t ab 1 ÀÁ and Dx j t ab 1 ÀÁ j ¼ 1 ; 2ðÞ. Finally, we can identify the sign of f 12 by integrating all these results. On the other hand, for identification of the sign of f 11 , we need to consider another time interval in which we can identify the remaining impossible sign combinations of f 11 , f 12 . Guidance for an experimental design and allowable measurement error ranges There have been some fundamental questions regard- ing an experimental design and measurements to iden- tify the functional interaction structure of a cellular network. For instance, how precise the measurement should be to infer the embedded true interaction lead- ing to the i th node x i from the measurement and how often the measurement should be taken to capture such a true interaction structure leading to the i th node x i from the measured dynamic profiles? Throughout the investigations into the present method, we can answer these questions as follows. The experimental A B Fig. 1. Postulated experimental data (A) and an illustration of impossible cases of the functional interaction structure with regard to x 1 (B). We can identify the impossible interaction structure by noting the temporal slope changes at some suitable time sets, e.g. t a 1 , t b 1 ÈÉ , t c 1 , t d 1 ÈÉ , from the postulated experimental data of the cel- lular network with two nodes. The arrow indicates an activation and the line with a bar at its end denotes an inhibition. Identification through temporal slope information K H. Cho et al. 3952 FEBS Journal 272 (2005) 3950–3959 ª 2005 FEBS measurement should be taken to provide time-series data containing some specific time intervals T i‘ with t a i ; t b i Âà & T i‘ where various sign combinations of D _ x j i t ab i ÀÁ and Dx j t ab i ÀÁ are included. Here T i‘ ¼ t s i‘ ; t e i‘ Âà is called an information interval at x i , which contains three sampling time points t s i‘ ; t c i‘ ; t e i‘ such that the temporal profile of x i is increasing on t s i‘ ; t c i‘ Âà and decreasing on t c i‘ ; t e i‘ Âà or vice versa, and it is also clear whether other temporal profiles x j (1 £ j £ n, j „ i) are increasing or decreasing over the same interval T i‘ .In the present method, it is assumed that the experimental measurements are taken only to ensure such informa- tion intervals for correct estimation of the functional interaction structure and thereby some measurement errors are allowed within the ranges accordingly. If a given experimental data set contains such dynamic information required to identify the functional inter- action leading to the i th node x i , we define the set of all the information intervals at x i as a fully excited information set at x i (Definition 5 in Supplementary material, Supplementary mathematical descriptions) and in this case we simply say that the experimental data set is a fully excited information set at x i . For instance, for the system in Eqns 1 and 2 with n ¼ 2, if there is an information interval T 11 where the sign combinations of D _ x 1 t ab 1 ÀÁ ; Dx 1 t ab 1 ÀÁ ; Dx 2 t ab 1 ÀÁÂà are (¯,¯,¯), (¯,¯,É), (¯,É,¯) for t a 1 ; t b 1 Âà & T 11 , then we learn that the signs of f 11 , f 12 cannot be any of the cases among (É,É), ( É,¯), (¯,É), (É,0), (0,É), (¯,0) and (0,¯). Hence, the signs of f 11 , f 12 turn out to be (¯,¯). The set {T 11 } is therefore a fully excited infor- mation set at x 1 in this case. Given an experimental data set, we can determine whether it is a fully excited information set at x i or not by applying the present method. If it is not the case then we can design a new experiment to complement the given experimental data by finding additional sampling points to be chosen for the required further information (Fig. 9). We can, of course, identify a par- tial interaction structure from the given experimental data set which is not a fully excited information set at x i without any further experiments (refer to the exam- ple results in Figs 7 and 8). Sign equations for identification of the functional interaction structure In order to identify the true interaction structure through the sign of f ij based on the information about temporal slope changes of time-series experimental data, we first formulate the algebraic Eqn. 2. Given a fully excited information set at x i , we formulate then the sign equations: sf i1 ðÞ6¼ S D _ x i t ab i ÀÁÂà S Dx 1 t ab i ÀÁÂà () ^ÁÁÁ^ sf in ðÞ6¼ S D _ x i t ab i ÀÁÂà S Dx n t ab i ÀÁÂà () summarized in Fig. 2 to obtain the impossible network signs (INS) at x i , i.e. the set of all [s( f i1 ) ÁÁÁ,s(f in )] satisfying the sign equations. We note here that s( f ij ) (1 £ j £ n) denotes one of the signs among ¯ (‘activa- tion’), É (‘inhibition’), and 0 (‘no interaction’) for the unknown sign of f ij . We can then obtain the feasible network signs (FNS) at x i by excluding the INS from the all network signs (ANS) as a complementary set of the INS in ANS at x i (Supplementary material, Supplementary mathematical descriptions). Note that S( f ij ) indicates the sign of f ij and [S( f i1 ) ÁÁÁ, S( f in )] is called the true network sign at x i , which is included in the FNS at x i . Based on the information of the FNS at x i , we can find the true network signs. For instance, if the non- zero s( f i‘ ) in the FNS are all determined as É then S( f i‘ ) ¼ ¯ (Supplementary material, Supplementary mathematical descriptions, Theorem 2) while the non- zero s( f i‘ ) in the FNS are all determined as É then s( f i‘ ) (Supplementary material, Supplementary mathe- matical descriptions, Theorem 3). Furthermore, if there exists both positive and negative signs among s( f i‘ ) in the FNS at x i , then S( f i‘ ) ¼ 0 (Supplementary material, Supplementary mathematical descriptions, Theorem 5). Fig. 2. Determination of the true network signs through investigation into the INS. Note the symbol ^ denotes the logical sum, and the information interval T i is a member of the fully excited information set at x i . K H. Cho et al. Identification through temporal slope information FEBS Journal 272 (2005) 3950–3959 ª 2005 FEBS 3953 Discussion Identification of the functional interaction structure of a cellular network from experimental data has crucial importance in improving our understanding on the biological function of a system. In spite of the recent advancements in this area based on parameter perturbation experiments such as [14–16], there has been a practical need to develop a new identification method without requiring parameter perturbation experiments but only utilizing the quali- tative information of time-series experimental data. In this paper, we have therefore presented a novel identification method based on temporal slope chan- ges of the experimental data profiles, which is distin- guished from any other approaches reported up to the present. One of the major characteristics of the presented method is that it can be rather robust to measurement noise or disturbances since it only makes use of the qualitative information about tem- poral ascending or descending slopes from the given time-series experimental data profiles and does not rely on the measured absolute value at each samp- ling time point. This also implies that we can get an insight into the allowable error ranges in the meas- urements and can further design a new experiment such that the required qualitative information regard- ing the dynamic pattern change of time-series profiles is guaranteed. The resulting experimental guideline is quite specific, e.g. by providing us with the informa- tion about the required sampling time points to cap- ture correct dynamic pattern changes. We stress here that only simple algebra on sign changes of the time-series profiles has been used in the present method without involving any other complicated computations on the measured absolute values of the experimental data. The result has been however, veri- fied through rigorous mathematical definitions and proofs. The proposed method cannot be applied to a case when the increasing ⁄ decreasing patterns are uncertain due to noisy variations in experimental data. Hence, in this case, we need some type of variability information to set the ‘threshold’ for judging a clear slope change. One way of dealing with this problem is as follows. We presume that the experimental data are represented by some error bars at each sampling time point. If the error bars of two adjacent sampling time points do not overlap each other (Fig. 9A) then we define the slope between these two time points as ‘clear’ since all poss- ible increasing ⁄ decreasing slope combinations between the two points should have an identical sign in this case regardless of the noisy variations. Figure 9A shows the time-series measurements with error bars and Fig. 9B illustrates one example profile obtained by connecting chosen sampled data within the error bars. We note that the slope information of all possible dynamic patterns does not change regardless of the chosen sampled data within the error bars since the error bars do not overlap in this case. If some two error bars overlap then the correspond- ing slope can be uncertain in that there can be a lot of different increasing ⁄ decreasing slope combinations (i.e. not all identical signs) between the two points. This leads then to multiple feasible network signs for each interaction and we should employ some statistical measure in this case (e.g. choosing a most frequently occurring one from the distribution) to decide the most feasible network sign or some partial interaction struc- ture, which remains as a further study. Experimental procedures Design The experimental design procedures we propose are as follows (these are further summarized as a flow diagram in Fig. 3). Step 1 – determination of the exact type of temporal slopes from given time-series experimental data To determine the correct INS of Fig. 2, the measured time- series data only need to be precise enough to guarantee the dynamic patterns, i.e. the temporal ascending or descending slopes. If the measurements do not satisfy this minimal requirement, then a new experiment should be designed. Step 2 – determination of T i‘ and G i First, we find an interval J i‘ over which the temporal profile of x i increases and then decreases or vice versa. Second, we choose an interval T i‘ (‘ ¼ 1 ÁÁÁ, c i ) among J i‘ , over which the signs of Dx j t ab i ÀÁ 1 j nðÞis distinct. Then G i becomes G i ¼ {1 ÁÁÁ, c i } and {T i‘ |‘ 2 G i } is called an infor- mation set at x i (Supplementary material, Supplementary mathematical descriptions, Definition 2). Step 3 – finding the INS at x i in Fig. 2 We find [s( f il ) ÁÁÁ, s( f in )] that satisfy the conditions of the INS at x i in Fig. 2 by choosing t a i ; t b i over T i‘ (‘ ¼ 1 ÁÁÁ, c i ) such that S _ x i t a i ÀÁÂà Á S _ x i t b i ÀÁÂà < 0 and Dx j t ab i ÀÁ 6¼ 0. In this way, we can exclude the corresponding impossible network structures. Identification through temporal slope information K H. Cho et al. 3954 FEBS Journal 272 (2005) 3950–3959 ª 2005 FEBS Step 4 – finding the true network signs at x i in Fig. 2 Provided that the given information set at x i is a fully exci- ted information set at x i (Supplementary material, Supple- mentary mathematical descriptions, Definition 5), we can find the FNS at x i by excluding the INS at x i obtained at Step 3 from the ANS at x i and thereby we can identify the true interaction structure at x i (Supplementary material, Supplementary mathematical descriptions, Theorem 2–5). In this way, we can still identify a partial interaction struc- ture even if the given information set at x i is not a fully excited information set at x i . Step 5 – repetition We can finally identify the overall interaction structure by repeating the above procedures from Step 2 to Step 4 for each network node, i(1 £ i £ n). Illustrative examples An in-numero example A network with four nodes In order to illustrate the present method and to verify its result, we assume a set of artificial time-series data gen- erated from a network with a known interaction struc- ture and known dynamics. For this purpose, we assume a network composed of four nodes for which _ m i ¼ f i mðÞi ¼ 1; 2; 3; 4ðÞwhere f 1 (m) ¼ 0.1(m 3 ) 1), f 2 (m) ¼ 0.1(m 4 ) 1), f 3 (m) ¼ –{0.189 + 0.2( m 2 ) 1)}(m 1 ) 1) and f 4 (m) ¼ ) 0.1(m 1 ) 1) 2 ) {0.15–0.1(m 2 ) 1)}(m 2 ) 1) (Fig. 4A). To identify the assumed functional interaction structure, we apply the present method to each time interval of the generated data profiles where the increasing and the dec- reasing patterns are distinct. We reorganize the functional interaction structure by integrating the analysis results and Fig. 3. Flow diagram of the proposed experi- mental design procedures (FEIS in Step 2 represents a fully excited information set and TNS in Step 4 denotes true network signs). K H. Cho et al. Identification through temporal slope information FEBS Journal 272 (2005) 3950–3959 ª 2005 FEBS 3955 then validate the identified structure through comparison with the assumed original structure. Applying the present method to this system, we obtain the INS at m i in Fig. 5B from the artificially generated fully excited information set (Supplementary material, Supplementary impossible network signs for a detailed example deriving [s( f i1 ) ÁÁÁ, s( f i4 )]. We note that the signs of f 1j , f 2j (1 £ j £ 4) are fixed in this case. Hence, we can identify the signs of f 1j , f 2j from the FNS sum- marized in Fig. 5B. That is, S( f 13 ) ¼ S( f 24 ) ¼ ¯ as s( f 13 ) and s( f 24 ) are all fixed with ¯ in the FNS (Supplementary material, Supplementary mathematical descriptions, Theo- rem 2), and S( f 1j ) ¼ 0( j ¼ 1, 2, 4) and S( f 2j ) ¼ 0( j ¼ 1, 2, 3) as the nonzero s( f 1j )( j ¼ 1, 2, 4) and s( f 2j )( j ¼ 1, 2, 3) are variant in the FNS (Supplementary material, Supplementary mathematical descriptions, Theorem 5). On the other hand, we note that the signs of f 31 , f 42 are all É and f 3j , f 4j ( j ¼ 3, 4) are zero as 0.1 < m 1 < 1.9, 0.1< m 2 < 1.5 in the temporal expression profiles over (0,300], but the signs of f 32 , f 41 are variant. Thus, we cannot apply the notion of the fully excited information set to m 3 , m 4 in this case. Nevertheless we can determine the fixed signs of f 3j , f 4j ( j ¼ 3, 4) using the FNS at m 3 , m 4 summar- ized in Fig. 5B. Applying the present method to the tem- poral expression profiles in Fig. 5A, we know S( f 3j ) ¼ S( f 4j ) ¼ 0( j ¼ 3, 4) as s( f 3j ), s( f 4j )( j ¼ 3, 4) vary with ¯ and É in the FNS, and S( f 31 ) ¼ S( f 42 ) ¼Éas s( f 31 ), s( f 42 ) are fixed with É in the FNS (Supplementary material, Supplementary mathematical descriptions, Theorem 3). The final, identified interaction structure based on f ij from the FNS is depicted in Fig. 6A and the original postulated interaction structure represented by f ij of _ m i ¼ f i mðÞi ¼ 1; 2; 3; 4ðÞis shown in Fig. 6B. We can confirm that the identified structure through the present method is well in accord with the true structure. Although this simple example illustrates only a four-node case, the proposed method can be applied to any larger cases in the AB Fig. 5. Temporal expression profiles (A) and a summary of the analysis results (B) for the artificial example system. The small circle points on each temporal profile of m i indicate that we can find the corresponding information intervals on the time axis, the number of which is equal to the number of members in the given fully excited information set at m i . The symbol p denotes the case that there is (or are) corresponding one(s) from (f i1 , f i2 , f i3 , f i4 ) (for simplicity, we have omitted the cases that some or all of the f ij values are zero in the ANS of f ij at m i ). AB Fig. 4. An artificial model network with four nodes for the in-numero example (A) and a simple real example for the partial inter- action structure of the HOG pathway from S. cerevisiae (B) for illustration of the present identification method. The dotted lines denote the presumed unknown interaction structure to be identified. Identification through temporal slope information K H. Cho et al. 3956 FEBS Journal 272 (2005) 3950–3959 ª 2005 FEBS same manner (Supplementary material, Supplementary feas- ible network signs: a larger scale artificial example system). We note however, that the identification of a true inter- action structure for such larger cases heavily depends on the available fully excited information set from given temporal profiles and any a priori biological information on the functional interactions. We should consider the dynamic range of the system when we define an appropriate sampling rate for applica- tion of the proposed method. The requirement for defining the sampling rate is to discern the increasing ⁄ decreasing patterns of the time course profiles. If the given time-series measurements are uncertain in this respect, another experi- ment should be designed such that additional sampling time points are chosen to clarify such uncertain increasing ⁄ decreasing patterns. We note here that the proposed method is relatively robust with respect to the particular time points sampled as compared to other methods that make use of the measured absolute values at each sampling time point. For instance, the result in Fig. 6 is the same even if we choose any sampling time points within the time intervals of (0, 9.6) (46.1, 59.4) (61.4, 75.4) (154.4, 166.1) (180.8, 193.2) (222.2, 234.1) (270.2, 284.2) rather than the sampling time points used (i.e. 3.7, 49.2, 69.1, 164.6, 187.2, 232.7, 277.7). A simple real example The partial interaction structure of the HOG pathway in S. cerevisiae MAP kinase cascades typically composed of three tiers of protein kinases, a MAP kinase (MAPK), a MAPK kinase (MAPKK) and a MAPKK kinase (MAPKKK), are common signalling modules in eukaryotic cells [20,21]. The budding yeast (S. cerevisiae) has several MAPK cascades Fig. 6. The identified interaction structure (A) vs. the original postulated interaction structure (B) for the artificial example sys- tem. The dotted lines in A indicate that the present method is not applicable for identi- fying these functional interactions. The dual indications for the signs of f 32 and f 41 in (B) mean that these functional interactions are postulated to vary from ¯ into É or vice versa and thereby S(f 32 ) and S(f 41 ) are not identifiable by the present method. AB Fig. 7. Temporal expression profiles (A) and a summary of the analysis results (B) for the simple real example system. Note that we cannot identify the signs of f 1j (j ¼ 1,2,3) by applying the present method since there is another node (not considered in this model) directly affecting YPD1 other than YPD1(m 1 ), SSK1(m 2 ), and SSK2 (m 3 ). These are denoted N.A. in (B). Moreover, we learn that additional experiments are needed to further identify the signs of f 3j (j ¼ 1,2,3) due to the insufficiently excited information set (IEIS) at m 3 (i.e., the given experimental data is not a fully excited information set at m 3 in this case). Fig. 8. The identified partial interaction structure (A) vs. the known interaction structure from literature (B) for the simple real example system. The dotted line in A indicates that this functional inter- action is not identifiable due to the insufficiently excited information set at m 3 . K H. Cho et al. Identification through temporal slope information FEBS Journal 272 (2005) 3950–3959 ª 2005 FEBS 3957 including the HOG response pathway for hyperosmolarity adaptation [22–26]. Yeast cells respond to increases in extra- cellular osmolarity by activating the HOG1 MAPK, the function of which is to elevate the synthesis of glycerol [22]. Extracellular hyperosmolarity in yeast is detected by two independent transmembrane osmosensors, SHO1 and SLN1. SHO1 activates the PBS2 (MAPKK) through the STE11 (MAPKKK). Once activated by phosphorylation, PBS2 activates the HOG1 MAPK, which induces glycerol synthe- sis and other adaptive responses [24,26,27]. SLN1 osmosen- sor, which is a homologue of prokaryotic two-component signal transducers, utilizes a multistep phosphorelation mechanism that involves His and Asp phosphorylation sites within SLN1, another His phosphorylation site in the inter- mediary protein YPD1 and an Asp in the receiver domain protein SSK1 [23,25,26]. SLN1 autophosphorylates under normal conditions and further phosphorylates YPD1, which again phosphorylates and de-activates SSK1. Under hyper- osmotic conditions, SSK1 is not phosphorylated, but acti- vates the two MAPKKK SSK22 and SSK2, which in turn activate PBS2 and thereby HOG1 [28]. In this paper, we consider identification of the two functional interaction structures between YPD1 and SSK1, and SSK1 and SSK2 ⁄ SSK22 as designated within the box in Fig. 4B (the dotted lines indicate the presumed unknown interaction structures). We assume here that the amount of each signalling protein is proportional to the corresponding mRNA expression level. Figure 7A shows the temporal gene expression profiles extracted from the Stanford Microarray Databases (http:// genome-www5.stanford.edu) [29] where m 1 , m 2 , m 3 denote YPD1, SSK1, SSK2, respectively. Each data point in Fig. 7A denotes the log 2 -ratio between the measurement and the reference pool (Experimental data 1) [30] or the fkh1, 2 asynchronous (Experimental data 2). Note that we consider only a subset of three molecules from the pathway for an illustration of applying the proposed method since the experimental data do not provide the fully excited infor- mation for the remaining molecules in the pathway. Note that we cannot identify the functional interactions leading to m i from the given experimental data of m i (i ¼ 1, 2, 3), as another node (e.g. SLN1 in Fig. 1) other than m i (i ¼ 1, 2, 3) also directly affects m 1 . Regarding the func- tional interaction leading to m 2 , we can successfully identify its interaction structure (i.e. the signs of f 2j ( j ¼ 1, 2, 3) from the given experimental data (Figs 7B and 8A) and can con- firm that the identified interaction structure is well in accord with the known result of [26] (Fig. 8 and Supplementary material, Supplementary impossible network signs for detailed computation procedures). We note here that the two experi- ments with different initial conditions contribute to making the given experimental data into a fully excited information set with regard to m 2 . We cannot apply the present method however, to the case of m 3 as there is no T 3‘ that satisfies that the profile of m 3 is increasing (or decreasing) on the sub- interval T 1 3‘ of T 3‘ and decreasing (or increasing, respectively) on the subinterval T 3‘ À T 1 3‘ , and each profile of m 1 and m 2 is also clearly increasing or decreasing on T 3‘ . Acknowledgements This work was supported by a grant from the Korea Ministry of Science and Technology (Korean Systems Biology Research Grant, M10503010001– 05 N030100111) and by the 21C Frontier Microbial Genomics and Application Center Program, Ministry of Science & Technology (Grant MG05-0204-3-0), Republic of Korea. References 1 Kerr MK, Martin M & Churchill GA (2000) Analysis of variance for gene expression microarray data. J Com- putational Biol 7, 819–837. 2 Vandesompele J, de Preter K, Pattyn F, Poppe B, van Roy N, de Paepe A & Speleman F (2002) Accurate normalization of real-time quantitative RT-PCR data by geometric averaging of multiple internal control genes. Genome Biol 3, 1–12. 3 Wolkenhauer O, Moller-Levet C & Sanchez-Cabo F (2002) The curse of normalization. Comparative Func Genomics 3, 375–379. Fig. 9. A postulated experimental time-series profile with error bars due to noisy variations (A) vs. one example profile obtained by connect- ing chosen sampled data within the error bars (B). The slope information of all possible dynamic patterns does not change regardless of the chosen sampled data within the error bars since the error bars do not overlap in this case. Each interval of (I), (II), and (III) illustrates a con- ceptual simplification of the temporal slope changes. Identification through temporal slope information K H. Cho et al. 3958 FEBS Journal 272 (2005) 3950–3959 ª 2005 FEBS 4 Yang YH, Dudoit S, Luu P, Lin DM, Peng V, Ngai J & Speed TP (2002) Normalization for cDNA microarray data: A robust composite method addressing single and multiple slide systematic variation. Nucleic Acids Res 30, e15. 5 Mestl T, Plahte E & Omholt SW (1995) A mathematical framework for describing and analysing gene regulatory networks. J Theor Biol 176, 291–300. 6 D’Haeseleer P, Wen X, Fuhrman S & Somogyi R (1999) Linear modeling of mRNA expression levels dur- ing CNS development and injury. Pac Symp Biocomput 4, 41–52. 7 Chen T, He HL & Church GM (1999) Modeling gene expression with differential equations. Pac Symp Bio- comput 4, 29–40. 8 Arkin A, Ross J & McAdams HH (1998) Stochastic kinetic analysis of developmental pathway bifurcation in phage lambda-infected Escherichia coli cells. Genetics 149, 633–648. 9 Weaver DC, Workman CT & Stormo GD (1999) Mod- eling regulatory networks with weight matrices. Pac Symp Biocomput 4 , 112–123. 10 D’Haeseleer P, Liang S & Somogyi R (2000) Genetic network inference: from co-expression cluster- ing to reverse engineering. Bioinformatics 16, 707–726. 11 Friedman N, Linial M, Nachman I & Pe’er D (2000) Using Bayesian networks to analyze expression data. J Comput Biol 7, 601–620. 12 Ong IM, Glasner JD & Page D (2002) Modelling regu- latory pathways in E. coli from time series expression profiles. Bioinformatics 18, S241–S248. 13 Pe’er D, Regev A, Elidan G & Friedman N (2001) Inferring subnetworks from perturbed expression pro- files. Bioinformatics 17, S215–S224. 14 Kholodenko BN, Kiyatkin A, Bruggeman FJ, Sontag E, Westerhoff HV & Hoek JB (2002) Untangling the wires: a strategy to trace functional interactions in signaling and gene networks. Proc Natl Acad Sci USA 99, 12841–12846. 15 Sontag E, Kiyatkin A & Kholodenko BN (2004) Infer- ring dynamic architecture of cellular networks using time series of gene expression, protein and metabolite data. Bioinformatics 20, 1877–1886. 16 Reference withdrawn. 17 Brown GC, Hoek JB & Kholodenko BN (1997) Why do protein kinase cascades have more than one level? Trends Biochem Sci 22, 288. 18 Bruggeman FJ, Westerhoff HV, Hoek JB & Kholodenko BN (2002) Modular response analysis of cellular regulatory networks. J Theor Biol 218, 507–520. 19 Kholodenko BN, Hoek JB, Westerhoff HV & Brown GC (1997) Quantification of information transfer via cellular signal transduction pathways. [Erratum (1997) FEBS Lett. 419, 150] FEBS Lett 414, 430–434. 20 Posas F & Saito H (1998) Activation of the yeast SSK2 MAP kinase kinase kinase by the SSK1 two-component response regulator. EMBO J 17, 1385–1394. 21 Waskiewicz AJ & Cooper JA (1995) Mitogen and stress response pathways: MAP kinase cascades and phospha- tase regulation in mammals and yeast. Curr Opin Cell Biol 7, 798–805. 22 Brewster JL, de Valoir T, Dwyer ND, Winter E & Gustin MC (1993) An osmosensing signal transduction pathway in yeast. Science 259, 1760–1763. 23 Maeda T, Wurgler-Murphy SM & Saito H (1994) A two-component system that regulates an osmosensing MAP kinase cascade in yeast. Nature 369, 242–245. 24 Maeda T, Takekawa M & Saito H (1995) Activation of yeast PBS2 MAPKK by MAPKKKs or by binding of an SH3-containing osmosensor. Science 269, 554–558. 25 Posas F, Wurgler-Murphy SM, Maeda T, Witten EA, Thai TC & Saito H (1996) Yeast HOG1 MAP kinase cascade is regulated by a multistep phosphorelay mechanism in the SLN1-YPD1-SSK1 ‘two-component’ osmosensor. Cell 86, 865–875. 26 Posas F, Witten EA & Saito H (1998) Requirement of STE50 for osmostress-induced activation of the STE11 mitogen-activated protein kinase kinase kinase in the high-osmolarity glycerol response pathway. Mol Cell Biol 18, 5788–5796. 27 Posas F & Saito H (1997) Osmotic activation of the HOG MAPK pathway via Ste11p MAPKKK: scaffold role of Pbs2p MAPKK. Science 276, 1702–1705. 28 Morris PC (2001) MAP kinase signal transduction pathways in plants. New Phytologist 151, 67–89. 29 Zhu G, Spellman PT, Volpe T, Brown PO, Botstein D, Davis TN & Futcher B (2000) Two yeast forkhead genes regulate the cell cycle and pseudohyphal growth. Nature 406, 90–94. 30 Gasch AP, Spellman PT, Kao CM, Carmel-Harel O, Eisen MB, Storz G, Botstein D & Brown PO (2000) Genomic expression programs in the response of yeast cells to environmental changes. Mol Biol Cell 11, 4241– 4257. Supplementary material The following supplementary material is available for this article online. Appendix S1 containing the sections: Supplementary mathematical descriptions; Supplementary impossible network signs: the artificial example system; Supple- mentary impossible network signs: the simple real example system; Supplementary feasible network signs: a larger scale artificial example system. K H. Cho et al. Identification through temporal slope information FEBS Journal 272 (2005) 3950–3959 ª 2005 FEBS 3959 . Unravelling the functional interaction structure of a cellular network from temporal slope information of experimental data Kwang-Hyun Cho 1,2 ,. available specific information from each part of experimental data in detail and show the functional interaction structure obtained by integrating such information.

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