Brain structures important for “timing” are also discussed to be critical for the deterioration of movements in Huntington's disease (HD). Changes in “timing processes” are found in HD, but no study has varied the degree of motor demands in timing functions in parallel in HD. It may be hypothesized that timing functions may be deteriorated to a different extent in motor and non-motor timing, because in motor timing the underlying brain structures may be more demanding than in non-motor timing.We assessed timing in two different experiments: a time-estimation (TE) and a time-discrimination (TD) task. The demand on motor functions is high in the TE-task and low in the TD-task. Furthermore, general motor ability was assessed at different complexity levels. A presymptomatic (pHD), a symptomatic (HD) and a control group were investigated. We found a decline in timing functions when demands on the motor system were high (TE-task), in HD and even in pHD, compared to controls. In non-motor timing (TD task) and in the assessment of general motor ability, performance in the pHD-group was comparable to the controls and better than in the symptomatic group. Performance in both timing tasks was related to the duration until the estimated age of onset in pHDs.The study shows a selective deterioration of time-estimation processes in symptomatic and even presymptomatic Huntington's disease. Time-discrimination processes were not affected in both patient groups. The relation of timing performance to the duration until the estimated age of onset in pHD is of clinical importance. Huntington's disease (HD) is an autosomal dominant, monogentic neurological disorder causing a degeneration of the neostriatum. The disease is genetically expressed by an extension of the CAG-repeat length at the 4th chromosome Optimal motor functioning requires a highly precise timing of the coordination of muscles involved in a movement. If timing functions are crucial, it may be speculated that deficits in cognitive timing functions may underly the voluntary movement disturbances in HD. This seems likely, since “time processing” is mediated via brain systems that are affected in HD and important for movement execution: i.e. cortico-striatal circuits, especially the MSPs, the SMA and the dopamine system In summary, the study aimed at assessing timing processes with varying demands on the motor system in presymptomatic (pHD), symptomatic HD (HD) and healthy controls. Furthermore the relation of possible timing dysfunctions to voluntary movement dysfunctions in HD will be investigated. Finally we want to explore the value of the assessment of timing for the diagnosis of HD.The results show a selective deterioration of time-estimation processes in presymptomatic Huntington's disease. Time-discrimination processes, as well as general motor performance were not dysfunctional, compared to healthy controls. In the pHD-group performance in the TE and TD-task was related to the estimated age of onset (eAO). In the HD-group, performance was poor in all tasks.The aim of this study was to assess to assess timing processes with varying demands on the motor system in presymptomatic (pHD), symptomatic HD (HD) and healthy controls to estimate if the assessment of timing of timing functions may be of clinical value in the diagnosis of HD.A group of twelve right-handed presymptomatic gene mutation carriers defined by a positive gene test and absence of clinical symptoms  from 22 to 52 years of age  were recruited. The mean CAG-length was 42.58 . Calculation of the estimated age of onset (eAO) Additionally, a group of ten unmedicated symptomatic gene mutation carries  from 21 to 57 years of age  were recruited. The mean CAG-length was 47.10 . The mean age of onset (AO) was 34.20 . All patients  agreed to be videotaped to document their neurological status. Scores for each group  of the UHDRS-motor score, UHDRS cognitive score, TFC, IS, MMSE, BDI and YMRS are given in In this study four different tasks were performened. Two tasks were designed to assess timing functions (time estimation (TE); time discrimination (TD), two tasks were designed to assess general motor functions (continuous reaction task (CRT); tapping (TA). The tasks are described in detail, below. The sequence of conducted tests  was counterbalanced in all groups. All four tasks were trained by the subjects until stable performance was reached.In this task subjects were presented with a white square (2.5×2.5 cm size) on a black computer screen. Subjects were required to press a response key as exactly as possible at 1200 ms after onset of a stimulus (a square). This time, which includes the estimated time for the motor response, had to be estimated by the subjects, i. e. the motor response defines the length of the time interval. Feedback was given if the response was made in a time range from 1000–1400 ms or not. If the response was made within that time range, positive feedback was given. If not, negative feedback was given; 120 trials were conducted.after the stimulus was presented. Thus the time interval is not defined by a precise motor response, making timing not as heavily dependent on precise motor function as in the TE-task.Time discrimination was assessed using a two-alternative-forced-choice paradigm. At the beginning of this task white letters “S” and “L” (2.5×2.5 cm size) were presented on a black computer screen. The stimuli differed in their duration of presentation. “S” was presented for 1000 ms and “L” was presented for 1200 ms. The subjects had to react with the left hand on “S”-presentations and with the right hand on “L”-presentations. After the subjects were familiar with the different presentation times of “S” and “L” they underwent 120 trials, in which the letter “H” was presented instead of “S” and “L”. The presentation times of “H” randomly changed between 1000 ms and 1200 ms  and the subjects had to decide using the appropriate response, if the presentation of the letter “H” was presented for a short (i.e. 1000 ms) or long period (i.e. 1200 ms). This decision had to be carried out In this task subjects were presented with white arrowheads , either pointing to the left or to the right. The subjects were required to react with the left hand when the arrowhead pointed to the left and with the right hand when the arrowhead pointed to the right. 60 arrowheads pointed in the left and 60 pointed to the right direction. The subjects were required to respond as fast as possible.This task assesses simple movements Asymptomatic gene carriers as well as symptomatic patients were recruited from the local Huntington Centre of the Neurological Clinic of the University of Bochum. They were selected on the basis of pre-clinical counselling in the case of asymptomatic gene mutation carries or on the basis of routine clinical counselling in the case of symptomatic patients. Healthy controls were recruited by newspaper announcements. All participants gave written informed consent. For the symptomatic individuals, a family member was aware of the recruitement for the study and was involved in the consent procedure. The study was approved by the ethics committee of the University of Bochum.P >.425; one-tailed). The TE- and TD-task were analyzed in a repeated-measures ANOVA using the within-subject factor “experiment” (TE vs. TD) and the between-subject factor group . Both the CRT- as well as the tapping task (TA) were analyzed using separate univariate ANOVAs with the between-subject factor group . The degrees of freedom were adjusted using the Greenhouse-Geisser-Correction when appropriate. Significances are given one-tailed, due to higher test-power. In addition effect sizes (η) are reported. The mean and standard deviation (±SD) are given. Post-hoc tests were calculated using the Bonferroni-correction. Due to higher test power, one-sided tests were performed. For statistical analysis SPSS 11.0 was used.Data from twelve presymptomatic gene mutation carriers (pHD) (N = 12), ten symptomatic subjects (HD) (N = 10) and twelve healthy controls (N = 12) was analyzed. There were no drop-outs. For the TE- and TD-task the number of error trials was analyzed. For the CRT-task the mean reaction time across all trials was analyzed. For the tapping task (TA) the mean number of performed taps as well as the intertap-interval was analyzed. Tests of normal distribution within each group using the Kolmogorov-Smirnov test revealed that each variable included to the ANOVAs and correlational analysis was normal distributed  showed that the number of errors increased from controls (36.58±16.23) to pHD (50.58±27.08) and HD (79.05±11.20). The main-effect “experiment”  = 166.29; p<.001; η = .84), showed poorer performance (i.e. more errors) in the TE-task (68.48±17.60) than in the TD-task (40.26±26.02). Both effects interacted with each other  = 37.05; p<.001; η = .70). Subsequent univariate ANOVAS revealed a group-effect in both tasks  = 20.83; p<.001; η = .57; TD:  = 128.01; p<.001; η = .89). Post-hoc tests for the TE-task showed that controls (50.33±8.02) performed better than both HD-groups  (p<.001), which did not differ from each other (p = .637) (p = .461), and both performed better than the HD-group (77.30±6.65) (P<.001) .Results of the performance in each test are given in  = .637) . For the(P<.001) . For thep<.001; η = .58) with the pHDs and controls not differing from each other  (p>.9), but performing better than the HD-group (518.6±44.21) (p<.001)  with pHDs and controls not differing from each other  (p >.9), and performing better than the HD-group (137±27.2) (p<.001). For the calculated inter-tap interval there was the same effect than for absolute frequencies  = 19.45; p<.001) with the pHDs and controls not differing from each other  (p>.9) and performing better than the HD-group (480.43±16.55) (p<.001).The CRT-task also revealed a group-effect  = 21.59; (p<.001) . The sam(p<.001)  P = .007; TD: r = −.502, p = .048) (refer: p = .391), tapping  was not.In pHDs performance in the TE and the TD tasks was related to the duration until the eAO in the pHDs ) refer: , while pIn sum, both HD-groups showed comparable performance in motor embedded timing (TE) and performed more poorly than the control group, while in TD controls and pHDs showed comparable performance The results suggest that in presymptomatic gene mutations carriers (pHD) performance in the TE-task is deteriorated, while performance in the TD-, CRT- and TA-task is still normal. Additionally, performance in the timing tasks (i.e. TE- and TD-task) was related to the estimated age of onset (eAO) in pHD. For the symptomatic group (HD) performance was generally poor.The result found for the TD-task is in line with the findings by Paulsen et al. Regarding the “neuroanatomy of timing” striatal medium spiny neurons (MSPs), the dopamine system and the SMA play a central role in a network mediating time control Despite performance in all tasks being poor in HD, only performance in timing-tasks was inversely related to the duration until the estimated age of onset (eAO) in pHDs Future studies may incorporate larger sample sizes in order to increase generalizability of results. What may also be critical in the current study is, that CAG-repeat sizes differed between the presymptomatic (pHD) and symptomatic group (HD), since different CAG-repeat size may be accompanied by different strengths of pathogenic processes. Yet, this was side effect of matching the groups with respect to their age, since tasks assessing cognitive timing functions are known to be age-sensitive. Since this study was restricted to behavioural data, the precise brain mechanisms underlying the observed pattern of results must remain hypothetical.In summary, the study shows a selective deterioration of time-estimation processes in presymptomatic Huntington's disease. Time-discrimination processes were not affected. It may be inferred that under special circumstances voluntary movements can be deteriorated even in the preclinical stage of HD. These may depend on the time-scale in which the movement takes place. Furthermore, performance in the TE- and TD-task is related to the duration until the likely onset of motor symptoms in HD. Future studies may be done in a longitudinal manner to better document the time course of preclinical disease progression and strengthen clinical importance. This may also be an important step in order to estimate if these easy and fast applicable tasks may be implemented in large clinical trials.


The algorithm outputs the minimum free energy structure.The analysis of sequence-structure relations of RNA is based on a specific notion and folding of RNA structure. The notion of coarse grained structure employed here is that of canonical RNA pseudoknot contact-structures with at most two mutually crossing bonds (3-noncrossing). These structures are folded by a novel, After giving some background on RNA pseudoknot structures and providing an outline of the folding algorithm being employed, we present in this paper various, statistical results on the mapping from RNA sequences into 3-noncrossing RNA pseudoknot structures. We study properties, like the fraction of pseudoknot structures, the dominant pseudoknot-shapes, neutral walks, neutral neighbors and local connectivity. We then put our results into context of molecular evolution of RNA.Our results imply that, in analogy to RNA secondary structures, 3-noncrossing pseudoknot RNA represents a molecular phenotype that is well suited for molecular and in particular neutral evolution. We can conclude that extended, percolating neutral networks of pseudoknot RNA exist. It is represented by drawing the vertices in a horizontal line and its arcs , where i <j, in the upper half-plane. The vertices and arcs correspond to nucleotides and Watson-Crick  and (U-G) base pairs, respectively. A diagram is k-noncrossing if it contains at most k - 1 mutually crossing arcs. Diagrams have the following three key parameters: the maximum number of mutually crossing arcs, k - 1, the minimum arc-length, λ, and minimum stack-length, τ, The length of an arc  is j - i and a stack of length τ is a sequence of "parallel" arcs of the formWe will show that the notion of diagrams  allows uλ a λ-arc. Biophysical constraints on the base pairings imply that in all RNA structures λ is greater than or equal to four. We call diagrams with a minimum stack-length τ, τ-canonical and if λ ≥ 4 we refer to diagrams as structures. To reiterate, in the simplest case we have 2-noncrossing RNA structures, i.e. the secondary structures in which no two arcs cross, see Figure O(n3)-time and O(n2)-space.see Figure ViennaRNA , having the following propertiesA ⟨(M1) it has a nonnesting coreτ = 3 and length λ = 4.(M2) all its arcs are contained in stacks of length exactly m-shadow is a k-noncrossing diagram obtained by successively increasing the stacks of m from top to bottom, see Figure A The key observation about motifs is that they can, despite the fact that they exhibit cross-serial dependencies, be generated inductively .ISi,j, over . ISi,j is a skeleton which has no nested arcs, see Figure Skeleta represent the non-inductive "frames" of pseudoknot RNA, i.e. skeleta entail exactly the cross-serial dependencies, that need to be considered exhaustively. A skeleton, S, is a 3-noncrossing structure, whose core has a connected L-graph. An L-graph is a diagram whose arcs are the vertices and two being adjacent if their corresponding arcs cross . An irreGiven a skeleton, cross saturates or "fills" via context-sensitive DP routines the skeleton-intervals. Note that, while the inserted substructures cannot cross any arc of the skeleton, they will in general contain crossing arcs within themselves.To summarize, first cross inductively constructs all roots of the skeleta-trees, second cross generates the skeleta-trees themselves and third it saturates the skeleta.α be an arc in the 3-noncrossing RNA structure, S and denote by AS(β) the set of S-arcs that cross β. Clearly, we have β ∈ AS(α) if and only if α ∈ AS(β). An arc α ∈ AS(β) is called a minimal, β-crossing arc if there exists no α' ∈ AS(β) such that α' ≺ α.We next discuss loops of 3-noncrossing RNA structures. Loops are not only the basic building blocks for the mfe-evaluation but also of importance for the coarse grained notion of pseudoknot-shapes, discussed in Subsection. Let i, j] denote the sequenceLet the interval  ) and subject to the following condition: if all ωh, τh), i.e. all substructures are simply arcs, for all h, then h = 2.where (4) a pseudoknot, see Figure (P1) a set of arcsi1 = min{is} and jt = max{js}, such thatwhere P is irreducible, i.e. the line-graph of P is connected and(i) the diagram induced by the arc-set is, js) ∈ P there exists some arc β (not necessarily contained in P) such that  is minimal β-crossing.(ii) for each  all vertices k.We now show that each 3-noncrossing RNA structure can uniquely be constructed by simpler substructures . FurtherTheorem. Suppose k ≥ 2,τ ≥ 3.Any k-noncrossing, t-canonical RNA structure corresponds to an unique sequence of shadows.(a) Any ⟨3. τ⟩-structure has an unique loop-decomposition.(b) In Figure Our results are organized in two sections. First we describe our findings with respect to the statistics of pseudoknot RNA structures and second we present our data with respect to the particular organization of the sequences in neutral networks.3, which generates 3-noncrossing, 3-canonical mfe-structures and second, cross4, which produces 3-noncrossing, 4-canonical mfe-structures.In this section we present some key statistics on pseudoknotted RNA structures. In order to put our findings into context we consider two variants of cross: first, cross3 and cross4. Figure n = 100, approximately 50% of the structures folded by both versions of cross contain pseudoknots.We next compute the fraction of RNA structures with pseudoknots within all structures for crossk-noncrossing cores  denote the set of S-compatible sequences, consisting of all sequences that have at any two paired positions one of the 6 nucleotide pairsLet us consider a fixed RNA structure, S motivates to consider a new adjacency relation within C [S]. Indeed, we may reorganize a sequence  into the pairThe structure uj denote the unpaired nucleotides and the pj =  all base pairs, respectively, see Figure vu =  ↦ )) or Hamming distance two  ↦ )). We call a u- or a p-neighbor, y, a compatible neighbor. If y is contained in the neutral network we refer to y as a neutral neighbor. This gives rise to consider the compatible- and neutral distance, denoted by C and N. These are the minimum length of a C[S]-path and path in the neutral network between v and v', respectively.Accordingly, there are two types of compatible neighbors in sequence space: u- and p-neighbors: a u-neighbor has Hamming distance one and differs exactly by a point mutation at an unpaired position. Analogously a p-neighbor differs by a compatible base pair-mutation, see Figure S(v). We then proceed inductively: assume vi is constructed. We randomly select some neutral (compatible) neighbor of vi, denoted by vi+1, subject to the condition dH > dH, where dH denotes the Hamming distance. If no such neighbor exists we choose some vi+1 ≠ vi with the property dH = dH. If all neutral vi-neighbors satisfy dH <dH we stop and output the integer dH. In Figure H-pseudoknot loop structure (a), second a hairpin-loop structure (b), third an interior-loop structure (c) and finally the phenylalanine tRNA structure (d), see Figure Our basic experiment is as follows: We select a (random) sequence, v and fold it into the structure v with respect to the structure S = S(v) is a u- or a p-neighbor, y, contained in the neutral network of S. It has Hamming distance one or two, depending on whether it is induced by a point or base pair mutation, see Figure S. The data presented here, are obtained in the course of the neutral walk experiments, displayed in Figure 4.Complementing the analysis of neutral walks, we study now the distribution of neutral neighbors. Recall that a neutral neighbor of a sequence n, of an n-cube alone does not imply that a small Hamming distance implies a small distance in Γn. For neutral sequences this means that two neutral sequences with Hamming distance less than four, are possibly connected via a neutral path of much greater length. Evidently, for molecular evolution it is therefore not connectivity but the existence of these short paths what matters. Local connectivity is a property which guarantees the existence of these short paths. If Γn is locally connected then a small Hamming distance does imply a Γn-distance scaled by at most a factor of Δ > 0. We shall begin by studying local connectivity for random induced subgraphs of n-cubes, i.e. where we select sequences with independent probability λn. Then we transfer the derived concepts to neutral networks of RNA pseudoknot structures.Connectivity of a subgraph, Γn is locally connected if and only if almost surely (a.s.)We call Γv, v' are in Γn. We immediately observe that, trivially, for any finite n such Δ exists. However, the key point is that (†) employs the notion "almost surely", i.e. it holds for arbitrary n.provided λn smaller than nδ/δ > 0 is arbitrarily small, there exists a.s. no finite Δ satisfying (†). On the other hand, for λn larger than or equal to nδ/n-cubes have giant components for λn = (1 + ε)/n, where ε > 0 [n-cubes.Random graph theory  shows thre ε > 0  we can cS and sequence v, contained in its neutral network. By construction, local connectivity refers to the two n-cubes S, see Figure Suppose we are given a structure S at v is given bybe the cardinality of the set of sequences in compatible distance two. Then the degree of local connectivity of DS(v) is the fraction of locally connected vertices of the compatible distance two neighbors of v, that can be obtained via a neutral path of length at most four.In other words, DS(vi) and the total number of locally connected sequences. Our findings are presented in Figure We perform the following experiment: we consider neutral walks for the UTR-pseudoknot structure of the mouse hepatitis virus displayed in Figure k-noncrossing canonical RNA structures and the resulting sequence to structure map employed for our experiments is new and represents a natural generalization of RNA secondary and bi-secondary structures. To be precise, bi-secondary structures are exactly planar 3-noncrossing RNA structures [RNA pseudoknot structures-in particular their statistical properties-are a fascinating and new territory. To our knowledge the only statistical data beyond RNA secondary structures were derived for bi-secondary structures in ,54. The ructures .n = 80 we have approximately 35% structures with nontrivial pseudoknots. In addition it is striking that basically all folded structures are irreducible, i.e. only a very small fraction can be decomposed into several independent substructures. This is of interest since decomposable structures can be folded much faster. It is known, [x-axis, decompose on average into three irreducible parts. This is of relevance, since a slight generalization of Dyck-path, the Motzkin-paths, having additional horizontal steps, correspond to secondary structures. Our findings suggest, that while secondary structures, decompose nontrivially, higher and higher crossing numbers change the picture. This complicates the computation of mfe-pseudoknot RNA due to their imminent irreducibility.It is clear, that for sequence-length less than or equal to 100 we only encounter pseudoknots of limited complexity. Our findings presented in Figure s known,  that Dyc3 exhibits more structural variety due to the fact that its minimum stack-length is only three. The ratio of pseudoknot structures shifts significantly from n = 80 to n = 100 to approximately 50%. We can conclude from this that pseudoknots cannot be ignored, they evidently become the dominant structure class for n greater than or equal to 100. Figure 3 and cross4 coincide, decreases linearly as a function of sequence length. This indicates that larger and larger sequences will exhibit more subtle structural elements whose emergence is facilitated by stabilizing large stacks.Both versions of cross produce analogous findings, confirming the generality of our results. The vast majority of pseudoknot-shapes is of a single type. As expected, crossk-noncrossing canonical RNA structures, given in Table 4 and cross5, have for n = 76 two stacks less than random 3-noncrossing structures with minimum stack-length greater than four and five, respectively. This deviation is significant and indicates that mfe-pseudoknot structures are far from "typical" random structures. We remark that, while it is straightforward to generate random RNA secondary structures, it is nontrivial to obtain random pseudoknot structures. In particular, at present time, no polynomial time algorithm is known which generates a random 3-noncrossing RNA structure with uniform probability.Furthermore, the mfe-pseudoknot structures generated by cross are far from being random. The central limit theorems for random The organization of the sequences contained in neutral networks of RNA pseudoknot structures seems to be very analogous to the neutral networks of RNA secondary structures . Figure n. However, the high neutrality degrees of RNA pseudoknot structures of Figure We discussed in Subsection local connectivity, a property of neutral networks which implies the existence of short, neutral paths. It is apparent that local connectivity is of central importance for molecular evolution and any type of evolutionary optimization. It has been shown in  that locUTR: Untranslated Region; HDV: Hepatitis Delta Virus; DP:dynamic program; lhs: left hand side; rhs: right hand side.The authors declare that they have no competing interests.All authors contributed equally to this paper.


To maximize the time-integrated X-ray flux from multiple X-ray sources and shorten the data acquisition process, a promising way is to allow overlapped projections from multiple sources being simultaneously on without involving the source multiplexing technology. The most challenging task in this configuration is to perform image reconstruction effectively and efficiently from overlapped projections. Inspired by the single-source simultaneous algebraic reconstruction technique (SART), we hereby develop a multisource SART-type reconstruction algorithm regularized by a sparsity-oriented constraint in the soft-threshold filtering framework to reconstruct images from overlapped projections. Our numerical simulation results verify the correctness of the proposed algorithm and demonstrate the advantage of image reconstruction from overlapped projections. Since the first computed tomography (CT) scanner was made , all the Interestingly, the technology of field-emission X-ray sources based on carbon nanotubes (CNT) is a recent invention with several intrinsic advantages over conventional X-ray tubes. To maximize the time-integrated X-ray flux from multiple sources and improve the signal-to-noise ratio (SNR), multiplexing was used, where multi-source are excited with different temporal modulations . If two  Here, we consider how to reconstruct an image from overlapped projections. Previously, our group developed a generalized simultaneous algebraic reconstruction technique (SART) algorithm to reconstruct an image from data collected with an X-ray line source . Assumin The paper is organized as follows. In the next section, we will formulate the overlapped projection model, which is not a sum of line integrals and quite different from that in the single-source case. In the third section, we will design a SART-type reconstruction algorithm for image reconstruction from overlapped projections. In the fourth section, we will report numerical results. In the last section, we will discuss related issues and conclude the paper. f =  ∈ ℝI × ℝJ, where the index 1 ≤ i ≤ I and 1 ≤ j ≤ J are integers. DefineFor CT reconstruction, a two-dimensional digital image can be expressed as n ≤ N and N = I × J, we have the image in a vector representation f = T ∈ ℝN. In this paper, we will use both the signs fi,j and fn for convenience. Let g = T ∈ ℝM be a measured vector with M being the product of the number of projections and the number of detector elements. They are linked by the following linear system: A =  ∈ ℝM × ℝN is a measurement matrix. In a typical fan-beam geometry, the nth pixel can be viewed as a rectangular region with a constant value fn, the mth measured datum gm as an integral of partially covered pixel areas by a narrow beam from an X-ray source to a detector element which are weighted by the corresponding X-ray linear attenuation coefficients. Thus, the component am,n in  or ordered-subset (OS) version of a general Landweber scheme , of whicAm is the mth row of A, k the iteration index, and 0 < λk < 2 a free relaxation parameter. Let ΛN+ ∈ ℝN × ℝN be a diagonal matrix with Λn,nN+ = 1/an+ and ΛM+ ∈ ℝM × ℝM be a diagonal matrix with Λm,mM+ = 1/am+, λk for k : = 0; initialize (S.2.)N,k+ and ΛM+,k; compute (S.3.) initialize the error image (S.4.)q = 1 to Q backproject the projection error;  for (S.4.1.) weight the projection error (S.4.2.)M+,k; weight the projection error (S.4.3.)qth X-ray source; backproject the weighted projection error towards the (S.4.4.)N,k+; weight the backprojected error image by Λ(S.4.5.) add the backprojected image to (S.5.) update the current estimated image by(S.6.)k : = k + 1;  set (S.7.) go to (S.2) until the convergence criteria are satisfied;(S.8.) scale the reconstructed image k, the relax parameter λ, and the initial image λk = 1 and k. (S.3)-(S.4) computes the current error image for one step iteration according to  initializes the iteration index rding to . BecauseThe conventional data acquisition is based on the Nyquist sampling theory, which states that for accurate reconstruction of a band-limited signal or image the sampling rate must be at least twice the highest frequency of the signal or image. However, the recently developed CS theory shows that a high-quality signal or image can be reconstructed from far fewer measurements than what is usually required by the Nyquist sampling theorem , 8. In l(SS.5.) perform a soft-threshold filtering operation for (SS.5.1.) compute the sparse transform;(SS.5.2.) estimate the optimal threshold;(SS.5.3.)perform the soft-threshold filtering;(SS.5.4.)perform the inverse sparse transform. In the above pseudo-code, the sparse transform in (SS.5.1) can be either any invertible lossless compressible transform such as wavelet transform  and FourTo verify the proposed SART-type algorithm for image reconstruction from overlapped projections, we implemented it in Matlab on a PC, with the computationally intensive segments coded in C and linked via the MEX mechanism. As illustrated in λk in the SART formula (I0 = 5 × 104 photons per detector element [ For each of our selected numbers of projections over a full-scan range, we first equiangularly acquired the corresponding projection dataset based on the aforementioned projection model in the single, dual, and triple source cases, respectively. Then, we reconstructed the images using our algorithm described in  formula  was set  element . The res In the CT field, the line integral model along an X-ray path has been widely used in consistency with Beer's law. However, it does not reflect the divergence due to the combination of the finite detector size and the source focal spot. As shown in  As far as the convergence of the proposed algorithm is concerned, it should converge to the least square solution in the cases of either noise-free and noisy projections. The reason is that the proposed method is in the framework of the general Landweber scheme, whose convergence has been well studied under quite general conditions , 13. Whe In conclusion, we have developed a SART-type algorithm for image reconstruction from overlapped projections. The algorithm has been verified and demonstrated in the numerical simulation. Our methodology has a potential to support more flexible designs of multi-source CT/micro-CT systems for better contrast and temporal resolution.


Eukaryotic cell cycle is a complex process and is precisely regulated at many levels. Many genes specific to the cell cycle are regulated transcriptionally and are expressed just before they are needed. To understand the cell cycle process, it is important to identify the cell cycle transcription factors (TFs) that regulate the expression of cell cycle-regulated genes.We developed a method to identify cell cycle TFs in yeast by integrating current ChIP-chip, mutant, transcription factor binding site (TFBS), and cell cycle gene expression data. We identified 17 cell cycle TFs, 12 of which are known cell cycle TFs, while the remaining five  are putative novel cell cycle TFs. For each cell cycle TF, we assigned specific cell cycle phases in which the TF functions and identified the time lag for the TF to exert regulatory effects on its target genes. We also identified 178 novel cell cycle-regulated genes, among which 59 have unknown functions, but they may now be annotated as cell cycle-regulated genes. Most of our predictions are supported by previous experimental or computational studies. Furthermore, a high confidence TF-gene regulatory matrix is derived as a byproduct of our method. Each TF-gene regulatory relationship in this matrix is supported by at least three data sources: gene expression, TFBS, and ChIP-chip or/and mutant data. We show that our method performs better than four existing methods for identifying yeast cell cycle TFs. Finally, an application of our method to different cell cycle gene expression datasets suggests that our method is robust.Our method is effective for identifying yeast cell cycle TFs and cell cycle-regulated genes. Many of our predictions are validated by the literature. Our study shows that integrating multiple data sources is a powerful approach to studying complex biological systems. Eukaryotic cell cycle is a complex process and is precisely regulated at many levels. One important aspect of this regulation is at the transcriptional level. That is, many genes specific to the cell cycle are transcribed just before they are needed . To haveet al. , where bi,j = 1 if (1) the p-value for TF j to bind the promoter of gene i is ≤ 0.01 in the ChIP-chip data and the promoter of gene i contains one or more binding sites of TF j or (2) the disruption of TF j results in a significant change of the expression of gene i and the promoter of gene i contains one or more binding sites of TF j. Otherwise, bi,j = 0.Using three data sources , we can construct a high-confidence TF-promoter binding matrix However, binding of a TF to the promoter of a gene does not necessarily imply regulation. Hence, additional information is required to solve this ambiguity inherent in the TF-promoter binding matrix. Using a time-lagged dynamic model of gene regulation, we can refine the TF-promoter binding matrix into a high-confidence TF-gene regulatory matrix. We consider the transcriptional regulatory mechanism of a target gene as a system with the regulatory profiles of several TFs as the inputs and the gene expression profile of the target gene as the output. The transcriptional regulation of a target gene is described by the following time-lagged dynamic system model -45y[t] represents the target gene's expression profile at time point t, k represents the target gene's basal expression level induced by RNA polymerase II, N denotes the number of TFs that bind to the promoter of the target gene (inferred from the TF-promoter binding matrix B), di indicates the regulatory ability of TF i, xi[t] represents the regulatory profile of TF i at time point t, τi indicates the time lag for TF i to exert a regulatory effect on the target gene's expression, λ indicates the degrading effect of the target gene's expression value y [t - 1] at time point t - 1 on the target gene's expression value y [t] at time point t and ε[t] denotes the stochastic noise due to the modeling error and the measuring error of the target gene's expression profile. ε[t] is assumed to be a Gaussian noise with mean zero and unknown standard deviation σ The biological meaning of Equation (1) is that y [t]  is determined by i at time point t - τi, where i = 1,..., N) and -λ·y[t - 1] (the degradation effect of the target gene at time point t - 1).where xi[t] (the regulatory profile of TF i at time point t) is defined as a sigmoid function of zi[t] (the gene expression profile of TF i at time point t) , y [v]} for i ∈ {1, 2, ..., N}, v ∈ {1, 2, ..., M}. Equation (2) at different time points can be put together as followsUsing the yeast cell cycle gene expression data from Spellman et al. , we can Y, Φ, θ and e to represent Equation (3) as followswhere Y = [y[w] ... y [M]]T, Φ is the system matrix, θ = [d1 ... dN k λ]T is the unknown parameter vector, and e is the error vector, The parameter vector θ can be estimated by the maximum likelihood (ML) method as follows ) [p-value computed by the t-distribution is then adjusted by the Bonferroni correction to represent the true alpha level in the multiple hypotheses testing [i is said to be a true regulator of the target gene if the adjusted p-value padjusted ≤ 0.05.Since se ε[t]) . The p-v testing . FinallyB =  into a TF-gene regulatory matrix C = . In this matrix, ci,j = 1 if bi,j = 1 and if TF j is shown by the time-lagged dynamic system model to exert a significant regulatory effect on the expression of gene i. Otherwise, ci,j = 0.From the above analysis, we can refine the TF-promoter binding matrix et al. [j is a cell cycle TF is as follows. Let S be the set of cell cycle-regulated genes identified by Spellman et al. [G be the set of genes that are regulated by TF j (inferred from the TF-gene regulatory matrix), T = S ∩ G be the set of cell cycle-regulated genes that are also regulated by TF j, and F be the set of all genes in the yeast genome. Then the p-value for rejecting the null hypothesis (H0: TF j is not a cell cycle TF) is calculated asFrom the high-confidence TF-gene regulatory matrix, the regulatory targets of each of the 203 TFs in yeast can be inferred. Then a TF is said to be a cell cycle TF if a statistically significant portion of its regulatory targets are in the set of 800 cell cycle-regulated genes identified by Spellman et al. ,47. The G| means the number of genes in set G. This p-value is then adjusted by the Bonferroni correction to represent the true alpha level in the multiple hypotheses testing [j is said to be a cell cycle TF if the adjusted p-value padjusted ≤ 0.05. This procedure is applied to each of the 203 TFs under study.where | testing . TF j isX phase  if a statistically significant portion of its regulatory targets belong to the X phase cell cycle-regulated genes identified by Spellman et al. [G and F are defined as before, S now denotes the set of X phase cell cycle-regulated genes identified by Spellman et al. [T = S ∩ G now denotes the set of X phase cell cycle-regulated genes that are also regulated by the cell cycle TF under study. The p-value computed by Equation (4) is then adjusted by the Bonferroni correction to represent the true alpha level in the multiple hypotheses testing. We say that a cell cycle TF functions in the X phase  if the adjusted p-value padjusted ≤ 0.05.For each of the 17 identified cell cycle TFs, we want to determine in which cell cycle phases it functions. We regard that a cell cycle TF functions in the WSW developed the algorithm, performed the simulation and wrote the manuscript. WHL conceived the research topic, provided essential guidance and revised the manuscript. All authors read and approved the final manuscript.Supplementary Table 1 contains the detailed information of the 17 identified cell cycle TFs, the 178 novel cell cycle-regulated gene (including 59 uncharacterized genes), the results of using different cell cycle gene expression datasets, and performance comparison between our method and four existing methods.Click here for file


Knowledge of the subcellular localization is also essential for genome annotation and drug discovery. Supervised machine learning methods for predicting the localization of a protein in a cell rely on the availability of large amounts of labeled data. However, because of the high cost and effort involved in labeling the data, the amount of labeled data is quite small compared to the amount of unlabeled data. Hence, there is a growing interest in developing unlabeled data. We compare semi-supervised AAMMs with: (i) Markov models (MMs) (which do not take advantage of unlabeled data); (ii) an expectation maximization (EM); and (iii) a co-training based approaches to semi-supervised training of MMs (that make use of unlabeled data).In this paper, we present an Abstraction Augmented Markov Model (AAMM) based approach to semi-supervised protein subcellular localization prediction problem. We investigate the effectiveness of AAMMs in exploiting The results of our experiments on three protein subcellular localization data sets show that semi-supervised AAMMs: (i) can effectively exploit unlabeled data; (ii) are more accurate than both the MMs and the EM based semi-supervised MMs; and (iii) are comparable in performance, and in some cases outperform, the co-training based semi-supervised MMs. The problem of predicting subcellular protein localization is important in cell biology, because it can provide valuable information for predicting protein function and protein-protein interactions. Furthermore, the location of proteins in their designated subcellular compartments is essential for the proper functioning of the cell. Abnormal subcellular localization has been correlated with diseases such as cancer .thk order Markov chain models, with k ranging from 1 to 8, and used an approximation technique to estimate the probability of each element in a sequence given the k contiguous previous elements.Many supervised machine learning methods have been successfully applied to the problem of predicting the subcellular localization of a protein, which can be formulated as a sequence classification problem , where tunlabeled data can provide valuable information, i.e., they contain information about the joint probability distribution over sequence elements. Consequently, there is a significant interest in semi-supervised algorithms ..abstractlabeled sequences LD using Equations (3), (4), and (5) . The current model is used to assign probabilistic labels to the  unlabeled sequences UD  using Equation (6) (E-step). Next, a new model is learned from originally labeled sequences EM applied to MMs (EM-MMs) involves an iterative process of E- and M-steps. Specifically, an initial Markov model is learned only from  another .LD be a set of labeled examples, and UD a set of unlabeled examples. A set U′D is obtained by sampling u examples from UD (we used u = 75 examples in experiments). Each example x has two views, i.e., can be encoded with two different sets of features, x(1) and x(2). First, use LD and the x(1) encoding to train a classifier h1, and LD and the x(2) encoding to train another classifier h2. Second, classify the examples in U′D using h1 and h2 separately. Select h1’s and h2’s top (jk)j=1,…,|C| most confident predictions from each class (corresponding to the underlying data distribution), add them to LD, and remove them from U′D. Sample 2 UD and move them to U′D. This process is repeated for a fixed number of iterations, or until all unlabeled data are used up   are clustered together if they induce similar conditional distributions of the next node, independent of the class. When the data are scarce, the estimates of joint probabilities are not reliable. However, the unlabeled data contain information about the joint probability distribution over sequence elements, and can help improve the statistical estimates of parameters. The abstraction hierarchy is subsequently used to learn a Markov model with abstract values of the parents.Our abstraction-based approach to learning classifiers for the protein subcellular localization prediction task exploits large amounts of m of an m-cut that defines an AAMM, an array of indices of size equal to the number of unique k-grams specifies the mapping between k-grams and abstractions . However, the number of parameters of AAMM (for a given class) based on such an abstraction hierarchy and an m-cut is m|X|, as opposed to |X|k|X| in the case of MMs, where m ≪ |X|k.It is worth mentioning that part of the AAMM is the representation of the clustering. Specifically, after the abstraction hierarchy is learned, for a given choice of the size abstraction. It is of interest to incorporate into AAMMs some means of gracefully trading off the complexity of the model against its predictive accuracy. One way to do this is to augment the algorithm, e.g., by designing an MDL-based scoring function to guide a top-down search for an optimal cut  is the number of times the symbol σ “follows” the k-gram s in the sequence xl, and p(ly =j c|xl) ∈ {0,1} is obtained based on the sequence label.where # is the number of times s occurs in xl.where # and # represent the number of times the symbol σ “follows” the k-gram s in the sequence xl, and xu, respectively.where #[k-grams, the context of an abstraction a = {s1,…, s|a|} is obtained by a weighted aggregation of the contexts of its k-grams. That is,Since an abstraction is a set of where a′ and a″ in D as follows:We identify the most “similar” abstractions as those that have the smallest weighted Jensen-Shannon divergence between their contexts. JS divergence  provideswhere  labeled training set θ from LD, denoted by k-gram s ∈ S . The estimates a = {s1,…,s|a|}, are a weighted aggregation of the estimates of a’s constituent k-grams, i.e.,Given awhere x0,…,nx−1 and an m-cut γm through T, k-gram ix−k…ix−1 find the abstractionw a ∈ γm it belongs to and retrieve the parameters associated withw a. Successively multiply i = k,…, n − 1 to obtain Given a new sequence x = As in MMs, apply Bayes rule to obtain The authors declare that they have no competing interests.All authors read and approved the final manuscript.


In this paper, the PI polynomial of the phenylenic nanotubes and nanotori are computed. Several open questions are also included.The PI polynomial of a molecular graph is defined to be the sum X G be a simple molecular graph without loops, directed and multiple edges. The vertex and edge sets of G are represented by V(G) and E(G), respectively. A topological index is a numeric quantity derived from the structural graph of a molecule. Usage of topological indices in chemistry began in 1947, when Harold Wiener developed the most widely known topological descriptor, the Wiener index, and used it to determine physical properties of the type of alkanes known as paraffins , see [Let G)], see  and 4]..G be a cet al. defined a new polynomial which they named the Padmakar-Ivan polynomial. They abbreviated this new polynomial as PI, for a molecular graph G. We define PI = ∑uv∈ V(G) XN where for an edge e = uv, N = neu(e|G) + nev(e|G) and zero otherwise. This polynomial is very important in computing the PI index. This newly proposed polynomial, PI, does not coincide with the Wiener polynomial ) for acyclic molecules.In a series of papers , 6 Ashraet al. investigated the structure and computed the Hosoya polynomial of some nanotubes and nanotori. Gutman et al.   et al. iAlso, in  and 12]et al. inn,2m]. We also denote a V-Phenylenic nanotorus by H=VPHY. Let G be an arbitrary graph. For every edge e, we definee), for every edge e ∈ E(G). From above the argument and mn–2n, |E(H)|=36mn and |V(T)| =24mn, |V(H)| =24mn. In the following theorem we compute the PI polynomial of the molecular graphT in Proof: To compute the PI polynomial of T, it is enough to calculate N(e). To do this, we consider three cases: that e is vertical, horizontal or oblique. If e is horizontal. a similar proof as Lemma 1 in [e)=8m. Also, if e is a vertical edge in one hexagon or octagon then N(e) = 4n, 2n, respectively.The novel phenylenic and naphthylenic lattices proposed can be contructed from a square net embedded on the toroidal surface. In this section, the PI polynomial of a V-Phenylenic nanotube and nanotorus are computed. Following Diudea  we denotrem 1 in  we have:e in A(T), we have two cases:We consider the set A(T) of oblique edges in T. For every Case 1:n.A similar argument as Lemma 2 in  gives thCase 2:th row of oblique edges in A(T) by Ai , by computations, we have N(e)=4n+2i-2, also if 2(m-|n-m|)+1 ≤ i ≤2m, then N(e)=8n-2. If m>n, then N(e)=8m. For n>m because of symmetry computations are similar to u er part of graph. So we have:We denote the iy Ai see . It is eAlso:Thus:Which completes the proof.H and calculate its Padmakar-Ivan polynomial, PI, In our next theorem we consider a V-Phenylenic nanotorus Proof: To prove the theorem, we apply a similar method as in Theorem 1. It is easily seen that N(e)=8n for each vertical edge in hexagons, that is two times more twice the tube case by horizontal symmetry. A vertical edge in an octagon has 2n parallels, as in Theorem 1. Also N(e) =8m for each horizontal edge. Let z = min{m,n}, for each oblique edge e we have N(e) = 6z − 2. So:We conclude our paper with the following open questions:Question 1: Let n. Is there a V-phenylenic nanotube or nanotorus T such that PI  = F(x)?Question 2: Is it true that for every polynomial F(x) with positive coefficients and of degree n, there exists a V-phenylenic nanotube or nanotorus T, such that PI  = F(x)?Question 3: What is the relation between the Hosoya polynomial and PI polynomial of a V-phenylenic nanotube or nanotorus?


We thank Tarone for his letter, as it provides an opportunity to elaborate on analytic strategies for the study of DDT associations with breast cancer.p,p′-DDT  is the primary ingredient of commercial grade DDT. p,p′-DDE , the most persistent DDT-related compound, is a metabolite of p,p′-DDT that is both made by humans during active exposure, and also ingested directly from food sources where it can be stored for long periods in fat -2-(o-chlorophenyl)ethane] is a low-concentration contaminant of commercial DDT that is eliminated by humans most quickly, making it a marker of recent exposure  of their exposure was only  Table 1 ]. Hence, Table 1 , suggestp,p′-DDT, p,p′-DDE, and o,p′-DDT allow evaluation of potential differences in the effects of these compounds. Other studies have also observed differing associations with cancer risk for p,p′-DDT and its metabolite, p,p′-DDE. p,p′-DDT association with risk of liver cancer was enhanced when p,p′-DDE was low. We also reported that a higher proportion of p,p′-DDT to p,p′-DDE in maternal serum samples was associated with longer time to pregnancy in their daughters 30 years after exposure in utero (p,p′-DDE—after adjustment for p,p′-DDT—for predicting breast cancer, particularly in postmenopausal women. We believe that simultaneous adjustment for DDT-related compounds is a strength of our study.Concurrent measurements of in utero . In anota priori and were a primary objective of our study. In this setting, subgroup analyses are a strength that enabled us to examine whether age at DDT exposure may be of importance in human breast cancer.Tarone suggests that subgroup analyses weaken the results of our article . Howeverin situ disease and its successful treatment, which would reduce incidence of invasive disease and mortality.The trends in breast cancer incidence in young women previously presented by Tarone in Table 1 of his article  do not r


Self-consistent field theory is used to study the self-assembly of a triblock copolymer melt. Two different external factors (temperature and solvent) are shown to affect the self-assembly. Either one or two-step self-assembly can be found as a function of temperature in the case of a neat triblock melt, or as a function of increasing solvent content (for non-selective solvents) in the case of a triblock-solvent mixture. For selective solvents, it is shown that increasing the solvent content leads to more complicated self-assembly mechanisms, including a reversed transition where order is found to increase instead of decreasing as expected, and re-entrant behavior where order is found to increase at first, and then decrease to a previous state of disorder. These molecules are much less studied than the less complicated diblock copolymers, because the addition of a third chemically distinct block greatly complicates the microphase separation properties of these self-assembling molecules. It is this increased morphological richness however that makes triblocks appealing, as they can form structures inaccessible to diblocks. It is not just a question of which structures will self-assemble, but also the manner in which the self-assembly takes place. Diblocks can only be in an ordered state or a disordered state, whereas triblock copolymers are the simplest possible copolymer than can be also in a “semi-ordered” state, where two of the blocks are mixed while the third alone is segregated. Triblock copolymers can therefore go from a completely disordered state to a completely ordered state through a multi-step process that holds promise for technological applications if it can be sufficiently understood and predicted.Triblock copolymers are linear polymers composed of three separate regions, or “blocks”, of monomers. Triblocks with three different chemical blocks are usually labeled as block-poly (D8-styrene)-block-poly(vinyl methyl ether) (PI-b-PDS-b-PVME) triblock terpolymers and found that ordering may proceed via a two-phase mechanism. These authors [Experimental studies of complex ordering in triblock systems were undertaken by Yamauchi , 2 who e authors  that seg authors  used sel authors , 2.Here, we build on the previous theoretical results  and show2.N, N, N, Af, Cf, Aa/Ca and Ba/Ca. These are, respectively, the segregation parameters between A and B segments, B and C segments, A and C segments, the volume fraction of A segments, the volume fraction of C segments, the ratio of the A statistical segment length to the C statistical segment length and the ratio of the B statistical segment length to the C statistical segment length. The segregation energies are standardly defined in terms of N, which is the product of the Flory-Huggins parameters N in a single triblock molecule, based on a reference segment volume Af and Cf need to be specified since through incompressibility Bf is not independent. For the current work, we will simplify the system further by restricting our attention to cases where Af C= f ≡ A/Cf = 0.25 (fB = 1 − Af − Cf = 0.5), in keeping with the previous work [Aa = Ba = Ca ≡ a. These restrictions will impose certain symmetries on the system, but should still allow the predominant structures and mechanisms to emerge as previously demonstrated for diblock systems [N = N = 50, but more general values will be considered here. In addition, we will also allow for the presence of solvent.Equilibrium morphologies for a neat, incompressible melt of monodisperse triblock copolymers can be calculated from the free energy functional given in ref. . The parous work . Similar systems . In our φ, with the volume fraction of the solvent being φs = 1 − φ. The second is the molecular volume ratio α, which is the ratio of the volume of a solvent molecule to the volume of a polymer molecule. This will be held fixed (α = 0.01) in this work. The third set is the interaction of the solvent with the polymer, which is determined by the three parameters N,N and N. The modified SCFT equations then areq = 1. s is the normalized contour length (0 ≤ s ≤ 1) of a polymer chain, and the field ω(r) is equal toq†  is also defined by the equationω(r) and the Lagrange multiplier ξ(r) (which is introduced to force incompressibility)j = A, B, C, and S. The single chain partition function and the monomer densities are thenF is given byIn order to introduce the effect of the solvent in the SCFT, we need to introduce more variables . The firare thenQ  = ∫dr   qφA(r)  = d2F/dφ2 > 0 for all φ; the mixture will macrophase separate if the curvature is negative [d2F/dφ2 is always positive so that macrophase separation is absent. Overall we have an eight dimensional parameter space: N, N, N, N, N, N, fa/c, φ. Obviously it is difficult to explore the whole space, so we will choose representative points that illustrate the wide variety of self-assembly mechanisms available to the triblock-solvent system. In particular, we wish to explicitly examine temperature dependence, and the effects of both non-selective and selective solvents in order to more closely parallel the experimental methods of Yamauchi et al. [The free energy curvature must be checked to ensure that negative . In the i et al. , 23.3.1.N = N fixed, and varying N. Although this shows in principle the possibility of engineering one or two-step self-assembly, it is not amenable to experiment nor is it the approach used by Yamauchi et al. [N) and temperature [A and B are constants related to the chemical interactions between species. Ignoring solvent (setting φ = 1), we drop B for qualitative analysis and we vary all A’s equally with temperature such that N = N = N ∝ T−1. This results in one-step self-assembly as shown in N = N = N ≲ 35.5 self-assembly is absent and the melt remains in a disordered state. At N = N = N ≃ 35.5 (marked as TP in N < 35.5 is the disordered phase where the free energy follows the expected linear dependence of N. Above TP (N > 35.5) the lowest free energy is realized by the hexagonal morphology illustrated in In Ref. , it was i et al. ,2. Here,perature , 10(13)A and B in ABA = BCA ≠ ACA in N, N and N to have temperature dependent and independent terms, but with N having a different temperature dependence factor. Since the relative values of the Flory-Huggins parameters vary with a different rate, the separation of different polymer components will occur at different temperatures and a two-step self-assembly will be observed. N = N = N. At high temperatures (low N) no phase separation occurs as all three components are able to mix. Lowering the temperature (increasing N) the A and C components becomes sufficiently immiscible with the B component to induce phase segregation (TP2 in TP1) to form the structure shown in If this symmetry in the interaction is now broken for any reason, then the transition will take place in two separate steps. There are many possible ways of lifting this symmetry as there are no known constraints on the values of n TP2 in , and the3.2.without using the dilution approximation. We fix N = N = N and N = N = N and vary the solvent volume fraction (1 − φ). The qualitative behavior of the melt with the solvent is found to be the same for a large range of the parameters describing the solvent-polymer interaction  all three components segregate into spatially iseparated domains. This scenario persists as small amounts of solvent is added to the mixture. Since the solvent is non-selective it distributes roughly uniformly throughout the system (see < 5 % above average solvent concentration) at the domain interfaces. This excess phenomena is known to be an energetic effect [ABA = BCA ≠ ACA and select any corresponding N, N and N  then upon adding non-selective solvent while holding these parameters fixed, we can induce two-step self-assembly just as if temperature were varied for components with an unequal temperature response. This is again consistent with dilution arguments. Thus either one or two-step self-assembly may be achieved through the use of non-selective solvent rather than through varying temperature.Similarly, if we choose 3.3.A and C components of the polymer than it is with the B component. Selecting N ≃ N < N, and keeping N = N = N we expect the presence of the selective solvent to produce a two-step self-assembly upon increasing solvent content. As the selective solvent is added to the polymer melt, it builds up primarily in the A and C rich domains (see φ ≃ 0.68 (TP1 in φ ≃ 0.64 causes a complete loss of structure as the order-disorder transition occurs (TP2).The symmetry in the interaction can also be broken by using a selective solvent. The selective solvent will have different concentration in regions occupied by different polymer components. This will affect the interactions between polymer components and break the symmetry. To demonstrate this effect we choose a solvent that is more miscible with the ains see , which i8 TP1 in  where thet al. [In general, the phase behavior of a triblock melt with solvent is very complicated. The complete phase diagram is embedded into a 6 + 2 dimensional parameter space (6 for the Flory-Huggins parameters plus two for the triblock composition and the volume fraction of polymer). So far we have dealt with simple transitions, which are straightforward extrapolations from a diblock to a triblock copolymer. Using polymers consisting of different components, and different types of solvent (which leads to different Flory-Huggins parameters) one might expect more complicated transitions. In the experiments by Yamauchi A and C component of our polymer should be mixed. To achieve miscibility of the A and C polymer components at low/no solvent density we have to chose a polymer with a weak interaction between these two components therefore we have to reduce the N parameter. Since we expect these two components to separate by adding solvent, this parameter cannot be much lower than the critical transition value. A selective solvent with a strong attractive interaction with the A component will tend to concentrate in the A rich domains. If at the same time this solvent strongly repels the C polymer component, it will cause an effective immiscibility of the A and C polymer components.Although it is very difficult to explore the full parameter space, we can determine a physical mechanism that explains this reverse transition. We assume that initially the N ≲ N ≲ N with N less than but very close to the order-disorder transition between A and C, and take φ = 1. A semi-ordered structure, results  induces re-entrant behavior in that the system becomes semi-ordered once again – A and C mix again as illustrated in A and C internal morphologies have reversed their patterns so that A now forms the cores in the mixed domains; this is similar to the reversal of morphologies seen in diblock solutions with strongly selective solvents [B-C interaction becoming increasingly diluted whereas both B and C interaction with the solvent remains very unfavorable. It therefore becomes attractive for the system to use the C monomers’ weak interactions with the A monomers to minimize the interaction with the solvent.Further increasing the solvent content  triblock melt, we find that one or two-step ordering is possible, the former occurring when N = N = N with all segregation being varied with temperature equally, and the latter occurring when N is lower than N or N.We have illustrated ordering mechanisms in a triblock copolymer system as they appear as a function temperature and (non-selective and selective) solvent content. For temperature dependent N values of A, B and C triblock constituents are held fixed at equal, or unequal values. For N = N = N, one-step ordering occurred, but for N = N ≠ N, two-step ordering resulted.The addition of non-selective solvent may also induce either one or two-step ordering processes depending on whether the Adding selective solvent causes two-step self-assembly when the solvent is sufficiently selective, and for certain choices of system parameters, a much more complicated route of self-assembly was found.


Their article should be interpreted with caution, particularly the estimated 5-fold increase in risk for women born after 1931 the authors reported without qualification in the “Abstract”; this value was repeated in the news article by p,p′-DDT, o,p′-DDT , and p,p′-DDE —by various categories of year of birth, yet they found no significantly increased risk estimates for any of the three DDT congeners in multiple comparisons that were not adjusted for the other DDT-related chemicals either in all women or in women born after 1931. The estimated 5-fold increase in risk for the upper tertile of p,p′-DDT serum levels was only observed in subgroup analyses that were both restricted to women born after 1931 and adjusted for serum level of o,p′-DDT. The impact of the adjustment for o,p′-DDT on the risk estimate for p,p′-DDT is remarkable in view of the low o,p′-DDT levels observed (35% were below the limit of detection). A significant inverse association between o,p′-DDT level and breast cancer risk, which was interpreted by Cohn et al. in terms of length of time since DDT exposure, became stronger after adjustment for p,p′-DDT levels; presumably this does not indicate a protective effect of recent DDT exposure.In a recent article, p,p′-DDE levels and breast cancer risk (In view of the absence of evidence for an association between cer risk , it seemcer risk . Women bcer risk , but thecer risk ; yet the


Several approaches, including metabolic control analysis (MCA), flux balance analysis (FBA), correlation metric construction (CMC), and biochemical circuit theory (BCT), have been developed for the quantitative analysis of complex biochemical networks. Here, we present a comprehensive theory of linear analysis for nonequilibrium steady-state (NESS) biochemical reaction networks that unites these disparate approaches in a common mathematical framework and thermodynamic basis.A obtained in the standard, linear-dynamic-stability analysis of the steady-state can be decomposed as A = SRT where R and S are directly related to the elasticity-coefficient matrix for the fluxes and chemical potentials in MCA, respectively; the control-coefficients for the fluxes and chemical potentials can be written in terms of RTBS and STBS respectively where matrix B is the inverse of A; the matrix S is precisely the stoichiometric matrix in FBA; and the matrix eAt plays a central role in CMC.In this theory a number of relationships between key matrices are introduced: the matrix One key finding that emerges from this analysis is that the well-known summation theorems in MCA take different forms depending on whether metabolic steady-state is maintained by flux injection or concentration clamping. We demonstrate that if rate-limiting steps exist in a biochemical pathway, they are the steps with smallest biochemical conductances and largest flux control-coefficients. We hypothesize that biochemical networks for cellular signaling have a different strategy for minimizing energy waste and being efficient than do biochemical networks for biosynthesis. We also discuss the intimate relationship between MCA and biochemical systems analysis (BSA). Developing methodologies for quantitative analysis, both experimental and mathematical, of complex biochemical networks has become one of the central themes of post-genomic biochemistry and mathematical biology. Several disparate approaches, including metabolic control analysis (MCA) ,2, flux The physical concept of a NESS  applies to that of a driven system, such as a living organism, with a constant flow of matter (transforming nutrients to waste) through the system and a corresponding dissipation of free energy . Even thA obtained in linear stability analysis, the flux control-coefficient matrix C in MCA, the stoichiometric matrix S of FBA, and the correlation matrix R from CMC are intimately related. We develop the relationships in the context of complex biochemical networks and their NESS thermodynamics. We show that all the key matrices in MCA and CMC can be computed if one knows the fluxes, chemical potential differences, and the stoichiometric matrix.We distinguish between two types of linear analysis: (i) linear stability analysis , and , fluxes (currents), and resistances. It becomes immediately clear that the resistances are related to reaction effectors, such as the expression levels of enzymes, that affect both the forward and backward fluxes of a given reaction, but do not change the chemical potential difference. It is this thermodynamic perspective of a NESS that can be used to develop a comprehensive theory that unites these approaches.J of each metabolic reaction in a NESS metabolic network is decomposed into J = φ+ - φ- and the chemical potential difference of the reaction in the NESS is Δμ = kBT ln(φ-/φ+), where kB is Boltzmann's constant and T is absolute temperature. Therefore, knowing the chemical potential difference and the flux of a reaction givesIn the context of the newly developed biochemical circuit theory (BCT) ,16, the μo is known from equilibrium thermodynamics, then Furthermore, if the standard-state free energy of reaction Δβ-cells and ketogenesis in hepatocytes, where cellular organelles, such as mitochondria, play a key role in the handshake between metabolism and signaling. Ultimately, our goal is to understand how these cellular systems behave in response to stress and disease [In many theoretical approaches, metabolic networks and signaling pathways are often separated into disparate mechanisms with only peripheral interactions. This is far from reality, however, as these mechanisms are intimately intertwined and often indistinguishable. For example, in whole-body metabolism, glucose and fatty acids serve as the fuel for cellular metabolism, but also as signals for insulin secretion in pancreatic  disease . It is iS, and measures the correlation matrix, R, to obtain the control coefficients, and vice versa . The jth biochemical reaction  is characterized by a set of stoichiometric coefficients Let us consider a network of ν's and κ's can be zero. The N × M matrix S = κ - ν is known as the stoichiometric matrix. Eq. (2) assumes that the forward and backward reaction kinetics are characterized by the constants Some of the integer s. Segel , for exaFor the reaction system (2), kinetic equations of the time-dependent concentration changes for each biochemical species can be written according to the law of mass action as ,20xℓto denote the concentration of respective Xℓ, and i = 1, 2, ..., N) are external injection fluxes to species i . We use the convention that subscripts index species and superscripts index reactions. To further simplify the notation, we introduce forward and backward fluxes where we use j=φ+j−φ−j:dx/dt = SJ + Je, in which x and Je are N-dimensional column vectors, and J is an M-dimensional column vector.Then Eq. (3) becomes closed reaction system, Je = 0. In this case, it can be shown that thermodynamic equilibrium is the only positive stationary solution to Eq. (3): the internal fluxes J = φ+ - φ- = 0 for each and every reaction. For a closed biochemical reaction system, since all the fluxes are necessarily zero in its unique equilibrium , all the control coefficients are necessarily zero. Therefore, in a NESS, at least one injection flux and one effux are nonzero , i.e., there is no unique stationary solution for different initial conditions.same NESS. We use the notations For an open system that approaches a NESS, the stationary solution is not usually degenerate. If we assume {In the linear regime near the NESS ,δxj = xj - in which A = {Aij} is called the linear stability matrix. We introduce the N × M matrix R = A = SRT. We also denote A-1 = B.S does not have full rank and A is singular [A into a nonsingular matrix by replacing its linearly dependent rows with vectors in its left null space (see Ref. 18 of [A is understood to be the transformed nonsingular matrix.If the nonlinear dynamics scheme of Eq. (3) conserves certain quantities, such as total element, motif, or enzyme concentrations, singular ,19,20. IEq. (3) is a general scheme for biochemical reaction networks. For metabolic reactions involving enzymes, the enzyme and the enzyme-substrate complexes are treated as additional "species." In this section, however, we study MCA, and assume that every reaction involves an enzyme which is not explicitly expressed as a species. Rather, we assume enzyme activity is absorbed into the rate constants for the forward and backward reactions. This is clearly not an accurate approximation for many enzymatic reaction mechanisms. Nevertheless, it provides a first-order approximation for treating a metabolic network. More complex rate laws for enzymatic reactions have been discussed for biochemical systems analysis ,22.th reaction or the Nth  species.MCA focuses on how the NESS {N is changed: XN. The immediate, local change is characterized byFirst, we consider the case where the concentration of external species ε = (dg J)-1 RT. Here, we use the same notation used by Heinrich and Schuster [v) denotes the diagonal matrix containing the components of the vector v along its diagonal. The unscaled, local, elasticity-coefficient matrix is given by ε' = (dg J) ε (dg x)-1 = RT (dg x)-1. The prime symbol is used to distinguish unscaled coefficients from scaled coefficients, both here and in what is to follow.which is called the scaled, local elasticity-coefficient . These cSchuster , where  satisfyThe coefficient response . When a BiN is the Nth column vector of the matrix A-1. The new NESS established near {Jm + δJm, are given bySolving Eq. (6), we have Hence, the scaled, steady-state elasticity-coefficient is ε = (dg J)-1 RTB (dg B)-1. Here, (dg B) denotes the diagonal matrix containing only the diagonal components of the matrix B along its diagonal. The unscaled, steady-state, elasticity-coefficient matrix is given by ε' = (dg J) ε (dg x)-1 = RTB (dg B)-1 (dg x)-1.which may also be known as a response coefficient ,25. Thesε and ε are different. In principle, the ε can be experimentally measured as a system response, but obtaining ε can be more difficult since it is a transient response that must be measured individually.As we shall show, some connectivity theorems for the two quantities th reaction EM is changed: EM → EM + δEM. Since we have assumed that the rate constants for reaction M, M, EM, we haveNext, we consider the case where the enzyme for the MEM → EM + δEM, the new NESS satisfies [When atisfies δxj/j = 1, 2, ..., N), we haveSolving for From Eq. (10), we get the scaled, concentration control-coefficient ,BS(dg J). The unscaled, concentration, control-coefficient matrix is x) J)-1 = -(dg x)BS.which is an element of the scaled, concentration, control-coefficient matrix given by Combining Eqs. (7) and 10), we get the scaled, flux control-coefficient [0, we getδmM is the Kronecker delta function. These coefficients are elements of the scaled, flux, control-coefficient matrix given by C = I - (dg J)-1 RTBS (dg J). The unscaled, flux, control-coefficient matrix is C' = (dg J) C (dg J)-1 = I - RTBS.where ε, ε, C, and Among between the scaled coefficients, and similarly,between the unscaled coefficients.SJ = -Je = 0, we haveSecond, from Eqs. (13)–(16), it is straightforward to show that if there is no injection flux, i.e., in NESS 1i × j and 0i × j denote the i × j matrices of ones and zeros, respectively. Eqs. (17) and (18) are known as the summation theorems for the scaled control-coefficients. For the unscaled control-coefficients, we have the generalized summation theorems, C'K = K and K is the M × (M - N) loop (or nullspace) matrix, i.e., its columns form a basis of the nullspace of S so that SK = 0N × (M - N) [where  (M - N) ,16.Euler's theorem on homogeneous functions can be used to understand the significance of the injection fluxes ,26. If tBSRT = I ⇒ (RTBS)(RTB) = RTB, which yields the connectivity theorems for the scaled coefficients:Third, and the generalized connectivity theorems for the unscaled coefficients :Finally, by combining the generalized summation and connectivity theorems, we getK ε') has been proved to be invertible [which is the central equation in MCA . The matvertible ,27. TherUsing the steady-state elasticity-coefficients, we haveK ε') is also invertible (see Methods). Therefore, if the steady-state elasticity-coefficients are known, the control-coefficients can be calculated. This is an important result because ε can be experimentally measured as a system response, but obtaining ε can be more difficult since it is a transient response and must be measured individually.The matrix , and vice versa. In general, the flux J and the chemical potential difference Δμ are nonlinearly related [Both  related :J and Δμ are linearly related with the biochemical conductance being φ+/kBT (= φ-/kBT) [φ- is relatively unperturbed. Then we have δJ/δΔμ = -φ+/kBT. Similarly, if the reaction is perturbed from its downstream, then δJ/δΔμ = -φ-/kBT.In the linear regime with small flux and chemical potential difference,  φ-/kBT) ,16. In tscaled, local elasticity-coefficients for chemical potentials [Combining BCT and MCA, we can define the tentials εμ Δ= kBT (dg Δμ)-1 ST, the scaled, steady-state elasticity-coefficients for chemical potentialsas elements of the matrix εμ Δ= kBT (dg Δμ)-1 STB (dg B)-1, and the scaled, potential control-coefficients [as elements of the matrix ficients Cμ Δ= -kBT (dg Δμ)-1 STBS (dg J), where reaction potentials and substrate concentrations in the reference state are used for normalization. The corresponding unscaled coefficients are given in matrix form asas elements of the matrix ST, STB, and STBS in the local and steady-state elasticity-coefficients and the control-coefficient for the potentials, as compared to the matricies RT, RTB, and RTBS in the coefficients for the fluxes. Furthermore, note the relationshipsNote the appearance of the matrices Eqs. (19) and (29) in combination yield the summation  and connEq. (31) can also be derived using the homogeneous function theory (discussed above with flux and concentration control-coefficients). For the unscaled coefficients, we get the generalized summation and connectivity theorems by combining Eqs. (21) and (30) so thatEq. (33) is a manifestation of Kirchhoff's loop law ,16.By combining the generalized summation and connectivity theorems, we getK ε') has been proved to be invertible [The matrix  has an inverse (see Methods). Therefore, if the steady-state elasticity-coefficients are known, the control-coefficients can be calculated.The matrix (xN ∈ Correlation metric construction (CMC) focuses on how the NESS {If the random perturbation has a sufficiently long correlation time compared to the relaxations of all the reactions in the system, then we are dealing with a shift in the NESS where, according to Eq. (6),δxi} with an uncorrelated perturbation on δxN, we haveFrom the time series of {i, j = 1, 2, ..., N.for δxN is a Markov process characterized byCMC extracts information from reaction systems by using perturbations with insufficiently long correlation times. kd controls the relaxation time of the perturbation, a is its random amplitude with Gaussian distribution, and ξ(t) is a uncorrelated noise. δxN in Eq. (38) is called an Ornstein-Uhlenbeck (OU) process with auto-covariance functionwhere With this type of random perturbation, the temporal correlations between the concentration fluctuations give information on the connectivity of a network following the kinetic equation (4)i = 1, 2, ...,(N - 1). More precisely , instead of Eq. (40) CMC follows an autoregressive (AR) process, the discrete version of an OU process, with discrete time T, 2T, ...,for in which A matrix into a (N - 1) × (N - 1) matrix Let's rewrite the which leads tog(τ) in Eq. (39) is ≫ all the relaxation times of Two limits of Eq. (42) are particularly interesting. If the correlation time of Rij(t) is the response curve, as a function of t, for the ith species to an impulse of the jth species at time 0. The relation between the correlation matrix and the fundamental solution is expected for a linear system. For directly connected species i and j, the i and j are not directly connected, The matrix analyzed .Both flux control-coefficients and biochemical conductances can be used as indicators for rate-limiting steps in a biochemical network . TraditiLet's consider the linear pathwayx0 and xn for X0 and Xn, respectively. Analogous to a continuous energy landscape, we have the discrete energy functionunder NESS with clamped concentrations j = 1, 2,..., n. If there is a rate-limiting step, say for example that reaction j is nearly irreversible, then Ej is maximal and that is where the highest energy barrier is located. We now show that it is also where the largest flux control-coefficient and the smallest conductance are.for The NESS flux of reaction system (44) can be solved for analytically to yield ,32Therefore, the flux control-coefficient isEj. The biochemical conductance, on the other hand, iswhich reaches its maximum at maximal Ej where concentration xℓis the lowest.which is at its minimum when J given, the smallest conductance also corresponds to the largest |Δμ|. In general, the rate-limiting step is not necessarily where the k's are smallest. The most efficient situation for reaction system (44) is when all the conductances are equal. This result is analogous to the constant torque principle suggested for molecular motors [Note that with r motors ,13,33.The analysis provided in the present paper poses the following biological question. When is a NESS of a biochemical network controlled by a constant flux injection and when is it controlled by a clamped concentration ? Clearly in real biological systems, the appropriate answer is a combination of both. Just as a real battery, having both finite internal resistance and conductance, in an electrical circuit is a combination of constant-current  and constant-voltage  ideal batteries, so too would a real biological system be a combination of constant flux injections and clamped concentrations . NeverthJ to a simple unimolecular reaction in a NESS, we have the heat dissipation rate (hdr)First, let's see what are the respective consequences of these two different types of "forcing" to a biochemical system. For a constant flux injection J a reaction with larger φ-  dissipates a smaller amount of heat. For a constant clamped chemical potential Δμ, we haveThis indicates that for the same injection flux hdr increases with φ- for the constant potential difference (concentration clamping) case. These results are analogous to an electric circuit in which the heat dissipation equals I2R and U2/R, respectively.We note that This observation leads us to the following hypothesis. In a biochemical network for biosynthesis, the fluxes are essential for supply and demand. Hence, such networks will in general have large conductances in order to minimize the energy waste and be efficient. On the other hand, in a biochemical network for cellular signaling, the concentrations are the essential signal. Thus, in this case the conductances are generally small in order to minimize the energy waste.J in the enzymatic reaction as a rational function of the concentration of a substrate x [MCA shares an intimate relationship with the biochemical systems analysis (BSA) originally proposed by M.A. Savageau . A critistrate x , such thK ≤ L due to substrate saturation or effector inhibition. The coefficients ai, bi, α0, and β0 contain the concentrations of all the other substrates. BSA provides insight to the properties of ai and bj. The network of metabolic reactions is then modeled [where  modeled  by a sysJ can be written asIn terms of Eq. (49) the local elasticity-coefficient for the flux which also has the standard form of a Pade approximation . Furtherwhich is essentially the integral form of the definition of local elasticity-coefficients.In contrast, MCA assumes no specific rate laws for an enzymatic reaction except that the rate is linearly proportional to the enzyme concentration. This assumption is certainly valid for most biochemical reactions except when a substrate concentration is significantly less than that of an enzyme.K ε'), that appears in the central, control-analysis equations, i.e., Eqs. (24) and (36), is invertible, we note that since we are dealing with systems without conserved quantities, rank(S) = N. Furthermore, we note that the matrix KTK is invertible because K contains linearly independent columns, and because all columns of K are in the right nullspace of S, the matrixTo demonstrate that the matrix,  loop (or nullspace) matrix (SK = 0); R: N × M correlation matrix defined such that A = SRT; S: N × M stoichiometric matrix; x:N-dimensional vector of species' concentrations; J: M-dimensional vector of reaction fluxes; Δμ: M-dimensional vector of reaction potentials; ε:M × N local, elasticity-coefficient matrix; ε: M × N steady-state, elasticity-coefficient matrix; N × M concentration, control-coefficient matrix; C: M × M flux, control-coefficient matrix; CμΔ: M × M biochemical potential, control-coefficient matrix; (dg v): If v is a vector, (dg v) denotes the diagonal matrix with the components of v along its diagonal. If v is a matrix, (dg v) denotes the diagonal matrix with only the diagonal components of v along its diagonal; *: Denotes steady-state; ': Distinguishes unscaled coefficients from scaled coefficients.BCT: Biochemical Circuit Theory; BSA: Biochemical Systems Analysis; CMC: Correlation Metric Construction; FBA: Flux Balance Analysis; MCA: Metabolic Control Analysis; NESS: Nonequilibrium Steady-State; HQ initiated this work. All authors contributed to and wrote the manuscript together. All authors read and approved the final manuscript.


With the abundant information produced by microarray technology, various approaches have been proposed to infer transcriptional regulatory networks. However, few approaches have studied subtle and indirect interaction such as genetic compensation, the existence of which is widely recognized although its mechanism has yet to be clarified. Furthermore, when inferring gene networks most models include only observed variables whereas latent factors, such as proteins and mRNA degradation that are not measured by microarrays, do participate in networks in reality.Motivated by inferring transcriptional compensation (TC) interactions in yeast, a stepwise structural equation modeling algorithm (SSEM) is developed. In addition to observed variables, SSEM also incorporates hidden variables to capture interactions (or regulations) from latent factors. Simulated gene networks are used to determine with which of six possible model selection criteria (MSC) SSEM works best. SSEM with Bayesian information criterion (BIC) results in the highest true positive rates, the largest percentage of correctly predicted interactions from all existing interactions, and the highest true negative (non-existing interactions) rates. Next, we apply SSEM using real microarray data to infer TC interactions among (1) small groups of genes that are synthetic sick or lethal (SSL) to SGS1, and (2) a group of SSL pairs of 51 yeast genes involved in DNA synthesis and repair that are of interest. For (1), SSEM with BIC is shown to outperform three Bayesian network algorithms and a multivariate autoregressive model, checked against the results of qRT-PCR experiments. The predictions for (2) are shown to coincide with several known pathways of Sgs1 and its partners that are involved in DNA replication, recombination and repair. In addition, experimentally testable interactions of Rad27 are predicted.SSEM is a useful tool for inferring genetic networks, and the results reinforce the possibility of predicting pathways of protein complexes via genetic interactions. While the existence of genetic compensation is widely accepted, the mechanism is largely unknown but important ,2. The pGraph models  depict Although each class of models has been shown to be informative for understanding genetic interactions, most of the models, except some Bayesian networks, have the estimation bias problem due to model mis-specification. The model mis-specification arises from the fact that microarrays measure the mRNA expressions only, while genetic interactions may be influenced by enzymes or proteins, for instance transcriptional factors. Furthermore, most genetic networks reconstructed in previous studies considered only a subset of the whole genome. Consequently, those genes that were left out may be regarded as latent factors influencing the genes of interest. Thus, ignoring latent factors in the models may cause bias when inferring the genetic interactions. Although a Bayesian network can also incorporate latent factors ,12, the et al. , where c ≥ 3. A large sample size can inflate a small difference between S and χ2 statistic. Numerous indices were proposed to remedy the bias, among them four have been assessed in our pilot studies, namely, χ2/df, χ2 - df [df denotes degree of freedom. The former two were found to be more effective than the latter two for network (model) selection.Among the six MSC's studied, the od ratio . Assumin χ2 - df , TLI [43 χ2 - df , where dT is the number of time points in the gene expression data. AIC [L/24.MSE between the observed and the predicted gene expressions is defined as ata. AIC  and BIC ata. AIC  are two  Schwarz  proposed, Sclove  suggestex(t), Λ and w. The significance of each link (λij and Wkl) is tested by its associated t-statistic. Let ti-window , respectively.A genetic network inferred from LDFM can be built by latent factor-gene and gene-gene interactions. A correct network is essential for estimation of gene-gene interactions using SEM. However, learning the optimal network from data subject to a goodness-of-fit index is NP-hard. Although global optimization techniques, such as simulated annealing and genetic algorithms, may be applied, the required computation time is not feasible. To make the learning process practical, we propose a stepwise approach. The key idea is to generate a set of candidate networks and retain plausible links by both using SEM and iteratively filtering with a moving window as follows. For any network generated in the iteration, we apply SEM to estimate Sc, Sp and Sf denote the sets of c-links, p-links and f-links, respectively. Suppose that EFA suggests k factors for a given data set. Given fixed k - 1, k or k + 1 factors, EFA is applied again to learn the associated latent (factor-gene) structures. SSEM begins with the aforementioned latent structure and a fully connected gene structure, namely, each gene is regulated by all genes and k - 1, k or k + 1 latent factors. To start the stepwise search, SEM is applied to the initial networks to estimate x(t), Λ and w. For a given initial network, first let the initial t0-window be 0 can be generated as follows. Checking the t-statistics of all links against the t0-window f-links, and retain all c-links, while considering all 0–1 combinations of p-links. Suppose there are l p-links in an initial model, then there are 2l combinations of each p-link being included in a model or not. Models including all c-links and each aforementioned combination are considered, and these 2l models can be viewed as generated by the t-window filtering. That is, the t-window serves as a filter to eliminate insignificant (the less-likely-to-exist) links. Specifically, Φ0 = {φ|φ ∈ Sc ∪ Lp, ∀ Lp ∈ P(Sp)}, where P(Sp) = {Lp|Lp ⊆ Sp} is the power set of Sp and Lp is a subset of Sp. Furthermore, we apply SEM to each candidate network in Φ0 to obtain the pre-specified goodness-of-fit index. To save computation time and to ensure that the superior networks are kept for the next iteration, only the top m networks (denoted by Ω0) are retained for the next iteration.Let i ≥ 1, SSEM generates a set of candidate networks by t-window filtering all networks generated by the top m networks from iteration(i-1), i.e., Ωi-1, with k - 1, k or k + 1 factors, respectively, to form Φi. So in total, there are 3 m seed models to generate networks. Among the networks in Φi ∪ Ωi-1, only the top m networks (Ωi) are retained by the specified goodness-of-fit index for iteration (i + 1). First, we let the ti-window equal to the ti-1-window +c. We use c = 0.1, but c can be other small constants. Again, given the ti-window, each link in the jth network in Ωi-1 can be discarded, retained or considered according to its t-statistic value. We denote the collection of these f-links, c-links, and p-links by Sjf, Sjc, and Sjp, respectively. A set of candidate networks is generated by retaining all c-links and considering all 0–1 combinations of p-links with k - 1, k or k + 1 factors in the model, and this set is denoted by φ|φ ∈ Sjc ∪ Lp, ∀ Lp ∈ P(Sjp)}. We combine all the generated sets to result in the ith set of networks Similar to the initial iteration, for each iteration i, we obtain the top m scored networks from Φi ∪ Ωi-1, which form Φi, to go to iteration (i + 1). The iteration terminates if the specified goodness-of-fit index can not be further improved.Evaluating the specified goodness-of-fit index for every network in Φk.Fit EFA to a given data set to determine the number of factors, say k - 1, k or k + 1 latent factors, respectively.• Apply EFA to generate three initial networks by estimating the latent structures with • For given k factors, obtain the latent structure by eliminating the links with factor loading less than a constant .• Specify an MSC.• For each initial network:i = 0, run SEM on the data set. Specify the t0-window = c-links and one of all the 0–1 combinations of p-links. Compute the MSC of all networks and select the top m models to form the candidate set Ω0.Step 1. Set iteration i = i + 1. Specify the ti-window Step 2. Set i-1, generate a set of networks, and form ΦiStep 3. Similarly to Step 1, for each network in Ωi, and choose the best m networks from Φi ∪ Ωi-1 to form the ith candidate set Ωi.Step 4. Evaluate the MSC for all networks in Φith top 1 MSC = the (i-1)th top 1 MSC, stop ; Otherwise, go to Step 2.Step 5. If the m networks from the union of all networks generated by different initial guesses.• Select the best Project home page is in . SSEM alGS and CC conceived the study, devised the method, and supervised methodology and implementation. CY carried out the method and part of simulation, and wrote an early draft of Shieh et al. in . JH wrotSSEM-TR. Technical Report of SSEM – Shieh et al. (2005).Click here for fileqRT-PCR. Description of the design of qRT-PCR experiments and how the experiments were conducted to confirm the predicted TC and TD interactions.Click here for fileSimulation. The description of the 6- and 10-gene networks, and the results of EBVM applied to the two networks.Click here for fileBayesianNW. The 6-gene network predicted by the three Bayesian network algorithms in Beal et al. (2005), Rangel et al. (2004) and Perrin et al. (2003).Click here for fileMulti-AR(1). The result of fitting multivariate AR(1) straightforwardly to the real data for the 6-gene network.Click here for fileSSL-TCNW. TC networks of SSL gene pairs. Description of how TC and TD interactions of SGS1 and RAD27 SSL gene pairs were predicted by SSEM.Click here for fileSSEM-algorithm. The zipped file consists of the standalone executable (.exe) file of SSEM.Click here for file