text:
- Methodology article
- Open Access
Inference of complex biological networks: distinguishability issues and optimization-based solutions
- Gábor Szederkényi1, 2Email author,
- Julio R Banga1Email author and
- Antonio A Alonso1
© Szederkényi et al; licensee BioMed Central Ltd. 2011
- Received: 31 May 2011
- Accepted: 28 October 2011
- Published: 28 October 2011
The inference of biological networks from high-throughput data has received huge attention during the last decade and can be considered an important problem class in systems biology. However, it has been recognized that reliable network inference remains an unsolved problem. Most authors have identified lack of data and deficiencies in the inference algorithms as the main reasons for this situation.
We claim that another major difficulty for solving these inference problems is the frequent lack of uniqueness of many of these networks, especially when prior assumptions have not been taken properly into account. Our contributions aid the distinguishability analysis of chemical reaction network (CRN) models with mass action dynamics. The novel methods are based on linear programming (LP), therefore they allow the efficient analysis of CRNs containing several hundred complexes and reactions. Using these new tools and also previously published ones to obtain the network structure of biological systems from the literature, we find that, often, a unique topology cannot be determined, even if the structure of the corresponding mathematical model is assumed to be known and all dynamical variables are measurable. In other words, certain mechanisms may remain undetected (or they are falsely detected) while the inferred model is fully consistent with the measured data. It is also shown that sparsity enforcing approaches for determining 'true' reaction structures are generally not enough without additional prior information.
The inference of biological networks can be an extremely challenging problem even in the utopian case of perfect experimental information. Unfortunately, the practical situation is often more complex than that, since the measurements are typically incomplete, noisy and sometimes dynamically not rich enough, introducing further obstacles to the structure/parameter estimation process. In this paper, we show how the structural uniqueness and identifiability of the models can be guaranteed by carefully adding extra constraints, and that these important properties can be checked through appropriate computation methods.
- Mixed Integer Linear Programming
- Chemical Reaction Network
- Core Reaction
- Network Inference
- Dense Realization
During the last decade, the wide availability of high-throughput biological data has made it possible to produce new knowledge via a systems biology approach [1–3]. The inference of biochemical networks (i.e. the mathematical mapping of the molecular interactions in the cell) is therefore a question of key importance in the field. During the last decade, many methods have been developed to solve the network-inference (sometimes called reverse-engineering [4]) problems arising in e.g. gene expression [5–13], signal transduction [14–17] and metabolic networks [18–25].
In this context, it is particularly worth mentioning the DREAM initiative (Dialogue for Reverse Engineering Assessments and Methods) [26], which targeted the problems of cellular network inference and quantitative model building in systems biology. DREAM tries to address two fundamental questions: (i) how can we assess how well we are describing the networks of interacting molecules that underlie biological systems? and (ii) how can we know how well we are predicting the outcome of previously unseen experiments from our models? Interestingly, one of the main conclusions of the DREAM3 event was that the vast majority of the teams' predictions were statistically equivalent to random guesses. Moreover, even for particular problem instances like gene regulation network inference, there was no one-size-fits-all algorithm [27].
The use of a performance profiling framework with the DREAM3 benchmark problems revealed that current inference methods are affected by different types of systematic prediction errors [6]. These authors conclude that reliable network inference from gene expression data remains an unsolved problem. Further, they highlight two major difficulties in the case of gene-network reverse engineering: limited data (which may leave the inference problem underdetermined), and the difficulty of distinguishing direct from indirect regulation. Prill et al [27] further explored the issue of intrinsic impediments to network inference, designating identifiability of certain network edges and systematic false positives as the main barriers. In this paper, we consider the widely used reaction kinetic formalism, where dynamic models of biological networks are described by a set of ordinary differential equations (see, e.g. [28–30] and the related literature). In particular, we consider the central question of the identifiability of such a network as understood in the systems and control area [31, 32].
Identifiability analysis studies whether there is a theoretical chance of uniquely determining the parameters of a mathematical model assuming perfect noise-free measurements and error-free modeling [33–35]. One of the early approaches for identifiability testing of nonlinear models is based on the Taylor-series expansion of the system output using the fact that the Taylor coefficients are unique [36]. A similar but more general method uses the generating series or Volterra-series coefficients of the system which is the nonlinear generalization of the Laplace-transform method used for linear systems [37]. In [38] a similarity transformation approach is proposed that gives necessary and sufficient conditions on local and global identifiability through the checking of nonlinear controllability and observability conditions. The appearance of differential algebra methods in systems and control theory [39, 40] opened the possibility for new types of identifiability tests that have gained significant popularity [41–43]. Further theoretical developments in the field include the identifiability conditions of rational function state space models [43], the possible effect of initial conditions on identifiability [44], and the application of Lie-algebras [45]. While identifiability is the property of a certain parameterized model, a related notion called distinguishability addresses the problem whether two or more parameterized models (with the same or with different structure) can produce the same output for any allowed input [46–48]. The literature about identifiability and distinguishability of biological and chemical system models is relatively wide: Compartmental systems (that form a special subclass of general mass-action networks) are studied in [38, 49, 50]. The authors treat general nonlinear CRNs in [51, 52] and [53] where it is shown that for thermodynamically meaningful models, nonlinearity reduces the chance of indistinguishability compared to the linear case [54]. Geometric conditions for the indistinguishability of CRNs are given in [55] with a related comment in [56]. Computer algebra tools can be successfully used for the symbolic computations needed for identifiability and distinguishability testing of complex models [57–60].
The importance of identifiability has been recognized previously in systems biology, too [14, 61–64]. However, and despite a number of works illustrating ways to test the structural and practical identifiability of models [65–67], a significant portion of modeling studies in systems biology continue to ignore this key property.
It has been known for long that chemical reaction networks with different structure and/or parametrization may produce the same dynamical models describing the time-evolution of species concentrations [28, 55]. A related problem, namely the non-unique structure of Petri nets associated to reaction network dynamics, is studied in [68]. Additionally, the value of prior information in biological network inference was clearly shown in [69, 70] by applying Bayesian network models. However, a constructive optimization-based approach for the study of dynamically equivalent (or similar) reaction networks is a recent development [71–74], which we further extend in this paper.
As a novelty, we present in this paper the definition and a computational method to find the so-called core reactions that are present in any dynamically equivalent reaction network if the set of complexes is given a priori. Moreover, a computationally improved method is introduced for the computation of dense realizations of CRNs together with a modified algorithm to check the uniqueness of a constrained reaction network structure. Structural non-uniqueness and the use of the proposed computational methods will be illustrated with the help of biological models known from the literature.
The structure of the paper is the following. The 'Methods' section introduces the notions of chemical reaction networks, structural identifiability and distinguishability of dynamical models. Moreover, it contains the procedures to obtain core reactions of a network and its sparse and dense representations, which rely on standard methods of linear programming (LP) and mixed integer linear programming (MILP) [75–78]. The analysis of four biological system models can be found in the 'Results and discussion' section, followed by the conclusions.
where x ∈ ℝ n is the state vector, y ∈ ℝ m is the output, u ∈ ℝ k is the input, and θ ∈ ℝ d denotes the parameter vector. We assume that the functions f and h are polynomial in the variables x, u and θ. Clearly, mass action type CRNs described in the following subsection (where θ is typically the set of reaction rate coefficients), and simple deterministic models of gene regulation such as the one in Example 4 belong to this model class.
Basic notions and known results related to mass-action models
In this subsection, the basic definitions for the description of CRNs will be given together with the already published results on finding dynamically equivalent network realizations with certain prescribed properties.
Structural and dynamical description of mass-action networks
- 1.
is the set of species or chemical substances.
- 2.is the set of complexes. Formally, the complexes are represented as linear combinations of the species, i.e.(2)
where α ij are nonnegative integers and are called the stoichiometric coefficients.
- 3.
, and C i is transformed to C j in the CRN} is the set of reactions. The relation will be denoted as C i → C j . Moreover, a nonnegative weight, the reaction rate coefficient denoted by k ij is assigned to each reaction C i → C j . Naturally, if the reaction C i → C j is not present in the CRN then k ij = 0.
The above characterization naturally gives rise to the following graph structure (often called 'Feinberg-Horn-Jackson graph' or simply reaction graph) of a CRN [29]. The weighted directed graph G = (V, E) of a CRN consists of a finite nonempty set V of vertices and a finite set E of ordered pairs of distinct vertices called directed edges. The vertices correspond to the complexes, i.e. V = {C1, C2, ... C m }, while the directed edges represent the reactions, i.e. (C i , C j ) ∈ E if complex C i is transformed to C j in the CRN. The positive reaction rate coefficients k ij are assigned as weights to the corresponding directed edges C i → C j in the graph. (Edges corresponding to zero rate coefficients are not drawn in the reaction graph.) A set of complexes {C1, ..., C k } is called a linkage class of a CRN, if the complexes of the set are linked to each other in the reaction graph but not to any other complex. It is remarked that loops (i.e. directed edges that start and end at the same vertex) are not allowed in reaction graphs.
Dynamical equivalence of mass-action networks
In this case, for i = 1, 2 are called realizations of a kinetic vector field f (see, e.g. [80] for more details). It is also appropriate to call a realization of and vice versa.
where and are valid Kirchhoff matrices and M is the invariant matrix containing the coefficients of the monomials.
Among the dynamically equivalent realizations, it is important to recall the following characteristic ones described in [71, 72]. A sparse realization contains the minimal number of reactions that is needed for the exact description of the corresponding dynamics (3). A dense realization contains the maximal number of reactions among dynamically equivalent realizations with a fixed complex set given by Y. While sparse realizations are generally structurally non-unique (as it will be illustrated for the constrained case, too, in Example 1), the structure of dense realizations with a given complex set is unique, and it contains every possible dynamically equivalent structure as a proper subgraph (i.e. a dense realization is a kind of super-structure) [71].
Known computation approaches for finding preferred CRN realizations
Here we briefly summarize the already published results corresponding to the computation of preferred dynamically equivalent CRN realizations (more details of these methods can be found in the publications [71–73, 81]). The computation of dense and sparse realizations can be traced back to mixed integer linear programming (MILP) where the decision variables are the non-diagonal elements of A k , the linear constraints encode the kinetic properties of the model, and the objective function contains integer variables for minimizing/maximizing the number of nonzero reaction rate coefficients [72]. It is remarked that the computation of sparse realizations is an NP-hard problem, where generally mixed integer linear programming cannot be avoided [82]. There exist certain conditions under which the problem can be solved in polynomial time [83] but these are often not fulfilled in the case of CRNs. Moreover, there are effective heuristics to address the problem [84], but convergence to one of the truly sparsest structures is not guaranteed. Luckily, the MILP-based computation of sparse CRN realizations can be parallelized effectively thus allowing a larger number of complexes to be treated. The computation of realizations having the minimal/maximal number of complexes or the reversibility property can also be solved in the MILP framework [71]. Moreover, it was shown in [73] that finding detailed balanced and complex balanced realizations of CRNs is a simple linear programming (LP) problem. Finally, weakly reversible dynamically equivalent CRN realizations can also be determined (if they exist) using MILP [85].
Constrained realizations of CRNs and testing their structural uniqueness
P1 Given a CRN (Y, A k ) and a constraint set , the unweighted reaction graph of any constrained realization is the subgraph of the unweighted reaction graph of the constrained dense realization.
P2 If the sets of complexes and constraints are fixed, then for any CRN, the structure of the constrained dense realization is unique.
P3 The reaction graph structure of a CRN with given sets of complexes and constraints is unique if and only if the unweighted directed graphs of its constrained dense and sparse realizations are identical.
The proofs of P1, P2 and P3 follow similar (although not completely identical) lines that were published in [71], and they are given for convenience in the Appendix at the end of the paper.
New concepts and computation results related to dynamically equivalent networks
This subsection contains new methodological contributions that extend the previously published results.
Making the computation of dense realizations more efficient
Computing dense realizations is treated originally also in a MILP-framework in [72]. However, using the structural uniqueness of such realizations given by P1, it is easy to give a polynomial-time algorithm based on a finite series of linear programming (LP) optimization steps. The idea of the improved algorithm is simple: the reaction C i → C j belongs to the (constrained) dense realization if and only if there exists any dynamically equivalent (constrained) realization where [A k ] j,i > 0. The result directly follows from the fact that the unweighted reaction graphs of (constrained) dense realizations give a unique super-structure. This allows us to formulate a polynomial-time method based on pure LP to determine (constrained) dense realizations as follows.
It is important to remark that the definition of ε ij in the form of (11) guarantees the solvability of (12). Naturally, the above described method can also be used for determining constrained dense realizations by adding constraints of the form (9) to the LP problems (10) and (12).
Using the notion and described properties of constrained realizations, we are now able to test the structural uniqueness of given CRNs. To accomplish this, only the (constrained) dense and sparse realizations have to be computed and compared (see P3). This method will be illustrated in Example 2.
Definition and computation of core and non-core reactions
where the matrix A k contains the decision variables, and the known matrices are Y and M. It is well-known that this task is equivalent to an LP problem where the objective function is an arbitrary linear function of the elements of A k [76]. Then, reaction C p → C q is a core-reaction if and only if the set defined by (13)-(17) is empty (i.e. the corresponding LP problem is infeasible), because in this case there is no dynamically equivalent CRN realization where C p → C q is not present. We remark here that the presented procedures for determining constrained dense realizations and computing core reactions are parallel in their original forms since the individual LP steps are independent of each other. Therefore the proposed methods can be very effectively implemented in a grid or multi-core hardware environment [86].
Basic concepts on structural identifiability and distinguishability
and x(t, θ) denotes the solution of (1) with parameter vector θ. According to (18), a structurally non-identifiable model can produce exactly the same observed output with different parametrization. This is clearly a fundamental obstacle of determining the true model parameters from measurements even if the selected model structure is considered to be correct.
Let us denote two parameterized models with possibly different structure by and , respectively, where θ i denote the parameter vector. Then is called distinguishable from if for any θ1 (possibly except for a finite number of values) there is no θ2 such that the input-output behaviour of and is the same [47]. Clearly, if and are indistinguishable and both model structures are feasible in a certain application, then there is no way to decide from input-output measurements to which one corresponds to the true model that generated the data.
In the case of CRNs, we will assume that all species concentrations are measured (i.e. y = x), the input is zero (i.e. we study autonomous systems), and that the set of possible chemical complexes is given. Generally, the model parameter vector θ is the set of reaction rate coefficients which are the off-diagonal elements of A k . Clearly, if a CRN has several different dynamically equivalent realizations, then these realizations are not distinguishable without additional constraints, and the model cannot be identifiable if all the rate coefficients are to be determined [55]. This situation can be improved by using prior knowledge in the form of adding further constraints on the model parameters such as the simple ones given by eq. (9). This way, the number of parameters to be estimated can be reduced and/or their feasibility region can be shrinked. It is important to note that although the structural uniqueness of a CRN definitely reduces the degree of non-identifiability (since zero and non-zero parameters are separated), it does not necessarily imply structural identifiability [53], and this latter property has to be checked by further numerical methods [31, 32].
Results and discussion
In this section, the application of the previously mentioned methods for finding different dynamically equivalent structures will be illustrated using biological models taken from the literature. The detailed numerical data corresponding to Examples 1-3 are contained in a standard spreadsheet form with brief explanations in Additional file 1: CRN_data.xls.
Example 1: a positive feedback motif
Our starting point is that we have a dynamic model of the process in the standard polynomial form of (20)-(24), the parameters of which are known from the results of identification and/or from literature. As we will see below, without well-defined constraints on the possible set of complexes and reactions, exactly the same dynamics can be realized in principle by a wide range of mechanisms.
It is visible that the structure of the dense realization is the same as in Figure 1b but the core reactions are different from the ones shown there. Here the degradation of mRNA is described by a core reaction but interestingly, the reaction corresponding to translation is not a core one. Naturally, this implies that the possible sparse realization structures with the second parametrization are different from the ones shown in Figure 2. Note that here the only goal was to illustrate the possible change of core and non-core reactions, and therefore the biological relevance of the parameter values in eq. (28) is not assumed in this case.
The dense CRN realization of the dynamics (20)-(24) with the updated Y' matrix given in eq. (29) using the original parameters described in (27) is shown in Figure 3b, where the core and non-core reactions are again indicated. It is apparent that now there are only 5 core reactions, and none of the remaining 12 reactions are essential to represent the dynamics (20)-(24). This means that the introduction of a new complex increased the flexibility of the network (i.e. mathematically, the majority of the reactions can be substituted by other ones and the network still maintains its original dynamics). Of course, not any combination of the non-core reactions can be omitted from the network, because the sparse realizations show that at least 9 reactions are needed to keep dynamical equivalence. It can be computed easily that the theoretical maximum number of sparse realizations with different structures is . However, as the numerical experiments show, majority of these structures do not give a practically feasible dynamically equivalent realization.
The above results clearly show that certain mechanisms may remain undetectable (or they are falsely detected) even if we have complete species concentration measurements and full information about possible complex formation, that are not very realistic assumptions. Moreover, the sparsest dynamically equivalent structure of mass-action models is not unique, therefore sparsity enforcing approaches for determining 'true' reaction structures are not enough in themselves without the necessary amount of prior information given in the form of additional constraints. The practical situation is most often even worse than that, since the measurements are typically incomplete, noisy and sometimes dynamically not rich enough, that may introduce further obstacles to the structure/parameter estimation process [66, 89].
Example 2: a biochemical switch in yeast cells
The following example is taken from [90] and it describes a 'switching device' in yeast cycle regulation. The detailed system description can be found in [90] and in the accompanying supporting information document. The order of state variables, corresponding to concentrations, is the same as in the original article, and is shown below:
The structure of the dense realization indicating the 12 core and 16 non-core reactions can be seen in Figure 4b.
It can be shown using the computational methods described in the 'Methods' section that the only possible sparse realization structure is identical to that of the original network. Therefore in this special case, there is only one possible reaction structure containing the minimal number of reactions. A straightforward approach to ensure the structural uniqueness of the whole network is to exclude all reactions that are not meaningful from the examined application's point of view or that are contradictory to modeling assumptions. For the current example, the removal of an unexpectedly low number of reactions is enough to obtain a unique structure. It can be shown by computing the corresponding constrained dense and sparse realizations, that excluding the reactions X5 → X3 + X5, X4 → X3 + X4, X2 + X3 → X3 + X5, and X3 + X1 → X3 + X4 is enough to make the reaction structure unique that is identical to the original structure shown in Figure 4. In other words, the exclusion of 4 well-selected reactions leads to the removal of an additional 6 reactions leaving only 18.
Example 3: a repressilator structure with 5 nodes and auto-activation
for the index pairs (i, j) ∈ {(1, 5), (2, 1), (3, 2), (4, 3), (5, 4)}. In eqs. (33)-(39), G i and P i represent the ith gene and protein, respectively. For the genes, superscripts A and R refer to activated and repressed states, respectively. Let us denote with r i,k the reaction with rate coefficient k i,k in eqs. (33)-(39).
In particular, it is easy to show (see also Additional File 1: CRN_data.xls) that reactions and are always indistinguishable. Similarly, the reaction can be substituted with the combination of reactions and . It can be seen from these results that in order to have a model with unique structure, it is very important to a priori exclude all reactions that are not meaningful for the particular application.
Example 4: sparse linear gene regulation network models
where A ∈ ℝ n×n contains the connectivity information of the network. A i,j > 0 indicates activation from node j to node i, while A i,j < 0 means repression, diagonal elements of A represent auto-activation or auto-repression depending on their sign. x ∈ ℝ n is the fully or partially measurable state of the system describing the time evolution of concentrations, and the input part Bu represents experimental perturbation (e.g. activation) of the genes. It is also a common assumption that the network is 'sparse' which means that there are only a limited number of activation or repression links between the nodes (i.e. the matrix A is 'sparse', too). But assuming sparsity can be a serious obstacle to identifiability as it will be shown.
where '+', '-' and '*' represent positive, negative and nonzero (but otherwise undefined) parameter values, respectively. If there are no prior assumptions about the structure of the interconnection matrix or about the relations between certain parameters, we can easily test the structural non-identifiability of the model by checking whether all nodes are reachable from the perturbed node on a directed path in the interconnection graph or not [33]. The reachability of nodes can be tested by several methods, e.g. a depth- or breadth-first-search (DFS or BFS) of the corresponding directed graphs that are fast polynomial-time algorithms [94]. To give a very simple example, it is clear from Figure 6a, that if nodes 1 or 2 were excited by an input signal, then the connections between the other nodes (3-10) would be undetectable by any method, however sophisticated it is. To examine whether situations like this one are common, we generated 10000 random state space models using the same method, and assuming zero initial conditions as in [93]. The connectivity of the corresponding directed graphs was tested using DFS. For 10 nodes and 2 nonzero elements in each row of A (i.e. N = 10, K = 2), we obtained that 73.38% of the generated models are structurally non-identifiable. The histogram showing the number of reachable states is shown in Figure 6b. The situation is dramatically improving if K is increased to 3 as shown in Figure 6c. In this case, around 17% of the models are structurally non-identifiable. When we have 20 nodes and 5 nonzero elements in each row of A (the second case investigated in [93]), then only 1.6% of the generated models are structurally non-identifiable. The results show that 'sparsity' has a clearly negative effect on structural identifiability because of limited information transmission between nodes. And finally, we did not speak at all about practical identifiability which is known to be a challenging issue even if the required structural properties are fulfilled [66].
Relation between high level networks and CRN structure
As shown in Example 3, the various possible dynamically equivalent CRN structures do not correspond to a different GRN structure, if all species concentration measurements are available and the mapping described in [92] is used for transforming the models into each other. Hence, exact matching of the dynamics of different GRN structures may generally be a too severe restriction. To extend this line of research, the relaxation of dynamical equivalence to 'close dynamical similarity' seems to be more meaningful but the corresponding definitions and computational methods are much more complex than in the case of dynamical equivalence. One promising recent approach to assess dynamical similarity of CRNs (that also adds more degrees of freedom to the computations) is the concept of 'linear conjugacy' [74]. However, it might happen that dynamically completely equivalent GRN structures will be shown in the future.
It has been shown in this paper using illustrative examples that biological network structures modeled by CRNs often cannot be uniquely determined even if the structure of the corresponding mathematical model is assumed to be known and all dynamical variables are measurable. The structural uniqueness and identifiability of the models often require additional constraints.
The main new contributions of the paper are the following. Firstly, core reactions present in any dynamically equivalent CRN realizations with a given complex set have been defined and a simple procedure with polynomial time-complexity has been given for determining them. Clearly, the core reactions are mandatory elements of every dynamically equivalent CRN realization assuming a fixed complex set. Secondly, a polynomial-time method based on linear programming for computing dense realizations has been outlined that is more scalable and therefore presents a clear improvement over the previously used MILP-based method. As an additional minor extension of previous results, constrained realizations of CRNs have been defined, and a computational method has been proposed to check the uniqueness of constrained realizations.
The presented concepts and algorithms were illustrated on previously published models describing biological processes. It was shown that the set of core reactions may change with the modification of the complex set. The examples also show that the frequently applied sparsity assumption alone is not enough for structural uniqueness of CRNs. Moreover, in the case of simple linear genetic network models, too sparse structures can degrade identifiability properties. The results further support the fact that as much prior information as possible should be incorporated in structural and parametric inference problems.
A Appendix
where and . Let us assume now that p is a dense solution for (48), i.e. it contains the maximal possible number of nonzero elements. If p has no zero elements, then the result to be proved is trivially satisfied. Therefore, without the loss of generality we can assume that the first l < m elements of p are nonzero, while the rest are zero, i.e. p j ≠ 0 for j = 1, ..., l, and p j = 0 for j = l + 1, ..., m. This can always be achieved by the appropriate reordering of the elements of p. Assume now that p' ∈ ℝ m is also a solution for (48), but for some c ∈ ℤ, l + 1 ≤ c ≤ m. Then p' = p + v, where , and v c ≠ 0. In this case, p″ = p + λ · v is also a solution for (48) for any λ ∈ ℝ and λ can always be chosen so that for j = 1, ..., l, and there is at least one index l + 1 ≤ c ≤ m for which . However, this contradicts to the assumption that p is a dense solution for (48).
Proof of P2. This is a straightforward consequence of P1, since the unweighted directed graphs of all constrained dense realizations must be identical.
Proof of P3. If the graph structure of the constrained realization is unique, then it trivially implies that the structures of the constrained dense and sparse realizations are identical, since there exists only one possible constrained reaction structure. If the structures of the constrained dense and sparse realizations are identical, then the number of nonzero reaction rates is the same in any constrained realizations including the constrained dense ones. Then it follows from P1 that the constrained reaction structure is unique.
This work was financially supported by project CAFE (Computer Aided Food Process Engineering) FP7-KBBE-2007-1 (Grant no: 212754). The authors acknowledge the support by the Spanish government, MICINN project 'MultiSysBio' (ref. DPI2008-06880-C03-02), and by CSIC intramural project 'BioREDES' (ref. PIE-201170E018). GS acknowledges the partial support of the Hungarian Scientific Research Fund through grant no. K83440. The authors thank Dr. Irene Otero Muras (Dept. of Computational Systems Biology, ETH Zurich) and Prof. Zsolt Tuza (MTA SZTAKI) for the fruitful discussions. The authors wish to thank the anonymous reviewers for their helpful comments.
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This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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The following is from BYU Conferences and Workshops.
One of the largest gatherings of family history experts in the world will take place in Salt Lake City during the last week of this month. The 13th Annual Conference on Computerized Genealogy and Family History, traditionally held at Brigham Young University in Provo in March, will take place at the Salt Palace on April 26 and 27, immediately followed by the four-day Annual Conference (April 28-May 1) of the National Genealogical Society, also at the Salt Palace. The conferences are designed to be how-to guides for beginning, intermediate and advanced researchers.
This is an exceptional opportunity for those with an interest in family history, an interest that already is booming due to the NBC TV show, "Who Do You Think You Are?" The experts at the Conference on Computerized Family History and Genealogy, sponsored by the Church of Jesus Christ of Latter-day Saints and BYU, will teach how to get started in family history research as well as how to make research faster, easier and more effective by using new computer technologies. Anyone with an interest in genealogy is invited to attend.
The two-day conference includes more than 50 classes to help participants learn about advances in computer programs, research techniques, managing genealogy databases, using New FamilySearch, publishing family histories, research methodology, digital photography, using social networking sites to do family history research, research in other countries, and using U.S. records to find ancestors. Vendors also will demonstrate how to use their programs.
Keynote presenter for the conference will be David Rencher, director of the Records and Information Division of the Family and Church History Department of the LDS Church. He has served as director of the Family History Library and has been a leader in providing records of value to family research, including acquiring the Social Security Death Index (SSDI) and making it available to genealogists. Other main presenters include Alan Mann of FamilySearch Community Services for the LDS Church, Kory Meyerink, adjunct religion and family history professor for BYU and vice president of ProGenealogists, and Rick Crume, an author and family history presenter. In addition, representatives from the Family and Church History Department of the LDS Church will discuss how to effectively use the Church's family history programs.
The two-day conference is sponsored by the Family History Department of the LDS Church, FamilySearch, the BYU History Department, the BYU Center for Family History, and BYU's Division of Continuing Education. More information can be found at http://
For pricing or additional program or registration information, contact:
BYU Conferences and Workshops
115 Harman Continuing Education Building Provo, UT 84602-1516
E-mail: firstname.lastname@example.com

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Canon Law
The question of canon, like the question of law, is one of authority. We who are Reformed spend a great deal of time and energy speaking about God's sovereign power. God's power is more than worthy of our attention and …Read More
By What Authority?
Can this be it? Centuries earlier the prophet Zechariah had declared, "Rejoice greatly, O daughter of Zion! Shout aloud, O daughter of Jerusalem! Behold your king is coming to you; righteous and having salvation is he, humble and mounted on a …Read More


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Posts Tagged "NCLY"
National Cyber Leap Year: Without a Good Running Start, There Might Be No Leap
by Russell on September 11, 2009
The National Cyber Leap Year (NCLY) report coming out in a few weeks might lead to more US government research funding for security metrics in coming years. But that depends on whether the report is compelling to the Feds and Congress. Given the flawed process leading up to the Summit, I have my doubts. Clearly, this NCLY process is not a good model for public-private collaboration going forward.


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State of the judiciary and access to justice: hearings before the Subcommittee on Courts, Civil Liberties, and the Administration of Justice of the Committee on the Judiciary, House of Representatives, Ninety-fifth Congress, first session ...
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Cheltenham Climate Conference (2016)
Thanks to James Streeter for taking these photographs.
All Intellectual Property Rights in the data, photographs and materials relating to the Model Climate Conference are the property of InterClimate Network or its licensors, as the case may be. The photographs may not be reproduced, distributed or modified without the express written permission of InterClimate Network.


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When links began circulating online over the Easter weekend about a reported sighting of the Loch Ness monster, it brought to my mind two interesting little tidbits about the beast that I have learnt during my time here at the Museum.
Satellite view of Loch Ness from Apple Maps, which Andy Dixon and Peter Thain claim provide evidence that the Loch Ness monster exists.
The first tidbit, told to me in the queue to get into our annual staff Christmas sale last year, was that the Loch Ness monster is one of only two scientifically described species that doesn't have a type specimen. A type is the particular specimen of an organism which is used to scientifically describe and name that specimen.
The Loch Ness monster was given a scientific name in the journal Nature on 11 December 1975, in an effort to protect it. Sir Peter Scott and Robert Rines wrote:
The Conservation of Wild Creatures and Wild Plants Act 1795 provides the best way of giving full protection to any animal whose survival is threatened. To be included, an animal should be given a common name and a scientific name. For the Nessie or Loch Ness monster, this would require a formal description, even though the creature's relationship with known species, and even the taxonomic class to which it belongs, remains in doubt.
Scott and Rines argued that it was better to be safe than sorry and, citing eye-witness accounts, photographs and sonar images as evidence of Nessie's existence, they proposed the name Nessiteras rhombopteryx. It combines the name of the Loch with the Greek word for wonder (teras or teratos), and the Greek for diamond shape (rhombos) and fin (pteryx), thus: the Ness monster with the diamond shaped fin.
An impression of the possible appearance of Nessiteras rhombopteryx from photographs, eyewitness accounts and sketches derived from a large number of sightings at the surface and from the accompanying underwater pictures, published alongside Scott and Rines' 1975 Nature paper.
Other than Nessie, the only other known species without a type specimen is actually us, Homo sapiens.
The second Nessie tidbit I've comes across during my time here is the arrangement the Museum has (or, at the least, used to have) with William Hill, to adjudicate on evidence presented to the bookmaker by anyone claiming to be able to prove the existence of the Loch Ness monster.
William Hill's Graham Sharpe first made the arrangement with Iain Bishop, the Museum's former deputy keeper of zoology, around 20 years ago. He says:
We quote odds about Nessie being proved to exist, which have fluctuated up to 1000/1 at times, depending on whether and when there have been sightings.
We once got Iain and a couple of members of Museum staff up to the Loch when we organised a Monster Hunt Weekend, offering, I think, £250,000 to anyone who could produce conclusive proof of the existence of Nessie. I remember those days very fondly, and will recall forever the sight of the eccentric rock 'n' roller and politician Screaming Lord Sutch trying to entice Nessie to emerge from the waters of the Loch by tempting her with British Rail sandwiches.
The William Hill-Museum-Loch Ness monster deal was last updated about 10 years ago, between the bookmaker's Rupert Adams and our fish curator, Oliver Crimmen.
The original offer had been based on 'capturing evidence of Nessie on film', but the last agreement stipulated the following identification guidelines:
- The Natural History Museum cannot undertake to identify photographs, sonar traces, written descriptions or other forms of indirect evidence, although would be interested in seeing them.
- The Natural History Museum hopes to be able to detect most hoaxes (eg man-made modifications to bones) but cannot guarantee to identify all fragments and would not regard a determination of 'animal bone unidentifiable' as positive identification of a monster (a small piece of horse rib would probably be impossible to allocate to a species).
- The Natural History Museum will make every effort to provide a positive identification of tangible adequate specimens.
- The Natural History Museum is unsure as to what might constitute a 'monster' though it might define it as a species hitherto unknown to science or widen the definition to include species previously known as fossils. It must be of a similar size to previous eye-witness accounts.
Of course the image visible on Apple Maps does not fall within the Museum's guidelines so, for the time being, William Hill's money goes unclaimed. However, Rupert Adam says:
It is the best evidence we have seen for a couple of years. We are now at 100/1 from 250/1 that Nessie will be found (confirmed by the Museum) before the end of 2014. It could cost us a six figure sum and we have to hope that Nessie does not pop up to trash our Christmas.
Others, though, are not convinced.


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DreamLine's SHDR-20427210S-09 is a Unidoor Hinged Door that will run the gamut from quiet simplicity to utter glamour and luxury. It is a 42" - 43" x 72" shower door enamored with a frameless design for a space-friendly feel while leaving zero distractions from any bathroom dcor. Graced with clean lines and updated accents, it will bring elegance even to the simplest bathroom theme. This DreamLine Shower Door appears to be the epitome of the all-American quality and durability. Not only does it come with an adorable, delicate design but it also has superior functionality about it to boot. It is reversible for a right or left door opening depending on your preference. A modern twist is its satin black-finished hardware that defines mystery and perfection. Another great thing about this hinged door are the luxurious, ANSI-certified tempered glass panels that are proven durable and practically unbreakable. Whats more, it parades DreamLines exclusive ClearMax coating that makes it resistant to water stains, making cleaning and maintenance a piece of cake. Another awesome feature is the self-closing solid brass hinges that enhances convenience so you are carefree each time you walk into the shower! Plus, it comes with 2 glass shelves for added comfort. This DreamLine Swing Door has a walk-in opening of 29" and a door width of 30". The inline panel measures 12" - 13". Add to this a matching SlimLine shower base and your bathroom becomes a fashion-forward haven. And as if these features are not awesome enough, this DreamLine Swing Door is backed by DreamLines 1-year manufacturer warranty.
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|292.58 lbs.


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Living in the Story
Telling a great story with your life is not easy. It's often exhausting, terribly so—and yet a growing number of evangelical writers passionately encourage us to "tell a great story with your life" or "stop trying to increase your …Read More
A Heart for Adoption
To be loved by the Father through the Son in the Spirit is to be caught up into an ever-flowing eternal love that progressively transforms, often painfully so, the one who is loved. As those who have been brought graciously …Read More
Dan Cruver
Dan Cruver is president and cofounder of Together for Adoption, and he is editor and coauthor of Reclaiming Adoption: Missional Living Through the Rediscovery of Abba Father.


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20755 Surnames | 136318 Records | 2776 Reunions
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Photo #23493
Last Name: Underwood
First Name : Mary
Middle Name: Elizabeth
Subject's City: Alto Pass
Subject's State: IL
Subject's County: Union
Subject's Country: United States
Date: 1851-1900
Photographer's City:
Photographer's State:
Nee: Ledbetter
Type** : TT
Comments: Mary Elizabeth Underwood (Nee: Ledbetter) | Alto Pass IL United States | 1851-1900 | Comments: This is a tintype of Mary Elizabeth Ledbetter Underwood, widow of Rev Jesse Underwood and mother of 14 children.
Archive Note:8/17/2006 Found! Discovery #930 Kim Writes** My husband descends from Jesse & Mary (Ledbetter) Underwood.Bless you for posting that photo**
Archive Note: 11/22/2005 Found! Discovery #652 finder writes **my g-ma was Myrtle Marie Underwood, daughter of Jesse James Underwood and Mary Josephine Harmon. so would that make Mary my gggg- gma? i have a photo of her brother Frank and his wife also. Frank being Francis Marion Underwood**
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MMA Affiliate Program
Our friends over at MMA Tycoon have asked us to post about their MMA Affiliate Program. If you run an MMA Site, you should give it a try by signing up to the game and getting your referral URL. They pay 20% on VIP purchases, which costs $30 per year, so if you refer someone who goes VIP for a year, you get $6. Give it a go with a banner ad - you can see we have and it's a nice little earner!


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Helped and Hobbled Causes: Colts
After the Redskins impressive 30-17 dismantling of the Indianapolis Colts on Saturday, Redskins coaches got a further look into how the Redskins 53-man roster is going to break down in a few weeks. Here are a few guys that either helped or hobbled their causes:
Chris Wilson – had the best game of any Washington Redskin on Saturday, at least in my opinion. After sitting out 2011 and not playing football, Wilson looks hell bent on being employed in 2012. On top of Wilson's sack that resulted in a safety, he also had three quarterback hurries and seem to be wreaking havoc in the Colts backfield nearly every time he rushed. If he wasn't on the fringe of people's 53-man before Saturday, he should be there firmly now. Wilson's raw pass rush ability is very valuable depth, especially with Brian Orakpo knicked with an injury to the same area that limited him last year. Read the rest of this entry »
Tags: Alfred Morris, Anthony Armstrong, Brandon Banks, chris cooley, Chris Wilson, Dez Briscoe, Logan Paulsen, Madieu Williams, Markus White, Rex Grossman, Tanard Jackson, Tristan Davis


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Ticketek launches App for iPhone with "Friend Spotter," a new mobile feature powered by CrowdOptic, to connect Facebook Friends at live events
Company combines Facebook data and augmented reality to create the Ultimate Fan enhancement for mobile discovery
Dec 12, 2012, 09:17 ET
SAN FRANCISCO, Dec. 12, 2012 /PRNewswire/ -- CrowdOptic, a maker of crowd-powered mobile applications, today announced that Ticketek, the leading ticketing partner to the sports and live entertainment industry in Australia and New Zealand, has launched a new App for iPhone that includes Friend Spotter™, a unique mobile feature which allows friends to locate each other instantly in a crowd using their smartphones. Australia's number-one Entertainment Events website and ticketing provider has made it possible for friends to spot each other in the stands at Ticketek events, simply by scanning the crowd through their camera viewer.
Ticketek's world-leading mobile platform already offers a complete end-to-end mobile ticketing solution, with the Ticketek mobile site and mobile ticket delivery giving the customer the option to browse, buy and deliver tickets entirely on their smartphone. Today, Ticketek adds the ability to share the live event experience with friends during the event itself, with merely a sweep of the smartphone.
"Smartphones are integral, both to event ticketing and during events themselves," said Cameron Hoy, Managing Director at Ticketek Australia. "By combining Facebook data and CrowdOptic's state-of-the-art visual augmented reality, we are the first to offer our customers this very simple yet impactful way to enhance the live event experience."
CrowdOptic's patent-pending technology platform for live events specializes in tracking dynamic activity in crowds, including everything from where friends are located in crowds, to where people in crowds are pointing their mobile phones moment to moment. The technology offers simple, intuitive visual augmented reality displays for search and discovery of fan activities.
"Visual augmented reality displays will supersede text-based search and discovery and change the way fans interact inside stadiums and arenas," said CrowdOptic CEO Jon Fisher. "We are very pleased to be working with a premier partner such as Ticketek to introduce this groundbreaking technology widely in sporting and entertainment arenas."
The Ticketek App is available for free from the Australian App Store on iPhone. For more information on Ticketek's mobile platform, visit ticketek.com.au. For more information about CrowdOptic, visit www.crowdoptic.com.
About Ticketek
Established in 1979, Ticketek is the leading ticketing partner to the sports and live entertainment industry in Australia and New Zealand. Ticketek processes over 18 million tickets to more than 13,000 events per year and provides exclusive ticketing services to Australia's premier venues. Ticketek is owned by Nine Entertainment Co., Australia's most diversified entertainment company. For more information, visit ticketek.com.au.
About CrowdOptic
CrowdOptic is a new crowd-powered "heat" signal – the next evolution of location-based services that recognizes the hottest crowd activity as it occurs. By tracking where smartphone cameras are located and where they are pointed, CrowdOptic can instantly filter mobile social media and create new opportunities for fan engagement. CrowdOptic identifies, tags and rebroadcasts the live event experience to the world, on mobile, social TV and second screens. CrowdOptic is a privately-held, venture-backed company based in San Francisco. For more information, visit www.crowdoptic.com.
SOURCE CrowdOptic


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Junk dimensions are dimensions that contain miscellaneous data such as flags and indicators. When designing a data warehouse, you might come across a source system that has a bunch of yes/no indicator fields. If those fields needs to be tracked in a fact table, the result could be many small dimension tables (each with just a few rows) along with much more information stored in the fact table, causing performance issues.
Instead, use a junk dimension that holds all the unique combinations of those indicator fields into a single dimension and assigns a unique key. This key is what is stored in the fact table. So you will have only one additional dimension table and will reduce the number of fields in the fact table. A key consideration when forming junk dimensions is how many combinations exist. If the number of combinations is too high the junk dimensions size may be unmanageable, in which case you might want to have more than one junk dimension.
More info:
Kimball Design Tip #48: De-Clutter With Junk (Dimensions)
Design Tip #113 Creating, Using, and Maintaining Junk Dimensions
Data Warehousing: Junk Dimensions
Mystery or Junk data warehouse dimensions
Junk Dimensions with no Loading Needed


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Simon Webbe
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About Simon Webbe
Simon Webbe was born March 30, 1979, in the Manchester working-class neighborhood of Moss Side. Though he dabbled with music in his early years, he was also an excellent amateur soccer player, and had intended to pursue athletics as a career before a serious injury in his early teens. Webbe was challenged by a series of troubled relationships during his teen years (some of which would be explored in later lyrics), one of which made him a father by the age of 17. Webbe channeled frustration into self-improvement, and modeling engagements paved the path for his musical career. Webbe was selected to join British boy band Blue, making him a multi-million record selling artist by his early twenties. Though successful, Blue's time line was limited, leaving Webbe to chart his own course by 2005. In August of that year, Webbe's urban folk-tinged single "Lay Your Hands" hit the airwaves and reached number four on the U.K. singles charts. The song "No Worries" followed, also reaching number four, priming the market for the release of Webbe's solo debut, Sanctuary. The record's alternative, rootsy style surprised and charmed audiences. Though it debuted at number 28, the record quickly climbed to number seven. The much anticipated follow-up release, Grace, was met with success in November of 2006, and peaked at number 11. With a strong fan base throughout Europe and Asia and a busy touring schedule, Webbe has an international reputation that continues to flourish. ~ Evan C. Gutierrez
- Manchester, England
- Pop
- Mar 30, 1979
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Containing the highest quality, most effective fulvic acid available, DIAMOND NECTAR can be used in hydroponic solutions or combined with soil or foliar applied fertilizers. It is rich in both organic and mineral substances. Natural buffering, chelating and extremely high ion-exchange properties convert many mineral elements into forms easier for plants to absorb, resulting in increased vitality, resistance to environmental stress and improved crop quality and yields.
This item is sold individually. The case size is listed for your convenience for instances where you wish to order in case quantities.
— Ships from H: WA, CA, CO, MI, TN, PA, FL USA.


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Things to do with kids in San Antonio, TX and nearby
In the surrounding of 100 km
Show me things to do with kids near city:
21 indoor/outdoor places to go with kids in and near San Antonio, TX:
In the area of 100 kms
|The Alamo
|San Antonio; US, Texas
|Originally named Mission San Antonio de Valero, the Alamo served as home to missionaries and their Indian converts for nearly seventy years.
|Sunset Station
|San Antonio; US, Texas
|Welcome to Sunset Station, San Antonio, Texas - one of the largest historic special event facilities in San Antonio.
|San Antonio Zoo
|San Antonio; US, Texas
|The Zoo is home to over 3,500 animals representing 600 species of animals on 56 acres. More than 1,000,000 guests visit the San Antonio Zoo annually.
|Days out in Toilet Seat Art Museum
|Toilet Seat Art Museum
|Alamo Heights; US, Texas
|Interesting museum
|Malibu Grand Prix San Antonio
|San Antonio; US, Texas
|Acres of Fun, So Close to Home! Race wheel-to-wheel into hairpin turns as you take the lead on the Go Kart Track. Hit one out of the park in our Batting Cages
|Bowlero - San Antonio
|San Antonio; Texas
|Welcome to Bowlero, San Antonio! It's going to be a little piece of bowling heaven situated in another one of the best communities in the country. At Bowlero, San Antonio, you will be able to experience all the amazing fun of the Bowlero brand.
|Brunswick Thousand Oaks Bowl
|San Antonio; Texas
|Bowling at Brunswick is family fun at its best. It's a great way to spend some quality time everyone will enjoy.
|Just Add Children
|San Antonio; US, Texas
|We are San Antonio's most fun Indoor Playground and Party Center for children ages 1 to 10. Rain or shine, it's party time!
|SeaWorld Adventure Park
|theme park
|San Antonio; US, Texas
|SeaWorld San Antonio, the world's largest marine life adventure park and family entertainment showplace, presents a splashy lineup of more than 26 sensational shows, thrilling rides, animal attractions and educational experiences for all ages.
|Friedrich Wilderness Park
|San Antonio; US, Texas
|The Friends of Friedrich Wilderness Park is a volunteer, non-profit organization founded in 1989 to support the preserve, which occupies land donated to the City of San Antonio in 1973.
|Natural Bridge Caverns
|San Antonio; US, Texas
|Chamber after chamber of spectacular formations, one the size of a football field. Gigantic towering columns. Emerald pools. Delicate, crystalline soda straws. All living and still growing.
|Clear Springs Aviaries and Zoological Gardens
|New Braunfels; US, Texas
|Clear Springs Aviaries and Zoological Gardens is a privately owned collection of rare and endangered exotic birds.
|Schlitterbahn Waterpark Resort
|water park
|New Braunfels; US, Texas
|Our family-owned, family-friendly park offers FREE parking, FREE tubes, FREE lifejackets, and allows you to bring your own picnic (no glass or alcohol please).
|Animal World and Snake Farm
|New Braunfels; Texas
|Animal World and Snake Farm (formerly known as Snake Farm) is the perfect place to spend an hour or a day!
|Dixie Dude Ranch
|Bandera; US, Texas
|For more than half a century, spending time at Dixie Dude Ranch in Bandera, Texas, has been a tradition for visitors from all over the world who are looking for that perfect western getaway
|The Exotic Resort Zoo
|Johnson City; US, Texas
|A one of a kind preserve where several endangered species of animals have a chance to increase their numbers and at the same time allowing people to enjoy their beauty and company.
|The National Museum of the Pacific War
|Fredericksburg; US, Texas
|The National Museum of the Pacific War is dedicated to perpetuating the memory of the Pacific Theater of WWII in order that the sacrifices of those who contributed to our victory may never be forgotten.
|Austin Zoo
|Austin; US, Texas
|With more than 300 animals across more than 100 different species, Austin Zoo and Animal Sanctuary is a private, non-profit zoo rescuing animals and serving the Austin community.
|The Millennium Youth Entertainment Complex
|Austin; Texas
|The mission, as recorded, is to provide a safe, secure, and comfortable environment (free from drugs, gangs, crime, and violence) where families can enjoy a wide range of affordable, high quality recreational and entertainment activities and attractions.
|Dart Bowl
|Austin; Texas
|Dart Bowl is a great place to have fun times with family and friends. We have plenty of bumper bowling lanes just for kids!
|Rockhoppers Family Fun Center
|Marble Falls; US, Texas
|Rockhopper's offers a completely stocked game room with new games as well as some of your old favorites. There's something to suit everyone including pinball, classic arcade games, a dance floor game, arcade games for kids, air hockey, and much more.
If you are searching for things to do with kids in some of USA country you can use links bellow:
Alabama -- Alaska -- Arizona -- Arkansas -- California -- Colorado -- Connecticut -- Delaware -- Florida -- Georgia
Hawaii -- Idaho -- Illinois -- Indiana -- Kansas -- Kentucky -- Louisiana -- Iowa -- Maine -- Maryland -- Massachusetts
To have fun with kids select some of the pages
Michigan -- Minnesota -- Mississippi -- Missouri -- Montana -- Nebraska -- Nevada -- New Hampshire -- New Jersey
New Mexico -- New York -- North Carolina -- North Dakota -- Ohio -- Oklahoma -- Wisconsin -- Wyoming -- Oregon
For family fun days out navigate to
Pennyslavia -- Rhode Island -- South Carolina -- South Dakota -- Tennessee -- Texas -- Utah -- Vermont -- Virginia
Washington -- West Virginia
On the following link you can find things to do with kids in UK.
As shortcut for things to do and where to go with the kids in the USA cities use following links:
Newark -- Phoenix -- Denver -- Reston -- Stockton -- Boston -- Washington -- Los Angeles -- Irvine
San Francisco -- Miami -- New York -- Chicago -- San Antonio -- San Diego -- San Diego -- Indianapolis
Albuquerque -- Anchorage -- Salt Lake City
As shortcut for family days out and day trips in the UK you can also use following links:
London -- South East - Milton Keynes -- North West - Manchester -- Scotland - Glasgow -- Ireland - Belfast
South West - Plymouth -- South East - Southampton -- Wales - Cardiff -- West Midlands - Birmingham
Scotland - Edinburgh