No scientist is simply involved in the single-minded pursuit of truth, he or she is also engaged in the passionate pursuit of research grants and professional success. Nutritionists may wish to attack malnutrition, but they also wish to earn their living in ways they find congenial.
However, to this day, there are still those obsessing about protein. Those promoting Paleolithic diets, for example, try to make the case for protein from an evolutionary perspective.
Okay, so what is the perfect food for human beings, the food that was fine-tuned just for us over millions of years to have the perfect amount of protein? Human breast milk. But this is patently not the case. Human breast milk is one the lowest-protein milks in the mammalian world. Adults require no more than 0. So, someone whose ideal weight is pounds may require up to 40 grams of protein a day.
On average, they probably only need about 30 grams a day, which is. But we say 0. People are more likely to suffer from protein excess than protein deficiency. The adverse effects associated with long-term high protein diets may include disorders of bone and calcium balance, disorders of kidney function, increased cancer risk, disorders of the liver, and worsening of coronary artery disease.
Therefore, there is currently no reasonable scientific basis to recommend protein consumption above the current recommended daily allowance, due to its potential disease risks. To see any graphs, charts, graphics, images, and quotes to which Dr. Greger may be referring, watch the above video. This is just an approximation of the audio contributed by Katie Schloer. Please consider volunteering to help out on the site. Image thanks to PublicDomainPictures via Pixabay.
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Although we may weigh ten times more than a baby, we may only eat four or five times more. So, our food needs to be more concentrated in protein. Nevertheless, people tend to get way more than we need.
Plant protein sources are preferable. Running times ranged from a few seconds per alignment in the local search implementation to several minutes for the CP implementation, which was set to time out at two minutes for each block. This means that our improvement was the equivalent of gaining one additional identity match for each column changed.
This suggests that, even with a simpler scoring function and no additional information such as structural or functional data, the CP implementation improves the ClustalW alignment. The local search implementation did not perform as well, with an insignificant average improvement when compared to BALiBASE alignments and improving only 14 out of 24 alignments. Nevertheless, our goal in this paper was to show that improving MSA by correcting mistakes in less conserved re- gions is a promising approach.
At this stage CP seems to give better results, but there is still work to be done optimizing the scoring functions and heuristics, and the local search implementation takes only milliseconds to find solutions, against several minutes for CP, so there is much room for improvement.
In addition, this gives us a flexible framework for using different scoring functions, not limited to the peculiarities of the underlying alignment algorithm. Of special interest is the inclusion of structural information, whether from determined structures or prediction algorithms, and also scoring functions adapted to coevolution studies, where the assumption of independent mutations does not hold.
A general method applicable to the search for similarities in [ Rapid and sensitive protein similarity searches. Basic local alignment search tool. The maximum weight trace problem in multiple sequence align- ment.
In Apostolico et al eds. Matching, pp — In: Rossi F. Handbook of Constraint Programming, Elsevier Multiple sequence alignment with the Clustal series of programs. Nucleic Acids Res 31 13 : — [8] Edgar, R. ProbCons: Probabilistic consistency-based multiple sequence alignment.
T-Coffee: A novel method for fast and accurate multiple sequence alignment. Profile hidden Markov models. Bioinformatics, 14 9 , Notredame, D. Higgins, Nucleic Acid Research, Vol. Multiple sequence alignment. Bioinformatics Oxford, England , 15 1 , [17] Will S. In 9th Int. Limited Discrepancy Search, In C. Exhaustive matching of the entire protein sequence database. Science, Nucleic acids research, Vol.
Palinkas fu-berlin. Bockmayr fu-berlin. Metabolic reactions and gene regulation are two closely re- lated cellular processes. In integrated models, both are considered to- gether. Metabolic and gene regulatory networks in isolation can be de- scribed by Petri nets. Although an integrated Petri net model for tryp- tophan synthesis has been proposed in the literature, a systematic and general method is not available so far. The goal of this paper is to present such a method, assuming that the stoichiometry of the metabolic reac- tions is known, the gene regulatory network is described in the Thomas formalism, and the interactions with the metabolism are given by logi- cal conditions.
For the gene regulatory network, the resulting Petri net shows exactly the asynchronous unitary dynamics given by the Thomas framework. On the one hand, genes regulate the synthesis of enzymes that catalyze metabolic reactions. On the other hand, metabolites may modulate directly or indirectly gene expression and activities.
In mathematical modeling, metabolism and gene regulation are often treated separately. The reason is that methods available for the isolated systems usually cannot be directly extended to integrated models. On the continuous side, ordi- nary differential equation models face the problem of different time scales. However, these methods heavily depend on the steady-state assumption of FBA. Among the different formalisms, Petri nets seem suitable for integrated mod- els since they can naturally represent not only metabolic networks e.
Next, biological knowl- edge was applied in order to add appropriate interactions with the metabolic part. However, no systematic method was given that would allow translating integrated models in general. In this paper, we present a new translation of GRNs to Petri nets. It is based on the more common description of GRNs by Boolean expressions, as it has been used for example for integrated models of E. Our translation has the advantage that the resulting Petri net is minimal and that it can easily be extended to translate integrated models which include also a metabolic part.
The resulting Petri nets, however, are of the same kind, i. Regarding the size of the models, the Petri net of the GRN grows expo- nentially in the number of interactions between the genes. But, this number is usually limited and in most cases one gene from the GRN can be represented by less than 10 nodes in the Petri net.
A more serious issue is the size of the reachability graph of the Petri net, which represents all the possible dynamics and is essential for most kinds of analysis, e. Its size can grow exponentially in the number of nodes as well as tokens. So far, we used our algorithm only for rather small models, where the GRNs did not have more than 8 genes. In these cases, it was most convenient to pur- sue the translation by hand. A computer implementation for larger models is currently under development.
The j-th column of a matrix A will be denoted by aj , the i-th row by ai , and the entry belonging to both by aji. A Petri net is a bipartite directed graph with N-weighted edges. Directed edges connect places and transitions, but never two nodes of the same kind. Furthermore the places can hold tokens. The dy- namics of a Petri net consists of tokens that are moved from one place to another via the transitions and along the directed edges. In both matrices, the rows represent the places, the columns the transitions, and the entries the weight of the edges.
In the input matrix R, edges lead from places to transitions, while in the output matrix Q, edges lead from transitions to places. Zero entries indicate missing edges see Fig. It changes when a transition fires, i. If tokens that are to be consumed are lacking on the corresponding places, the firing cannot take place. However, the input matrix R is still needed to determine which transitions are enabled. For our translation method, we will describe how to obtain the columns of the matrices R and D.
Z Z fire! Example for the firing of a transition. We need two additional notions [14, 8] that will play a central role further on. A test-edge is an abbreviation for a pair of equally weighted edges that connect the same two nodes in opposite directions. Test-edges implement further constraints on the dynamics because they add conditions for transitions to be enabled.
In the incidence matrix they are invisible. Complementary places is the name for a pair of places such that every token leaving one place moves to the other one see Fig. In other words, in the incidence matrix, the row for the first place is the negative of the row for the second place.
The sum Max of the tokens on complementary places thus remains constant. We use this to represent inhibition in the GRN. For the translation of a metabolic network, we only need the stoichiometric matrix, which will be interpreted as the incidence matrix D of the Petri net. While this is not true in general, it holds here because a metabolic Petri net does not contain test-edges. We will come back to this in Sect. Note that this translation only works if reversible reactions are split into a forward and a backward reaction, see Fig.
In the graphical representation, places correspond to circles, with small dots inside indicating the tokens. Transitions correspond to rectangles.
Edges have the weight 1 unless another weight is indicated. Example of a metabolic network with three reactions. In the stoichiometric matrix, the reversible reaction is already split up, so we can directly interpret it as the incidence matrix of a Petri net structure shown on the right.
Basically these can be described by regulatory compo- nents 1,. Each component i can take as value an integer from 0 to Max i. We assume that the target functions fi are given by Boolean expressions. Therefore, we first describe the states using Boolean variables. To describe a particular state it is not necessary to define the assignment of all variables, because some implications hold.
Example of a logical network. They are shown here in minimised DNF. But, this number can be reduced to two. In this case where all components are 2-valued, we have some simplifications: first, the Boolean variables xA B C D 1 , x1 , x1 , x1 can be abbreviated by A, B, C, D resp. Given such a network, we may define different dynamics see e. Richard [15]. Here, we focus on the asynchronous unitary dynamics [9]. If no component can change its value, we are in a fixpoint and no updates are possible.
This dynamics can be represented in the state transition graph. We follow them in the way the resulting Petri net represents the components of the logical network. Each gene is represented by a pair of complementary places. The transitions are executing an unitary update if the component is fired, i. Most discrete models of regulatory networks are described by Boolean expres- sions. In integrated models, the dependencies of reactions on gene expression can be formulated with Boolean expressions, as done e.
Therefore, we propose here a new translation method that is working with Boolean expressions. These have the additional advantage that their disjunctive normal form DNF can be minimised using e. Thus, we can get a Petri net that is minimal in the number of transitions, which is not the case in the translation of Chaouiya et al.
Note that Steggles et al. Its current value is 1. In grey, the test-edges are shown that will connect the components and therewith implement the Boolean expressions defining the networks dynamics.
Next we need transitions that shift one token between the complementary places of component i see Fig.
Evaluating these Boolean expressions will be achieved with test-edges that can test each of the Boolean variables xiw. In order to have at most one transition enabled at each component Fig. The numbering of the places starts with all positive places. For this al- gorithm it is convenient to construct the incidence matrix D and the input matrix R. As explained before, this pair gives a complete description of the Petri net structure.
In D, test-edges are not visible, so we only have to en- code the actual shifting of tokens which, in our case, is just one token that is shifted between the negative and positive place of one component. All other edges are test-edges. The test-edges at one transition combine as a conjunction of the literals, since the transition is only enabled if all test-edges return a positive result.
Implementing a sep- arate transition for each minterm yields the disjunction of these, since every transition can be enabled independently of the others.
After the firing, some transitions might not be enabled anymore. Similar to the STG of logical networks, we can represent all possible dynamics in a directed graph.
The Petri net consists solely of elementary building blocks Fig. When component i of the logical network has maximal value Max i , then the tokens in the corresponding elementary building block should sum up to Max i.
Any marking that fulfills this is called valid. Analysing the full translation algorithm shows that the firing of any transition leads to an asynchronous unitary update and that a transition is only enabled if the target function implies this update.
Conversely, for every update that is prescribed by the target function, there is a transition that implements it. This can be formally verified and leads to the following statement. Theorem 1. The asynchronous unitary STG of a logical network and the global reachability graph of the Petri net obtained by Algorithm 1 are isomorphic as directed graphs. We start with a formal defi- nition that includes all information necessary for the translation.
Definition 3. An integrated model consists of metabolites M 1 ,. It is described by: 1. Some models for rFBA have exactly this form, see e. There are now two kinds of components: genes and metabolites. The classical Petri net of metabolism as in Fig.
To implement regulation in the GRN, we always have to test for the absence of tokens. This can only be done with complementary places as explained in Sect. Therefore, both kinds of components will be implemented as pairs of complementary places. For gene-components, the tokens on the positive place represent the level of gene expression.
For the metabolic components, they represent the concentration. For the logical network in isolation, we first set the places and then just apply Algorithm 1 to add transitions and edges. This implements the unitary asynchronous dynamics defined by the Boolean expressions. The procedure for the integrated model is given by the following extension: 1.
A pair of complementary places is implemented for each component, whether gene or metabolite. Algorithm 1 is applied to implement all regulatory dynamics. An example of an integrated model In Fig. The fat arrows pointing to the reactions R1 and R2 stand for the enabling of these reactions by the genes which code for enzymes. Maximal values are defined for metabolites and genes. But only x1 and x1 play a role, so we abbreviate them as B, C respectively, as we did with the other, 2-valued components.
An example of a tiny integrated model Fig. Petri net of the tiny model Pseudocode for the integrated Petri net In the Petri net of an integrated model as in Def.
Again we assume all places to be already implemented. The code below adds the transitions and edges by creating columns dn of the inci- dence and rn of the input matrix. The first part of the code is again Algorithm 1. In the new part, the stoichiometric matrix is included. For shifting metabolite tokens according to the stoichiom- etry, the columns of the stoichiometric matrix S have to be included in D. For the complementary negative places the negative of the S-column is inserted.
The input part of these shifts has to be implemented in R as well. They identified fixpoints, cyclic attractors and inter- preted them biologically. Their reachability graphs had the very small size of markings.
The reachability graphs had already about nodes and identification of attractors seemed not sufficient to draw interesting conclusions. Here, model checking seems to be a well suited tool to answer specific ques- tions about the dynamics of the Petri net model.
It is also powerful enough to handle much larger reachability graphs. Petri nets are very flexible objects. For example, by assigning a rate to each transition, we get a continuous time Markov chain, which allows applying prob- abilistic model checking or stochastic simulation. As already mentioned, the genome-scale model of E. However, the reach- ability graph would be much too big to be computed in practice. Further research is needed to develop analysis methods that are suitable for such networks.
References [1] C. Integration of Metabolic Reactions and Gene Regulation. Molecular Biotechnology, —82, Covert, C. Schilling, and B. Regulation of gene expression in flux balance models of metabolism. Journal of theoretical biology, 1 —88, Covert, N. Xiao, T. Chen, and J. Integrating metabolic, tran- scriptional regulatory and signal transduction models in Escherichia coli.
Bioin- formatics, 24 18 , Lee, E. Gianchandani, J. Eddy, and J. Dynamic analysis of integrated signaling, metabolic, and regulatory networks. PLoS computational biology, 4 5 :e, Shlomi, Y. Eisenberg, R. Sharan, and E. A genome-scale computa- tional study of the interplay between transcriptional regulation and metabolism. Molecular systems biology, 3 1 , Reddy, M. Liebman, and M. Qualitative analysis of biochemical reaction systems.
Computers in biology and medicine, 26 1 :9—24, Molecular Informatics, 29 12 —, Chaouiya, A. Naldi, E. Remy, and D. Petri net representation of multi-valued logical regulatory graphs. Natural Computing, pages 1—24, Thomas and M. Multistationarity, the basis of cell differentiation and memory. Logical analysis of regulatory networks in terms of feedback circuits.
Chaos, 11 1 —, Remy, D. Thieffry, and C. Qualitative modelling of regu- lated metabolic pathways: application to the tryptophan biosynthesis in E.
Bioinformatics, 21 suppl 2 , Covert, E. Knight, J. Reed, M. Herrgard, and B. Inte- grating high-throughput and computational data elucidates bacterial networks. Nature, —96, Lee, V. Portnoy, and B. Integrated analysis of regulatory and metabolic networks reveals novel regulatory mechanisms in Sac- charomyces cerevisiae.
Genome research, 16 5 , Goelzer, F. Brikci, I. Martin-Verstraete, P. Noirot, P. Aymerich, and V. Reconstruction and analysis of the genetic and metabolic reg- ulatory networks of the central metabolism of Bacillus subtilis. BMC systems biology, 2 1 , Petri nets: Properties, analysis and applications. Proceedings of the IEEE, 77 4 —, Negative circuits and sustained oscillations in asynchronous automata networks. Advances in Applied Mathematics, 44 4 —, Steggles, R. Banks, O. Shaw, and A.
Qualitatively modelling and analysing genetic regulatory networks: a Petri net approach. Bioinformatics, 23 3 , Gay inria. This paper presents a constraint program for checking whe- ther one graph can be obtained from another graph by using node dele- tions and node mergings. This NP-complete problem is equivalent to the existence problem of a subgraph epimorphism between two graphs, This differs from the well-known subgraph isomorphism problem by the exis- tence of merge in addition to delete operations.
Subgraph epimorphisms allow us to identify biologically meaningful reduction relationships be- tween biochemical reaction graphs in large model repositories such as biomodels. This concept thus offers a computational tool for studying model reductions in systems biology by considering solely the structure of biochemical networks. In this setting, one can define a very general notion of model reduction as a particular form of graph transformation and use it to compare models in systems biology model repositories [3].
For instance, the classical reduction of Michaelis-Menten consists in reducing a system of three reactions, where an en- zyme E binds in a reversible manner to a substrate S to form a complex ES and release a product P , to a single reaction catalyzed by the enzyme, as depicted by the following graphs: E c ES p P E d S S c P The reduced graph can be obtained from the detailed graph by a sequence of delete and merge operations on either species or reaction nodes.
This operational view of graph reduction is equivalent to the existence of an induced subgraph corresponding to delete operations epimorphism i. Subgraph epimorphisms SEPI differ from minors [4] by the possibility to merge non adjacent nodes, by creating a loop when merging two adjacent nodes and by the impossibility to delete an arc without merging the nodes.
In this paper, after a presentation of the basic definitions and properties, we describe the constraint model and search strategy we use to compute subgraph epimorphisms in the biomodels. This benchmark consists in curated models of up to several hundreds of molecular species. Here we chose to use the underlying directed graph. Now something a biologist modeller may want to do is to check whether some model is a reduced version of another model.
On a larger scale, the mod- eller wants to get a hierarchy of models where each model is a refinement or a simplification of the surrounding ones. One way to relate two models is to define graph editing operations which make it possible to transform one reaction graph into another.
A simple thing to do when trying to reduce models is to consider that two species are variants and treat them as equivalents, and to merge every interaction any of the two species had into a new species. The same merging operation can be generalized for reactions. Another natural operation is node deletion. It may be useful for instance to remove intermediate species, or species whose concentration is constant, or reactions that have become trivial after a molecular merging, or reverse reac- tions that occur in a much slower rate than their forward counterpart.
Model refinement proceeds with the dual operations of node addition and splitting and is thus also covered by this approach. Model Reductions in Systems Biology 61 2. There are two operations: node deletion and node merging. The merge operation removes two nodes from a graph and replaces them with a new one inheriting all incident arcs.
The problem of interest is now: given two graphs G and G0 , is there a reduc- tion from G to G0? First, let us find a way to express a reduction as a one-step operation.
An epimorphism from G to G0 is a morphism that is surjective on both the nodes and the arcs of G0. As shown below, graph epimorphisms relate graphs that can be obtained by only merge operations.
Nevertheless, the practical instances of such problems may well be solved by efficient algorithms and it is the purpose of this section to describe a con- straint program for the SEPI problem. For this work, we developed a GNU- prolog [5] program dedicated to our particular subgraph epimorphism problems, using finite domain constraints and a simple search strategy for enumerating all solutions by backtracking.
Graph morphisms can be modeled quite naturally by introducing one variable per node of the source graph, with, as domain, one integer value per node of the target graph.
A variable assignment thus represents a mapping from the source nodes to the target nodes [6]. In this representation, the morphism condition itself, stating that the arcs must be preserved by the mapping, can be written using the primitive tabular constraint of GNU-Prolog fd relation integer list list, variable list This constraint states that the tuple of variables in the second argument here two variables representing the image of an arc in the source graph is equal to one element of the list in the first argument representing all the arcs of the target graph.
The surjectivity property could be represented using the primitive cardinality constraint fd at least one applied to the arcs of the target graph. However, a more efficient modeling was found by introducing antecedent variables for the target arc variables, i.
For each arc in the target graph, this constraint is actually used on the first and second nodes of the arc to state that the antecedent variables correspond to an arc in the source graph. Actually, enumerating only one of the sets is enough. Suppose we have tried to enumerate the source node variables, and failed. Then, there is clearly no SEPI from the source graph to the target graph. If on the contrary the enumeration succeeded, then there is obviously a mor- phism.
Is it surjective? Since every antecedent variable has a non-empty domain, we know that every pair of antecedents of the corresponding target pair of nodes comes from an arc in the source graph. Thus, an enumeration of the source node variable is enough to enforce arc surjectivity. However, compared to enumerating the antecedents variables be- forehand, this choice of variables checks the surjectivity quite late.
Now, suppose we have enumerated only the antecedent variables, and failed. Once again, it is obvious that there is no SEPI from the source graph to the target graph. If the enumeration succeeded, then some source node variables have been determined by the process, i. Then, it can be proved that the valuation of the source node variables is a SEPI from the source graph to the target graph.
Indeed, the valuation is always a morphism because of the way dummy values are used, and the antecedent arc variables being determined, it is surjective on the arcs. For brevity, it supposes that the target graph has no isolated nodes. This allows us to enforce surjectivity on arcs only, as in this case it entails surjectivity on nodes. Some matchings between unrelated model classes were found. These biolog- ically false positive matchings typically arise with small models that formally appear as reductions of large models without any biological meaning.
This class contains the family of models of [8] numbered 26 to 31 in biomod- els. In this family, models 27, 29 and 31 are the simpler ones: they have few molecules because the catalyses are represented with only one reaction. The epi- morphism exhibited from model 31 to 27 corresponds to the splitting of two variants of MAPKK in Model 29 distinguishes between the sites of phos- phorylation of Mp, yielding a model with two molecules MpY and MpT.
The subgraph epimorphism found from 29 to 27 corresponds to the deletion of one variant of Mp. Conversely, this distinction prevents the existence of an epimor- phism from 31 or 27 to Models 26, 28 and 30 have more detailed catalysis mechanisms and differ as previously by the phosphorylation sites of Mp. However, some epimorphisms from big models to small ones may have no biological meaning.
Still, model 26 with non-differentiated Mp does not reduce to model 29 since that model indeed distinguishes MpY and MpT variants. Indeed, they only differ by molecule names and parameter values. They do not reduce to 28 and 30, which are models that do not differentiate sites of phosphorylation. They do not reduce to 26 either, which uses a more detailed mechanism for dephosphorylations.
Model 10 is another 3-step MAPK with no catalysts for dephosphorylations. This is shown here as a reduction of the previous models obtained by merging the output of the third level with the catalyst of the first level. Finally, models 49 and are bigger than the others and can easily be matched by them.
There were a few comparisons for which no result was found before the timeout. However some optimizations, such as redundant constraints, should result in even better be- haviour for bigger instances. Furthermore, the handling of labels and annotations attached to molecular species nodes would drastically reduce the search space for the labeling. Future work includes another harder problem to tackle, the problem of great- est common epimorphic subgraph, i.
Given two graphs G and G0 , what are the greatest smallest graphs G00 such that both G and G0 reduce to resp. Such graphs may however be not unique. References [1] Reddy, V. In Hunter, L. Volume of Lecture Notes in Computer Science.
The discovery of regulatory networks is an important aspect in the post genomic research. Among structure learning approaches we are interested in local search methods in the Bayesian network framework. We propose a new local move operator to escape more efficiently from local maxima in the search space of directed acyclic graphs. This operator allows to overtake the acyclic constraint of Bayesian networks and authorizes local moves previously banned with classic operators. First results show improvements of learnt network quality.
Our algorithm uses Comet language providing abstraction for local search and constraint programming. Keywords: structure learning, Bayesian networks, local search, Comet language, gene regulation inference, genetical genomics. Currently, integrative approaches are developed to combine several sources of information in order to improve prediction quality. One of these approaches consists in using genetical genomic data combining gene expressions and sequence polymorphisms observed by genetic markers [1] [2] Chap.
Among the many existing frameworks used to infer GRN, we choose probabilistic graphical models and more specifically static Bayesian Networks BN [3].
Learning BN structures from data is a NP-hard problem [4] and several approaches have been proposed to solve it. One of them consists in exploring the space of BN structures using local search methods and evaluating each structure with a specific scoring criterion in order to select the structure which maximizes the score. Then we report in Section 3 our preliminary work using this operator inside the Comet local search platform and give some positive results on simulated genetical genomic data.
We assume each gene Gi is co-located with a single genetic marker Mi. Each marker may explain the variation of its associated gene activity or the variations of other regulated genes. An example is given in Figure 1. Some of the techniques listed in Fiasco may require a sound knowledge of Hypnosis, users are advised to either leave those sections or must have a basic understanding of the subject before practicing them.
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