{
  "id": "0904.2960",
  "version": "v1",
  "published": "2009-04-20T07:50:51.000Z",
  "updated": "2009-04-20T07:50:51.000Z",
  "title": "Sign patterns for chemical reaction networks",
  "authors": [
    "J. William Helton",
    "Igor Klep",
    "Vitaly Katsnelson"
  ],
  "comment": "23 pages",
  "journal": "J. Math. Chem. 47 (2010), no. 1, 403-429",
  "doi": "10.1007/s10910-009-9579-4",
  "categories": [
    "math.DS"
  ],
  "abstract": "Most differential equations found in chemical reaction networks (CRNs) have the form $dx/dt=f(x)= Sv(x)$, where $x$ lies in the nonnegative orthant, where $S$ is a real matrix (the stoichiometric matrix) and $v$ is a column vector consisting of real-valued functions having a special relationship to $S$. Our main interest will be in the Jacobian matrix, $f'(x)$, of $f(x)$, in particular in whether or not each entry $f'(x)_{ij}$ has the same sign for all $x$ in the orthant, i.e., the Jacobian respects a sign pattern. In other words species $x_j$ always acts on species $x_i$ in an inhibitory way or its action is always excitatory. In Helton, Klep, Gomez we gave necessary and sufficient conditions on the species-reaction graph naturally associated to $S$ which guarantee that the Jacobian of the associated CRN has a sign pattern. In this paper, given $S$ we give a construction which adds certain rows and columns to $S$, thereby producing a stoichiometric matrix $\\widehat S$ corresponding to a new CRN with some added species and reactions. The Jacobian for this CRN based on $\\hat S$ has a sign pattern. The equilibria for the $S$ and the $\\hat S$ based CRN are in exact one to one correspondence with each equilibrium $e$ for the original CRN gotten from an equilibrium $\\hat e$ for the new CRN by removing its added species. In our construction of a new CRN we are allowed to choose rate constants for the added reactions and if we choose them large enough the equilibrium $\\hat e$ is locally asymptotically stable if and only if the equilibrium $e$ is locally asymptotically stable. Further properties of the construction are shown, such as those pertaining to conserved quantities and to how the deficiencies of the two CRNs compare.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-04-20T07:50:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "80A30",
      "15A48",
      "37N25"
    ],
    "keywords": [
      "chemical reaction networks",
      "sign pattern",
      "equilibrium",
      "stoichiometric matrix",
      "construction"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0904.2960H"
    }
  }
}