### Abstract

When bivalent ligands capable of bridging binding sites on two different receptors interact with bivalent receptors, aggregates form. The aggregates can be of two types: chains (open structures containing n receptors, n - 1 doubly bound ligands and 0, 1, or 2 singly bound ligands) and rings (closed structures containing n receptors and n doubly bound ligands). Both types of aggregates have been detected experimentally. In general, to determine the time dependence of the concentration of any particular aggregate requires solving an infinite set of coupled ordinary differential equations (ODEs). Perelson and DeLisi [19] showed that great simplification results if all receptor binding sites are equivalent, i.e., the binding properties of a site on a receptor are independent of the size of the aggregate the receptor is in. If only chains form, the problem reduces to solving two coupled ODEs for the concentrations of singly and doubly bound ligands. From the solutions to these ODEs, the time dependence of the entire aggregate size distribution can be determined. We show that the equivalent site approximation as formulated by Perelson and DeLisi [19] is incompatible with ring formation. We then present a modified equivalent site approximation that is useful if chains of any size can form but rings above a certain size (k) cannot. We show how to reduce the resulting infinite set of coupled ODEs to a closed system of at most 4k + 2 ODEs for the ligand concentrations, the ring concentrations, and the concentrations of all chains up to size k. Although we can only predict the kinetics of aggregate formation for aggregates of size k or less, at equilibrium the modified equivalent site approximation yields the complete aggregate size distribution.

Original language | English (US) |
---|---|

Pages (from-to) | 171-190 |

Number of pages | 20 |

Journal | Mathematical Biosciences |

Volume | 126 |

Issue number | 2 |

DOIs | |

State | Published - 1995 |

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### ASJC Scopus subject areas

- Agricultural and Biological Sciences(all)
- Biochemistry, Genetics and Molecular Biology(all)
- Immunology and Microbiology(all)
- Applied Mathematics
- Modeling and Simulation
- Statistics and Probability
- Medicine(all)
- Ecology, Evolution, Behavior and Systematics

### Cite this

*Mathematical Biosciences*,

*126*(2), 171-190. https://doi.org/10.1016/0025-5564(94)00045-2

**The kinetics of bivalent ligand-bivalent receptor aggregation : Ring formation and the breakdown of the equivalent site approximation.** / Posner, Richard G; Wofsy, Carla; Goldstein, Byron.

Research output: Contribution to journal › Article

*Mathematical Biosciences*, vol. 126, no. 2, pp. 171-190. https://doi.org/10.1016/0025-5564(94)00045-2

}

TY - JOUR

T1 - The kinetics of bivalent ligand-bivalent receptor aggregation

T2 - Ring formation and the breakdown of the equivalent site approximation

AU - Posner, Richard G

AU - Wofsy, Carla

AU - Goldstein, Byron

PY - 1995

Y1 - 1995

N2 - When bivalent ligands capable of bridging binding sites on two different receptors interact with bivalent receptors, aggregates form. The aggregates can be of two types: chains (open structures containing n receptors, n - 1 doubly bound ligands and 0, 1, or 2 singly bound ligands) and rings (closed structures containing n receptors and n doubly bound ligands). Both types of aggregates have been detected experimentally. In general, to determine the time dependence of the concentration of any particular aggregate requires solving an infinite set of coupled ordinary differential equations (ODEs). Perelson and DeLisi [19] showed that great simplification results if all receptor binding sites are equivalent, i.e., the binding properties of a site on a receptor are independent of the size of the aggregate the receptor is in. If only chains form, the problem reduces to solving two coupled ODEs for the concentrations of singly and doubly bound ligands. From the solutions to these ODEs, the time dependence of the entire aggregate size distribution can be determined. We show that the equivalent site approximation as formulated by Perelson and DeLisi [19] is incompatible with ring formation. We then present a modified equivalent site approximation that is useful if chains of any size can form but rings above a certain size (k) cannot. We show how to reduce the resulting infinite set of coupled ODEs to a closed system of at most 4k + 2 ODEs for the ligand concentrations, the ring concentrations, and the concentrations of all chains up to size k. Although we can only predict the kinetics of aggregate formation for aggregates of size k or less, at equilibrium the modified equivalent site approximation yields the complete aggregate size distribution.

AB - When bivalent ligands capable of bridging binding sites on two different receptors interact with bivalent receptors, aggregates form. The aggregates can be of two types: chains (open structures containing n receptors, n - 1 doubly bound ligands and 0, 1, or 2 singly bound ligands) and rings (closed structures containing n receptors and n doubly bound ligands). Both types of aggregates have been detected experimentally. In general, to determine the time dependence of the concentration of any particular aggregate requires solving an infinite set of coupled ordinary differential equations (ODEs). Perelson and DeLisi [19] showed that great simplification results if all receptor binding sites are equivalent, i.e., the binding properties of a site on a receptor are independent of the size of the aggregate the receptor is in. If only chains form, the problem reduces to solving two coupled ODEs for the concentrations of singly and doubly bound ligands. From the solutions to these ODEs, the time dependence of the entire aggregate size distribution can be determined. We show that the equivalent site approximation as formulated by Perelson and DeLisi [19] is incompatible with ring formation. We then present a modified equivalent site approximation that is useful if chains of any size can form but rings above a certain size (k) cannot. We show how to reduce the resulting infinite set of coupled ODEs to a closed system of at most 4k + 2 ODEs for the ligand concentrations, the ring concentrations, and the concentrations of all chains up to size k. Although we can only predict the kinetics of aggregate formation for aggregates of size k or less, at equilibrium the modified equivalent site approximation yields the complete aggregate size distribution.

UR - http://www.scopus.com/inward/record.url?scp=0028931731&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0028931731&partnerID=8YFLogxK

U2 - 10.1016/0025-5564(94)00045-2

DO - 10.1016/0025-5564(94)00045-2

M3 - Article

C2 - 7703593

AN - SCOPUS:0028931731

VL - 126

SP - 171

EP - 190

JO - Mathematical Biosciences

JF - Mathematical Biosciences

SN - 0025-5564

IS - 2

ER -