Welcome to Receptors

Welcome to Receptors#

This Jupyter Book explores how cell surface receptors can be modeled using SageMath, a free open-source mathematics software system (see https://www.sagemath.org).

The focus is Markov chain models of receptors and symbolic algebraic calculations of equilibrium and steady-state kinetic occupancy measures (i.e., the probability of each state), as opposed to numerical methods to calculate steady states or time-dependent dynamics.

After a brief introductory tutorial, we illustrate an approach to modeling conformational coupling of receptor dimers and higher-order oligomers that was introduced in the following two papers.

  • Gregory Douglas Conradi Smith, Allostery in oligomeric receptor models, Mathematical Medicine and Biology: A Journal of the IMA, 37(3):313-333, 2020. doi: 10.1093/imammb/dqz016

  • Richard H. Hammack and Gregory D. Smith, Cycle bases of reduced powers of graphs, ARS Mathematica Contemporanea, 12(1):183–203, 2017. doi: 10.26493/1855-3974.856.4d2

Comments and suggestions are welcome!

Greg Conradi Smith
Professor and Associate Chair
Department of Applied Science
Programs in Neuroscience and CAMS Mathematical Biology
William & Mary

greg.conradi.smith@gmail.com
0000-0002-1054-6790

Author of Cellular Biophysics & Modeling: a primer on the computational biology of excitable cells. Cambridge University Press, 2019.

Doodle

Table of Contents#

Receptor Modeling

Receptor Dimers

Fitting Allosteric Parameters

Nonequilibrium steady states

Cycle Fluxes

Wang algebras

Appendix

Document

Modified

Method

Run Time (s)

Status

bfs_spanning_tree

2026-06-21 19:50

cache

2.83

❌

cycle_fluxes

2026-06-21 19:50

cache

1.88

❌

cycle_fluxes_derived_chain_part1

2026-06-21 19:50

cache

1.84

❌

docstring_test

2026-06-21 19:50

cache

2.92

✅

enumerate_allosteric_parameters

2026-06-21 19:51

cache

1.89

❌

equilibrium

2026-06-21 19:51

cache

1.74

❌

example-lmfit

2026-06-21 19:51

cache

2.12

✅

example-minimize-linear-regression

2026-06-21 19:51

cache

1.96

❌

example_graphs

2026-06-21 19:51

cache

1.79

❌

example_reduced_graph_powers

2026-06-21 19:51

cache

1.8

❌

fitting_allosteric_parameters

2026-06-21 19:51

cache

1.88

❌

hill_diagrammatic_method

2026-06-21 19:51

cache

1.92

❌

ligands

2026-06-21 19:51

cache

2.14

❌

markov_chains

2026-06-21 19:51

cache

2.14

❌

minimize-receptor-model-pade

2026-06-21 19:51

cache

2.08

❌

my_struggle_with_docstrings

2026-06-21 19:51

cache

2.32

✅

nonequilibrium

2026-06-21 19:51

cache

1.77

❌

plotting-hypercubes

2026-06-21 19:51

cache

1.75

❌

practicing-python

2026-06-21 19:51

cache

0.95

✅

receptor_modeling_overview

2026-06-21 19:51

cache

1.56

✅

receptor_tools

2026-06-21 19:51

cache

1.8

✅

receptor_tools_cycle_fluxes

2026-06-21 19:51

cache

1.74

✅

reduced_graph_powers

2026-06-21 19:51

cache

1.75

❌

states_and_transitions

2026-06-21 19:51

cache

1.76

❌

thermodynamic_constraints_and_allosteric_parameters

2026-06-21 19:51

cache

2.15

❌

tools_and_examples_part1

2026-06-21 19:51

cache

1.87

❌

tools_and_examples_part2

2026-06-21 19:51

cache

1.86

❌

tools_and_examples_part3

2026-06-21 19:51

cache

1.87

❌

tree_polynomials

2026-06-21 19:51

cache

1.87

❌

wang_algebra_directed_graph

2026-06-21 19:51

cache

1.77

❌

wang_algebra_overview

2026-06-21 19:51

cache

1.56

✅

wang_algebra_undirected_graph

2026-06-21 19:51

cache

1.77

❌