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 an 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.

Table of Contents

Receptor Modeling

• Overview
• States and transitions
• Ligand binding
• Thermodynamic constraints
• Equilibrium formalism
• Tools and Examples - Part 1

Receptor Dimers

• Overview
• Reduced Graph Powers
• Tools and Examples - Part 2
• Enumerating allosteric parameters

Fitting Allosteric Parameters

• Overview
• Fitting allosteric parameters
• Example lmfit
• example-minimize-linear-regression.ipynb
• Minimize receptor model pade

Nonequilibrium steady states

• Markov chain receptor models
• Non-equilibrium steady states
• Hill’s diagrammatic method
• Tools and Examples - Part 3

Cycle Fluxes

• Overview
• Cycle fluxes in receptor models
• Reduced Graph Powers
• Derived Markov chains
• Example 1 - Cycle fluxes in the graph C3
• Example 2 - Cycle fluxes in the House graph
• Double Derived Markov chain w/ contexts
• Double Derived Markov chain w/ contexts
• Further Reading

Wang algebras

• Wang algebra
• Wang algebra - undirected graph
• Wang algebra - directed graph

Appendix

• Example graphs
• Example Reduced Graph Powers
• BFS Spanning Tree
• docstring test
• My struggle with docstrings
• Plotting hypercube graphs
• practicing python
• Receptor Tools
• Receptor Tools - Cycle Fluxes and Derived Markov chain

Document	Modified	Method	Run Time (s)	Status
2023-08-20-example-lmfit	2026-04-27 23:18	cache	9.1	✅
2023-08-21-practicing-python	2026-04-27 23:18	cache	4.69	✅
2023-08-24-example-minimize-linear-regression	2026-04-27 23:18	cache	5.92	✅
2023-08-24-example-minimize-receptor-model-pade	2026-04-27 23:18	cache	5.03	✅
2023-08-25-plotting-hypercubes	2026-04-27 23:18	cache	4.09	✅
EnumerateAllostericParameters	2026-04-27 23:18	cache	5.02	✅
MyStruggleWithDocstrings	2026-04-27 23:18	cache	6.16	✅
MyTools	2026-04-27 23:18	cache	3.64	✅
MyToolsExamplesEnumerateAllostericParameters	2026-04-27 23:18	cache	5.02	✅
MyToolsExamplesVertexAndEdgeLabels	2026-04-27 23:18	cache	5.07	✅
PlotBindingCurves	2026-04-27 23:19	cache	6.09	✅
ReducedFromCartesianPower	2026-04-27 23:19	cache	6.3	✅
bfs_spanning_tree	2026-04-27 23:19	cache	4.82	✅
cycle_flux_derived_chain	2026-04-27 23:19	cache	23.64	✅
cycle_fluxes	2026-04-27 23:19	cache	6.06	✅
cycle_fluxes_derived_chain_part1	2026-04-27 23:20	cache	23.0	✅
docstring_test	2026-04-27 23:24	cache	4.26	✅
enumerate_allosteric_parameters	2026-04-27 23:24	cache	5.13	✅
enumerate_allosteric_parameters-2	2026-04-27 23:24	cache	4.95	✅
equilibrium	2026-04-27 19:36	cache	10.0	✅
example-lmfit	2026-04-27 23:24	cache	3.8	✅
example-minimize-linear-regression	2026-04-27 23:18	cache	5.92	✅
example_graphs	2026-04-27 19:37	cache	18.12	✅
example_reduced_graph_powers	2026-04-27 19:37	cache	4.96	✅
fitting_allosteric_parameters	2026-04-27 23:25	cache	16.25	✅
hill_diagrammatic_method	2026-04-27 19:37	cache	6.26	✅
ligands	2026-04-27 19:37	cache	4.68	✅
markdown-notebooks	2026-04-27 19:37	cache	3.94	✅
markov_chains	2026-04-27 19:37	cache	4.77	✅
minimize-receptor-model-pade	2026-04-27 23:25	cache	5.0	✅
my_struggle_with_docstrings	2026-04-27 23:25	cache	4.96	✅
my_tools_examples_enumerate_allosteric_parameters	2026-04-27 23:25	cache	4.9	✅
my_tools_examples_vertex_and_edge_labels	2026-04-27 23:25	cache	4.86	✅
nonequilibrium	2026-04-27 19:37	cache	4.79	✅
plot_binding_curves	2026-04-27 23:25	cache	5.98	✅
plotting-hypercubes	2026-04-27 23:25	cache	4.09	✅
practicing-python	2024-07-08 23:00	cache	4.43	✅
receptor_modeling	2026-04-27 19:37	cache	4.4	✅
receptor_modeling_equilibrium_formalism	2026-04-27 19:37	cache	3.99	✅
receptor_modeling_ligands	2026-04-27 19:37	cache	4.89	✅
receptor_modeling_nonequilibrium	2026-04-27 19:38	cache	4.81	✅
receptor_modeling_overview	2026-04-27 19:38	cache	3.52	✅
receptor_modeling_scraps	2026-04-27 19:38	cache	4.36	✅
receptor_modeling_states_and_transitions	2026-04-27 19:38	cache	4.26	✅
receptor_tools	2026-04-27 19:38	cache	3.69	✅
receptor_tools_cycle_fluxes	2026-04-27 19:38	cache	3.64	✅
reduced_graph_powers	2026-04-27 19:38	cache	4.98	✅
states_and_transitions	2026-04-27 19:38	cache	4.16	✅
thermodynamic_constraints_and_allosteric_parameters	2026-04-27 19:38	cache	3.97	✅
tools_and_examples_part1	2026-04-27 23:25	cache	4.86	✅
tools_and_examples_part2	2026-04-27 19:38	cache	5.39	✅
tools_and_examples_part3	2026-04-27 23:25	cache	4.9	✅
tree_polynomials	2026-04-27 19:38	cache	6.96	✅
wang_algebra	2026-04-27 23:38	cache	3.52	✅
wang_algebra_directed_graph	2026-04-27 19:38	cache	4.1	✅
wang_algebra_overview	2026-04-27 23:38	cache	3.52	✅
wang_algebra_undirected_graph	2026-04-27 19:38	cache	4.04	✅