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.

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
bfs_spanning_tree	2026-07-11 00:51	cache	3.9	❌
cycle_fluxes	2026-07-11 00:51	cache	2.0	❌
cycle_fluxes_derived_chain_part1	2026-07-11 00:51	cache	1.87	❌
docstring_test	2026-07-11 00:52	cache	3.77	✅
enumerate_allosteric_parameters	2026-07-11 00:52	cache	1.89	❌
equilibrium	2026-07-11 00:52	cache	1.77	❌
example-lmfit	2026-07-11 00:52	cache	2.28	✅
example-minimize-linear-regression	2026-07-11 00:52	cache	2.17	❌
example_graphs	2026-07-11 00:52	cache	1.8	❌
example_reduced_graph_powers	2026-07-11 00:52	cache	1.79	❌
fitting_allosteric_parameters	2026-07-11 00:52	cache	2.25	❌
hill_diagrammatic_method	2026-07-11 00:52	cache	1.79	❌
ligands	2026-07-11 00:52	cache	2.18	❌
markov_chains	2026-07-11 00:52	cache	2.3	❌
minimize-receptor-model-pade	2026-07-11 00:52	cache	2.11	❌
my_struggle_with_docstrings	2026-07-11 00:52	cache	2.34	✅
nonequilibrium	2026-07-11 00:52	cache	1.79	❌
plotting-hypercubes	2026-07-11 00:52	cache	1.8	❌
practicing-python	2026-07-11 00:52	cache	1.24	✅
receptor_modeling_overview	2026-07-11 00:52	cache	1.6	✅
receptor_tools	2026-07-11 00:52	cache	1.98	✅
receptor_tools_cycle_fluxes	2026-07-11 00:52	cache	1.91	✅
reduced_graph_powers	2026-07-11 00:52	cache	1.93	❌
states_and_transitions	2026-07-11 00:52	cache	1.91	❌
thermodynamic_constraints_and_allosteric_parameters	2026-07-11 00:52	cache	2.18	❌
tools_and_examples_part1	2026-07-11 00:52	cache	2.02	❌
tools_and_examples_part2	2026-07-11 00:52	cache	2.03	❌
tools_and_examples_part3	2026-07-11 00:52	cache	2.01	❌
tree_polynomials	2026-07-11 00:52	cache	1.99	❌
wang_algebra_directed_graph	2026-07-11 00:52	cache	1.93	❌
wang_algebra_overview	2026-07-11 00:52	cache	1.6	✅
wang_algebra_undirected_graph	2026-07-11 00:52	cache	1.91	❌