Abstract
Human health and physiology is strongly influenced by interactions between human cells and intestinal microbiota in the gut. In mammals, the host-microbiota crosstalk is mainly mediated by regulations at the intestinal crypt level: the epithelial cell turnover in crypts is directly influenced by metabolites produced by the microbiota. Conversely, enterocytes maintain hypoxia in the gut, favorable to anaerobic bacteria which dominate the gut microbiota. We constructed an individual-based model of epithelial cells interacting with the microbiota-derived chemicals diffusing in the crypt lumen. This model is formalized as a piecewise deterministic Markov process (PDMP). It accounts for local interactions due to cell contact (among which are mechanical interactions), for cell proliferation, differentiation and extrusion which are regulated spatially or by chemicals concentrations. It also includes chemicals diffusing and reacting with cells. A deterministic approximated model is also introduced for a large population of small cells, expressed as a system of porous media type equations. Both models are extensively studied through numerical exploration. Their biological relevance is thoroughly assessed by recovering bio-markers of an healthy crypt, such as cell population distribution along the crypt or population turn-over rates. Simulation results from the deterministic model are compared to the PMDP model and we take advantage of its lower computational cost to perform a sensitivity analysis by Morris method. We finally use the crypt model to explore butyrate supplementation to enhance recovery after infections by enteric pathogens.
Similar content being viewed by others
References
Alex S, Lange K, Amolo T, Grinstead JS, Haakonsson AK, Szalowska E, Koppen A, Mudde K, Haenen D, Al-Lahham S, Roelofsen H, Houtman R, van der Burg B, Mandrup S, Bonvin AMJJ, Kalkhoven E, Muller M, Hooiveld GJ, Kersten S (2013) Short-Chain Fatty Acids Stimulate Angiopoietin-Like 4 Synthesis in Human Colon Adenocarcinoma Cells by Activating Peroxisome Proliferator-Activated Receptor. Mol Cell Biol 33(7):1303–1316. https://doi.org/10.1128/MCB.00858-12
Almet AA, Maini PK, Moulton DE, Byrne HM (2020) Modeling perspectives on the intestinal crypt, a canonical system for growth, mechanics, and remodeling. Current Opinion in Biomedical Engineering 15:32–39
Aregba-Driollet D, Natalini R, Tang S (2003) Explicit diffusive kinetic schemes for nonlinear degenerate parabolic systems. Math Comput 73(245):63–94. https://doi.org/10.1090/S0025-5718-03-01549-7
Awad M, Senga Kiesse T, Assaghir Z, Ventura A (2019) Convergence of sensitivity analysis methods for evaluating combined influences of model inputs. Reliability Engineering & System Safety 189:109–122. https://doi.org/10.1016/j.ress.2019.03.050
Bansaye V, Méléard S (2015) Stochastic models for structured populations: scaling limits and long time behavior. Springer International Publishing
Barker N (2014) Adult intestinal stem cells: Critical drivers of epithelial homeostasis and regeneration. Nat Rev Mol Cell Biol 15(1):19–33. https://doi.org/10.1038/nrm3721
Barker N, van Es JH, Kuipers J, Kujala P, van den Born M, Cozijnsen M, Haegebarth A, Korving J, Begthel H, Peters PJ, Clevers H (2007) Identification of stem cells in small intestine and colon by marker gene Lgr5. Nature 449(7165):1003–1007. https://doi.org/10.1038/nature06196
Barker N, van de Wetering M, Clevers H (2008) The intestinal stem cell. Genes & Development 22(14):1856–1864. https://doi.org/10.1101/gad.1674008
Batlle E, Henderson JT, Beghtel H, Van den Born MM, Sancho E, Huls G, Meeldijk J, Robertson J, Van de Wetering M, Pawson T, Clevers H (2002) \(\beta \)-catenin and TCF mediate cell positioning in the intestinal epithelium by controlling the expression of EphB/EphrinB. Cell 111(2):251–263. https://doi.org/10.1016/S0092-8674(02)01015-2
Bourgin M, Labarthe S, Kriaa A, Lhomme M, Gérard P, Lesnik P, Laroche B, Maguin E, Rhimi M (2020) Exploring the bacterial impact on cholesterol cycle: A numerical study. Front Microbiol 11:1121
Bravo R, Axelrod DE (2013) A calibrated agent-based computer model of stochastic cell dynamics in normal human colon crypts useful for in silico experiments. Theor Biol Med Model 10(1):66
Buchwald DS, Blaser MJ (1984) A review of human salmonellosis: Ii. duration of excretion following infection with nontyphi salmonella. Rev Infect Dis 6(3):345–356
Buckwar E, Riedler MG (2011) An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution. J Math Biol 63(6):1051–1093
Buske P, Galle J, Barker N, Aust G, Clevers H, Loeffler M (2011) A comprehensive model of the spatio-temporal stem cell and tissue organisation in the intestinal crypt. PLoS Comput Biol 7(1):e1001045. https://doi.org/10.1371/journal.pcbi.1001045
Buske P, Przybilla J, Loeffler M, Sachs N, Sato T, Clevers H, Galle J (2012) On the biomechanics of stem cell niche formation in the gut - Modelling growing organoids. FEBS J 279(18):3475–3487. https://doi.org/10.1111/j.1742-4658.2012.08646.x
Byndloss MX, Bäumler AJ (2018) The germ-organ theory of non-communicable diseases. Nat Rev Microbiol 16(2):103–110. https://doi.org/10.1038/nrmicro.2017.158
Campolongo F, Cariboni J, Saltelli A (2007) An effective screening design for sensitivity analysis of large models. Environmental Modelling & Software 22(10):1509–1518. https://doi.org/10.1016/j.envsoft.2006.10.004
Cherbuy C, Darcy-Vrillon B, Morel MT, Pégorier JP, Duée PH (1995) Effect of germfree state on the capacities of isolated rat colonocytes to metabolize n-Butyrate, glucose, and glutamine. Gastroenterology 109(6):1890–1899. https://doi.org/10.1016/0016-5085(95)90756-4 **
Clausen MR, Mortensen PB (1994) Kinetic studies on the metabolism of short-chain fatty acids and glucose by isolated rat colonocytes. Gastroenterology 106(2):423–432. https://doi.org/10.1016/0016-5085(94)90601-7
Collins JW, Keeney KM, Crepin VF, Rathinam VAK, Fitzgerald KA, Finlay BB, Frankel G (2014) Citrobacter rodentium: infection, inflammation and the microbiota. Nat Rev Microbiol 12(9):612–623. https://doi.org/10.1038/nrmicro3315
Cremer J, Arnoldini M, Hwa T (2017) Effect of water flow and chemical environment on microbiota growth and composition in the human colon. Proc Natl Acad Sci 114(25):6438–6443
Curtain RF, Zwart HJ (1995) An introduction to infinite dimensional linear systems theory. Springer
Daley DJ, Vere-Jones D (2003) An Introduction to the Theory of Point Processes. Probability and Its Applications. Springer (second edition). URL https://books.google.fr/books?id=GqRXYFxe0l0C
Darrigade L (2020) Modélisation du dialogue hôte-microbiote au voisinage de l’épithélium de l’intestin distal. Ph.D. thesis, université Paris-Saclay
Darwich AS, Aslam U, Ashcroft DM, Rostami-Hodjegan A (2014) Meta-analysis of the turnover of intestinal epithelia in preclinical animal species and humans. Drug Metab Dispos 42(12):2016–2022. https://doi.org/10.1124/dmd.114.058404
Degirmenci B, Valenta T, Dimitrieva S, Hausmann G, Basler K (2018) GLI1-expressing mesenchymal cells form the essential Wnt-secreting niche for colon stem cells. Nature. https://doi.org/10.1038/s41586-018-0190-3. URL http://dx.doi.org/10.1038/s41586-018-0190-3www.nature.com/articles/s41586-018-0190-3
Fan YY, Davidson LA, Callaway ES, Wright GA, Safe S, Chapkin RS (2015) A bioassay to measure energy metabolism in mouse colonic crypts, organoids, and sorted stem cells. American Journal of Physiology - Gastrointestinal and Liver Physiology 309(1):G1–G9. https://doi.org/10.1152/ajpgi.00052.2015
Fletcher AG, Murray PJ, Maini PK (2015) Multiscale modelling of intestinal crypt organization and carcinogenesis. Math Models Methods Appl Sci 25(13):2563–2585. https://doi.org/10.1142/S0218202515400187
Hannezo E, Coucke A, Joanny JF (2016) Interplay of migratory and division forces as a generic mechanism for stem cell patterns. Phys Rev E 93(2):022405. https://doi.org/10.1103/PhysRevE.93.022405
Hannezo E, Prost J, Joanny JF (2011) Instabilities of Monolayered Epithelia: Shape and Structure of Villi and Crypts. Phys Rev Lett 107(7):078104. https://doi.org/10.1103/PhysRevLett.107.078104
Herman J, Usher W (2017) SALib: An open-source Python library for Sensitivity Analysis. The Journal of Open Source Software 2(9):97. https://doi.org/10.21105/joss.00097
Jacobsen M (2006) Point process theory and applications: marked point and piecewise deterministic processes. Birkhaüser
Kaiko GE, Ryu SH, Koues OI, Collins PL, Solnica-Krezel L, Pearce EJ, Pearce EL, Oltz EM, Stappenbeck TS (2016) The Colonic Crypt Protects Stem Cells from Microbiota-Derived Metabolites. Cell 165(7):1708–1720. https://doi.org/10.1016/j.cell.2016.05.018
Kelly CJ, Zheng L, Campbell EL, Saeedi B, Scholz CC, Bayless AJ, Wilson KE, Glover LE, Kominsky DJ, Magnuson A, Weir TL, Ehrentraut SF, Pickel C, Kuhn KA, Lanis JM, Nguyen V, Taylor CT, Colgan SP (2015) Crosstalk between microbiota-derived short-chain fatty acids and intestinal epithelial HIF augments tissue barrier function. Cell Host Microbe 17(5):662–671. https://doi.org/10.1016/j.chom.2015.03.005
Krndija D, El Marjou F, Guirao B, Richon S, Leroy O, Bellaiche Y, Hannezo E, Matic Vignjevic D (2019) Active cell migration is critical for steady-state epithelial turnover in the gut. Science 365(6454):705–710. https://doi.org/10.1126/science.aau3429
Labarthe S, Polizzi B, Phan T, Goudon T, Ribot M, Laroche B (2019) A mathematical model to investigate the key drivers of the biogeography of the colon microbiota. J Theor Biol 462:552–581. https://doi.org/10.1016/j.jtbi.2018.12.009
Langlands AJ, Almet AA, Appleton PL, Newton IP, Osborne JM, Näthke IS (2016) Paneth Cell-Rich Regions Separated by a Cluster of Lgr5+ Cells Initiate Crypt Fission in the Intestinal Stem Cell Niche. PLoS Biol 14(6):1–31. https://doi.org/10.1371/journal.pbio.1002491
Lasiecka I (1980) Unified theory for abstract parabolic boundary problems-a semigroup approach. Appl Math Optim 6(1):287–333
Litvak Y, Byndloss MX, Bäumler AJ (2018) Colonocyte metabolism shapes the gut microbiota. Science 362(6418):eaat9076. https://doi.org/10.1126/science.aat9076
Lopez CA, Miller BM, Rivera-Chávez F, Velazquez EM, Byndloss MX, Chávez-Arroyo A, Lokken KL, Tsolis RM, Winter SE, Bäumler AJ (2016) Virulence factors enhance citrobacter rodentium expansion through aerobic respiration. Science 353(6305):1249–1253. https://doi.org/10.1126/science.aag3042
Lopez-Garcia C, Klein AM, Simons BD, Winton DJ (2010) Intestinal stem cell replacement follows a pattern of neutral drift. Science 330(6005):822–825. https://doi.org/10.1126/science.1196236
Lushnikov PM, Chen N, Alber M (2008) Macroscopic dynamics of biological cells interacting via chemotaxis and direct contact. Physical Review E 78(6):061904
Martin-Gallausiaux C, Marinelli L, Blottière HM, Larraufie P, Lapaque N (2020) Scfa: mechanisms and functional importance in the gut. Proceedings of the Nutrition Society 80(1):37–49. https://doi.org/10.1017/s0029665120006916
McMurtrey RJ (2016) Analytic Models of Oxygen and Nutrient Diffusion, Metabolism Dynamics, and Architecture Optimization in Three-Dimensional Tissue Constructs with Applications and Insights in Cerebral Organoids. Tissue Eng Part C Methods 22(3):221–249. https://doi.org/10.1089/ten.tec.2015.0375
Moorthy AS, Brooks SP, Kalmokoff M, Eberl HJ (2015) A spatially continuous model of carbohydrate digestion and transport processes in the colon. PLoS ONE 10(12):e0145309
Morris MD (1991) Factorial Sampling Plans for Preliminary Computational Experiments. Technometrics 33(2):161–174. https://doi.org/10.1080/00401706.1991.10484804
Murray PJ, Edwards CM, Tindall MJ, Maini PK (2009) From a discrete to a continuum model of cell dynamics in one dimension. Phys Rev E 80(3):031912
Murray PJ, Walter A, Fletcher AG, Edwards CM, Tindall MJ, Maini PK (2011) Comparing a discrete and continuum model of the intestinal crypt. Phys Biol 8(2):026011. https://doi.org/10.1088/1478-3975/8/2/026011
Muñoz-Tamayo R, Laroche B, Éric Walter, Doré J, Leclerc M (2010) Mathematical modelling of carbohydrate degradation by human colonic microbiota. Journal of Theoretical Biology 266(1): 189–201. https://doi.org/10.1016/j.jtbi.2010.05.040
Naito T, Mulet C, De Castro C, Molinaro A, Saffarian A, Nigro G, Bérard M, Clerc M, Pedersen AB, Sansonetti PJ, Pédron T (2017) Lipopolysaccharide from crypt-specific core microbiota modulates the colonic epithelial proliferation-to-differentiation balance. mBio 8(5):1–16. https://doi.org/10.1128/mBio.01680-17
Neumann PA, Koch S, Hilgarth RS, Perez-Chanona E, Denning P, Jobin C, Nusrat A (2014) Gut commensal bacteria and regional Wnt gene expression in the proximal versus distal colon. Am J Pathol 184(3):592–599. https://doi.org/10.1016/j.ajpath.2013.11.029
Osbelt L, Thiemann S, Smit N, Lesker TR, Schröter M, Gálvez EJC, Schmidt-Hohagen K, Pils MC, Mühlen S, Dersch P, Hiller K, Schlüter D, Neumann-Schaal M, Strowig T (2020) Variations in microbiota composition of laboratory mice influence Citrobacter rodentium infection via variable short-chain fatty acid production. PLoS Pathog 16(3):e1008448. https://doi.org/10.1371/journal.ppat.1008448
Pearce SC, Weber GJ, van Sambeek DM, Soares JW, Racicot K, Breault DT (2020) Intestinal enteroids recapitulate the effects of short-chain fatty acids on the intestinal epithelium. PLoS ONE 15(4):e0230231. https://doi.org/10.1371/journal.pone.0230231
Pianosi F, Sarrazin F, Wagener T (2015) A Matlab toolbox for Global Sensitivity Analysis. Environmental Modelling & Software 70:80–85. https://doi.org/10.1016/j.envsoft.2015.04.009
Purcell EM (1977) Life at low Reynolds number. Am J Phys 45(1):3–11. https://doi.org/10.1119/1.10903
Rivera-Chávez F, Lopez CA, Bäumler AJ (2017) Oxygen as a driver of gut dysbiosis. Free Radical Biol Med 105:93–101. https://doi.org/10.1016/j.freeradbiomed.2016.09.022
Rogan MR, Patterson LL, Wang JY, McBride JW (2019) Bacterial Manipulation of Wnt Signaling: A Host-Pathogen Tug-of-Wnt. Front Immunol 10:2390. https://doi.org/10.3389/fimmu.2019.02390
Saltelli A, Aleksankina K, Becker W, Fennell P, Ferretti F, Holst N, Li S, Wu Q (2019) Why so many published sensitivity analyses are false: A systematic review of sensitivity analysis practices. Environmental Modelling & Software 114:29–39. https://doi.org/10.1016/j.envsoft.2019.01.012
Sancho R, Cremona CA, Behrens A (2015) Stem cell and progenitor fate in the mammalian intestine: Notch and lateral inhibition in homeostasis and disease. EMBO reports 16(5):571–81. https://doi.org/10.15252/embr.201540188
Sarrazin F, Pianosi F, Wagener T (2016) Global Sensitivity Analysis of environmental models: Convergence and validation. Environmental Modelling & Software 79:135–152. https://doi.org/10.1016/j.envsoft.2016.02.005
Sasaki N, Sachs N, Wiebrands K, Elenbroek SIJ, Fumagalli A, Lyubimova A, Begthel H, van den Born M, van Es JH, Karthaus WR, Li VSW, López-Iglesias C, Peters PJ, van Rheenen J, van Oudenaarden A, Clevers H (2016) Reg4+ deep crypt secretory cells function as epithelial niche for Lgr5+ stem cells in colon. Proc Natl Acad Sci 113(37):E5399–E5407. https://doi.org/10.1073/PNAS.1607327113
Serra D, Mayr U, Boni A, Lukonin I, Rempfler M, Challet Meylan L, Stadler MB, Strnad P, Papasaikas P, Vischi D, Waldt A, Roma G, Liberali P (2019) Self-organization and symmetry breaking in intestinal organoid development. Nature 569(7754):66–72. https://doi.org/10.1038/s41586-019-1146-y
Shyer AE, Tallinen T, Nerurkar NL, Wei Z, Gil ES, Kaplan DL, Tabin CJ, Mahadevan L (2013) Villification: How the gut gets its villi. Science 342(6155):212–218. https://doi.org/10.1126/science.1238842
Snippert HJ (2016) Colonic Crypts: Safe Haven from Microbial Products. Cell 165(7):1564–1566. https://doi.org/10.1016/j.cell.2016.06.003
Snippert HJ, van der Flier LG, Sato T, van Es JH, van den Born M, Kroon-Veenboer C, Barker N, Klein AM, van Rheenen J, Simons BD, Clevers H (2010) Intestinal crypt homeostasis results from neutral competition between symmetrically dividing Lgr5 stem cells. Cell 143(1):134–144. https://doi.org/10.1016/j.cell.2010.09.016
Sunter JP, Appleton DR, de Rodriguez MS, Wright NA, Watson AJ (1979) A comparison of cell proliferation at different sites within the large bowel of the mouse. J Anat 129(4):833–842
Tetteh PW, Basak O, Farin HF, Wiebrands K, Kretzschmar K, Begthel H, Van Den Born M, Korving J, De Sauvage F, Van Es JH, Van Oudenaarden A, Clevers H (2016) Replacement of Lost Lgr5-Positive Stem Cells through Plasticity of Their Enterocyte-Lineage Daughters. Cell Stem Cell 18(2):203–213. https://doi.org/10.1016/j.stem.2016.01.001
Thalheim T, Quaas M, Herberg M, Braumann UD, Kerner C, Loeffler M, Aust G, Galle J (2017) Linking stem cell function and growth pattern of intestinal organoids. Developmental Biology 433(2):254–261. https://doi.org/10.1016/j.ydbio.2017.10.013
Tomas J, Reygner J, Mayeur C, Ducroc R, Bouet S, Bridonneau C, Cavin JB, Thomas M, Langella P, Cherbuy C (2015) Early colonizing Escherichia coli elicits remodeling of rat colonic epithelium shifting toward a new homeostatic state. ISME J 9(1):46–58. https://doi.org/10.1038/ismej.2014.111
Tóth B, Ben-Moshe S, Gavish A, Barkai N, Itzkovitz S (2017) Early commitment and robust differentiation in colonic crypts. Molecular Systems Biology 13(1):902. https://doi.org/10.15252/msb.20167283
Twarogowska M (2011) Numerical approximation and analysis of mathematical models arising in cells movement. Ph.D. thesis, Università degli studi de l’Aquila
Uchiyama K, Sakiyama T, Hasebe T, Musch MW, Miyoshi H, Nakagawa Y, He TC, Lichtenstein L, Naito Y, Itoh Y, Yoshikawa T, Jabri B, Stappenbeck T, Chang EB (2016) Butyrate and bioactive proteolytic form of Wnt-5a regulate colonic epithelial proliferation and spatial development. Sci Rep 6(1):32094. https://doi.org/10.1038/srep32094
Wang Y (2018) Bioengineered Systems and Designer Matrices That Recapitulate the Intestinal Stem Cell Niche. Cell Mol Gastroenterol Hepatol 5(3):15
Zhou Z, Cao J, Liu X, Li M (2021) Evidence for the butyrate metabolism as key pathway improving ulcerative colitis in both pediatric and adult patients. Bioengineered 12(1):8309–8324. https://doi.org/10.1080/21655979.2021.1985815
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Marie Haghebaert’s contribution was permitted in part by funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement ERC-2017-AdG No. 788191 - Homo.symbiosus).
Appendices
Jump rates definitions
We detail the jump rates used in the model as summed up in Table 1.
1.1 Jump rates for Sects. 4.3 and 4.4
For stem cells:
For progenitor cells:
For enterocytes:
For goblet cells:
1.2 Jump rate definition of the complete model
The jump rate dependency to butyrate, space and cell density is the following.
For stem cells:
For progenitor cells:
For enterocytes:
For goblet cells:
Existence and regularity of deterministic trajectories
The global existence and \(C^1\) regularity of the ODE part describing cells trajectories comes from the Cauchy-Lipschitz theorem. We briefly sketch existence and regularity results for the PDE part describing concentrations. For the butyrate concentration, the operator \(A_1=\sigma \partial _{zz}\) with domain \({\mathscr {D}}(A_1)=\{c \in H^2(\left]0,z_{\text {max}}\right[_a),\, \partial _z c(-a/2)=0,\,c(z_{\text {max}}+a/2)=0\}\) is a Riesz spectral operator on \(\left]0,z_{\text {max}}\right[_a\), that generates a strongly continuous, analytic semigroup \(S_1\) on \(L^2(\left]0,z_{\text {max}}\right[_a)\) (see e.g. Curtain and Zwart 1995). For the oxygen concentration, the same holds for \(A_2=\sigma \partial _{zz}\) with domain \({\mathscr {D}}(A_2)=\{c \in H^2(\left]0,z_{\text {max}}\right[_a),\,c(-a/2)=0,\, \partial _z c(z_{\text {max}}+a/2)=0 \}\), with associated semigroup \(S_2\). It fits the framework proposed in Lasiecka (1980), so that setting
where \(H^{\alpha }(\left]0,z_{\text {max}}\right[_a)\) are the standard Sobolev spaces, and denoting \(S=(S_1,S_2)\) the semigroup induced on the product space \(L^2(\left]0,z_{\text {max}}\right[_a)^2\), the following property holds:
Proposition 1
Let \(c_0=(c_{01},c_{02}) \in H^1(\left]0,z_{\text {max}}\right[_a)^2\), \((c_{\text {lum}},c_{\text {bot}}) \in \left( H^1[0,T]\right) ^2\) and \(u=(u_1,u_2) \in \left( L^2([0,T],L^2(\left]0,z_{\text {max}}\right[_a)\right) ^2\), such that \(c_{01}(z_{\text {max}}+a/2)=c_{\text {lum}}(0)\) and \(c_{02}(-a/2)=c_{\text {bot}}(0)\), then the evolution problem
admits a unique weak solution in \(\left( W^{2,1}_{L^2(\left]0,z_{\text {max}}\right[_a)} \cap C([0,T],H^1(\left]0,z_{\text {max}}\right[_a))\right) ^2\) given by
Now the semilinear PDEs (6) can be viewed as (53) with u replaced by a nonlinear perturbation taking the form of an operator defined on \(L^2(\left]0,z_{\text {max}}\right[_a)^2 \times [0,T]\) by
This operator is locally Lipschitz in c on \(L^2(\left]0,z_{\text {max}}\right[_a)^2\), uniformly in t on [0, T], when \(\nu _t\) is a fixed, measurable function from [0, T] to \({\mathscr {M}}_{F^+}({\mathscr {X}})\) endowed with the weak convergence topology and the associated Borel \(\sigma \)-algebra, such that \(\sup _{t \le T} \langle \nu _t,1\rangle < \infty \). Using proposition 1, together with a standard fixed point argument we obtain the local existence (in time) of the solutions of PDEs (6), that can be extended to a global existence on [0, T]. More precisely,
Proposition 2
Let \(\nu _t\) as above, and \(c_0=(c_{01},c_{02}) \in H^1(\left]0,z_{\text {max}}\right[_a)^2\), \((c_{\text {lum}},c_{\text {bot}}) \in \left( H^1[0,T]\right) ^2\) such that \(c_{01}(z_{\text {max}}+a/2)=c_{\text {lum}}(0)\) and \(c_{02}(-a/2)=c_{\text {bot}}(0)\). The non-linear evolution problem (6) admits a unique solution c in \(\Big (W^{2,1}_{L^2(\left]0,z_{\text {max}}\right[_a)} \cap C([0,T],H^1(\left]0,z_{\text {max}}\right[_a)) \Big )^2 \).
PDMP stability
We aim at showing that an almost surely finite number of jump occurs during any finite time interval. We first introduce the independent Poisson probability measures \(N_k(d\theta ,i,dt)\) associated to each jump type \(k \in {\mathscr {E}}\), defined on \([0,1]\times {\mathbb {N}}^* \times {\mathbb {R}}_+\) with intensity
The process introduced in Sect. 2 is solution of the following SDE, which is called the pathwise representation of the process
Fig.18 sketches the rationale behind this formula, built upon the remark that if \(T_n\) and \(T_{n-1}\) are successive jump times, then for \(t>T_n\),
Note that, when possible without ambiguity, we simplify the notation \(\mathbbm {1}_{\{ \theta \le q_k(x^i_{s^-},\nu _{s-}*D_k (x^i_{s^-}),c_s*\psi _{a}(z^i_{s^-})) \}}(\theta )\) by \(\mathbbm {1}_{\{ \theta \le q_k(\cdot ) \}}(\theta )\).
For all \(n \in {\mathbb {N}}\), we define the stopping time \(\tau _n = \inf \{ t \ge 0, \langle \nu _t ,1 \rangle = n \} \). Let
The stopped process at n cells is \((c_{T_n(t)},\nu _{T_n(t)})_{t \ge 0}\). \({\mathscr {C}}^{p,q,\bullet }({\mathbb {R}}^+ \times {\mathscr {X}},{\mathbb {R}})\) denotes the space of functions \(f(t,z,\ell )\) that are p times derivable with respect to t and q times derivable with respect to z for all \(\ell \in {\mathscr {T}}\).
Lemma C1
Let \(f \in {\mathscr {C}}^{1,1,\bullet }({\mathbb {R}}^+ \times {\mathscr {X}},{\mathbb {R}})\), \(g \in H^1(\left]0,z_{\text {max}}\right[_a)^{2}\) , \(\psi \in {\mathscr {C}}^1({\mathbb {R}}^2)\), \(t \in {\mathbb {R}}\). We denote by \(\langle c,g \rangle _{L^2}\) the scalar product in \(L^2\). Then, for \(t>0\) and \(n \ge \langle \nu _0,1 \rangle \),
Proof
Using the pathwise representation of the PDMP (55), we get
where the stochastic integral is correctly defined since the population size is majored by n, and f is continuous on the compact \({\mathscr {X}}\).
Then, applying formula (12), the first term of the RHS can be written
and the second one
Due to the smoothness of f, \(\partial _t f\) and \(\nabla _z f\), Fubini’s theorem and formula (55) are applied to the last integral term to get:
We then have:
We also have
Then, multidimensional Itô’s formula with jumps is applied to get the desired expression. \(\square \)
We now have a tool to study the stability of the PDMP. We must show that the population does not explode in finite time.
Lemma C2
For any \(t \ge 0\), if an integer \(p \ge 1\) such that \({\mathbb {E}} \left[ \langle \nu _0,1 \rangle ^p \right] < \infty \) exists then
where \(C(p,t) < \infty \) is a constant that only depends on p and t.
Proof
For all \(n \in {\mathbb {N}}\), let \(T_n(t)\) be as defined in (56). As the deterministic evolution does not impact the population size and the stochastic events modify it with at most one individual by jump, formula (57) applied to \(\langle \nu _{T_n(t)},1 \rangle ^p\) gives
The last integral grows with time since the integrand is non-negative, so that the supremum can be applied in the left hand side. Furthermore, for any non-negative x, \((1 + x)^p -x^p \le C(p) (1 + x^{p-1})\). Hence,
Switching to the expectations:
Recalling that \(n + n^p \le 2n^p\) for all \(p\ge 1\), Fubini’s theorem leads to
and Gronwall lemma provides an estimate that does not depend on n:
Hence \(\tau _n \rightarrow \infty \) a.s. Indeed, suppose that there exists \(T>0\) such that, for all \(n\in {\mathbb {N}}\), \({\mathbb {P}}\left( \tau _n \le T\right)> \varepsilon > 0\). We remark that
Hence, by Markov inequality, for all \(n\in {\mathbb {N}}\)
which is in contradiction with (62).
Hence, \(\underset{n \rightarrow \infty }{\lim }\ \underset{s \le T_n(t)}{\sup }\langle \nu _s,1 \rangle ^p = \underset{s \le t }{\sup }\langle \nu _s,1 \rangle ^p\) a.s. Fatou’s lemma finally gives
The stability of the PDMP is a direct consequence of this lemma.
Corollary C3
If there exists an integer \(p \ge 1\) such that \({\mathbb {E}} \left[ \langle \nu _0,1 \rangle ^p \right] < \infty \), then:
-
1.
The PDMP is stable.
-
2.
\(T_n(t)\) can be replaced by t in formula (57).
Proof
(1): Let \((J_n)_n\) be the sequence of jump times. We aim at showing that \(J_n \underset{n \rightarrow \infty }{\rightarrow } \infty \) a.s. Assume that there exists \(T < +\infty \) and a set \(\varOmega _T\) of non-null probability such that, \(\forall \ \omega \in \varOmega _T\), \(\sup \limits _{n \in {\mathbb {N}}} J_n(\omega ) < T \). We recall that, as proven in the proof of Lemma C2, that \({\mathbb {E}} \left[ \sup \limits _{s \le t} \langle \nu _s,1 \rangle \right] < \infty \) implies that \(\tau _n \rightarrow \infty \) a.s. Hence, it exists \(N \in {\mathbb {N}}\) and a set \(\varOmega _{T,N}\subset \varOmega _T\) of non-null probability such that, \(\forall \ \omega \in \varOmega _{T,N}\), \( T < \tau _N(\omega )\). Let \(C(N)\) be an upper bound of the jump rate on \({\mathscr {M}}_{P^+}({\mathscr {X}})_N = \left\{ \nu \in {\mathscr {M}}_{P^+}({\mathscr {X}}), \langle \nu ,1 \rangle \le N \right\} \subset {\mathscr {M}}_{P^+}({\mathscr {X}})\). Such an upper bound exists since the individual jump rates are bounded by \(q_k^\infty \) by definition. Hence, for all \(\omega \in \varOmega _{T,N}\), the jump time sequence \((J_n(\omega ))_{n \in {\mathbb {N}}}\) can be built as an extraction of the sequence \((J'_n(\omega ))_{n \in {\mathbb {N}}}\) of the jump time of a Poisson process with jump rate \(C(N)\). But the sequence \((J'_n(\omega ))_{n \in {\mathbb {N}}}\) is a.s. divergent, which comes in contradiction with \(\sup \limits _{n \in {\mathbb {N}}} J_n(\omega ) < T \) for all \(\omega \in \varOmega _T\).
(2): As the stochastic integrals in the proof of Lemma C1 are correctly defined (see e.g. theorem A.3 Bansaye and Méléard 2015) due to the hypothesis on \(\nu _0\) and Lemma C2. \(\square \)
A case of bistability
Here we explain the unexpected simulations values we obtained for \(K_{\text {div}}[dens] = - 50 \%\) in Sect. 4.2. Starting from an initial number of 700 cells, we see in Fig. 19a that either the number of cells drops below 400 cells and activity in the crypt goes on, or the number of cells stabilises around 500 and the crypt freezes, meaning that no additional extrusion or division events are observed, so that the total cell population does not evolve any longer. We evaluated the fraction of crypt that freezes upon variations of \(K_{\text {div}}[dens]\), always starting with 700 cells (Fig. 19b). This fraction increases when \(K_{\text {div}}[dens]\) decreases. Indeed, for low values of \(K_{\text {div}}[dens]\), cell division is totally inhibited by contact inhibition and the division rate is 0. At first, cells are extruded at the top of the crypt as density of cells (starting from 700 cells) is way above what \(K_{\text {div}}[dens]\) prescribes. Once exceeding cells have been extruded, the population can reach a mechanical equilibrium where no extrusion occurs at the top of the crypt (as density equilibrium has been reached). Meanwhile, no division occurs in the lower part of the crypt as density is still too high there, and the division rate is therefore zero.
Parameters values for the different models
PDMP model parameters are given for the different sub-models of increasing complexity studied during the PDMP exploration. Namely, the parameters used for the model with the sole deterministic part, studied in Sect. 4.1, are given in Table 7. When the spatialized division and extrusion events are added in Sect. 4.2, parameters of Table 8 are used. In Sect. 4.3 where cell types are introduced, parameters of Table 9 are used. Adding \(\hbox {O}_2\) and butyrate in Sect. 4.4, we use parameters presented in Table 10. Finally, the complete model including the feedback of butyrate on stem cell division as studied in Sect. 4.5 is implemented with the parameters presented in Table 11.
Parameters used for the deterministic PDE approximation of the PDMP are given in Table 12.
1.1 Parameters of chemical kinetic
We could not find value for the molar concentration of oxygen. Therefore, we switched to normalized concentrations. We set \(\bar{c_o} = \frac{c_o}{C}\) with C a normalizing constant. The evolution equation of oxygen concentration 6 becomes
We suppose that experimental measures of the reaction speed in Clausen and Mortensen (1994) are done at a standard (normoxic) oxygen concentration \(c_o^*\). The half of the maximal reaction speed is reached experimentally at a butyrate concentration of 0.184 mole/L. Therefore,
We suppose as well that oxygen concentration at the bottom of the crypt is \(c_o^*\). The exact value of \(c_o^*\) is unknown to us, but we set \({\bar{c}}_o^* = 10\). Then, \(C = \frac{c_o^*}{{\bar{c}}_o^*} = \frac{c_o^*}{10}\). Therefore, we can find a value for the parameter \(K_{\beta }^5/C^4\) of (63):
As C is unknown, the quantity \(s_0/C\) in equation (63) is still unknown, but we decided to use \(C=1\) in all simulations.
Rights and permissions
About this article
Cite this article
Darrigade, L., Haghebaert, M., Cherbuy, C. et al. A PDMP model of the epithelial cell turn-over in the intestinal crypt including microbiota-derived regulations. J. Math. Biol. 84, 60 (2022). https://doi.org/10.1007/s00285-022-01766-8
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00285-022-01766-8
Keywords
- Piecewise deterministic Markov processes
- Porous media equation
- Crypt model
- Host-microbiota interactions