 Research note
 Open Access
 Published:
Mathematical analysis of the effect of portal vein cells on biliary epithelial cell differentiation through the DeltaNotch signaling pathway
BMC Research Notes volume 14, Article number: 243 (2021)
Abstract
Objective
The DeltaNotch signaling pathway induces finegrained patterns of differentiation from initially homogeneous progenitor cells in many biological contexts, including Drosophila bristle formation, where mathematical modeling reportedly suggests the importance of production rate of the components of this signaling pathway. In contrast, the epithelial differentiation of bile ducts in the developing liver is unique in that it occurs around the portal vein cells, which express extremely high amounts of Delta ligands and act as a disturbance for the amount of Delta ligands in the field by affecting the expression levels of downstream target genes in the cells nearby. In the present study, we mathematically examined the dynamics of the DeltaNotch signaling pathway components in disturbancedriven biliary differentiation, using the model for finegrained patterns of differentiation.
Results
A portal vein cell induced a high Notch signal in its neighboring cells, which corresponded to epithelial differentiation, depending on the production rates of Delta ligands and Notch receptors. In addition, this epithelial differentiation tended to occur in conditions where finegrained patterning was reported to be lacking. These results highlighted the potential importance of the stability towards homogeneity determined by the production rates in Delta ligands and Notch receptors, in a disturbancedependent epithelial differentiation.
Introduction
Coordinated positioning of cells is one of the fundamental features of organs. The liver consists of bileproducing hepatocytes and biletransporting cholangiocytes (epithelial cells of the intrahepatic bile duct; IHBD). The IHBD runs parallel to the portal vein [1] and its development starts from the differentiation of cholangiocytes between embryonic days 13.5 (E13.5) and E18.5 in mice [2]. During this period, portal vein smooth muscle cells induce cholangiocyte differentiation of progenitor cells (hepatoblasts) through the DeltaNotch signaling pathway [3].
The DeltaNotch signaling pathway consists of two transmembrane components: Delta ligands and Notch receptors. On the one hand, upon the interaction of Delta ligands and Notch receptors at the adjacent cell surfaces (transinteraction), the Notch intracellular domain (NICD) is cleaved and translocated to the nucleus, which induces transcriptional suppression of Delta production (lateral inhibition) [4]. In addition, NICD overexpression induces the expression of cholangiocyte markers in vitro [5]. On the other hand, Delta ligands and Notch receptors bind and inactivate each other on the same cell surface (cisinteraction) to bias the effect of lateral inhibition [6] (Fig. 1). Because these interactions were significant in multiple biological contexts [6,7,8], we considered that they are fundamental and at work also in the developing liver. A mathematical model which incorporates both trans and cisinteraction, called the Lateral Inhibition with Mutual Inactivation (LIMI) model, has been proposed for studying the dynamics of DeltaNotch signaling components [9]. This model yields a finegrained pattern of differentiation owing to lateral inhibition, which is facilitated by mutual inactivation. The production rates of Delta ligands and Notch receptors were deterministic for the patterning from an initially homogeneous state in this model.
Finegrained patterning of differentiation from uniform cells via the action of the DeltaNotch signaling pathway, as seen in Drosophila bristle formation, has been extensively documented. In contrast, the differentiation of cholangiocytes is unique in that it occurred around the portal smooth muscle cells with extremely high Jagged1 expression (one of the mammalian homologs of Delta ligands) [3, 5, 10]. Although an in vivo study of NICD overexpression or Notch receptor knockout showed a positive correlation between the amount of Notch signal and cholangiocyte differentiation [11], it is mathematically unclear what patterns would be formed through the action of trans and cisinteractions with such a disturbance. Here, the effect of a disturbance on cholangiocyte differentiation was examined using the LIMI model [9].
Main text
Methods
LIMI model
A 20 × 20 twodimensional field with square cells was created, mimicking a planar crosssection of the liver perpendicular to the intrahepatic part of the portal veins. For computational purposes, toroidal boundary conditions were imposed on the 20 × 20 field. Moreover, when applicable, portal vein (PV) cells were indicated with a cross. Each cell had three parameters: D(t) for Delta ligands, N(t) for Notch receptors, and R(t) for reporter target genes. The initial values for D(t) and N(t) were set to follow the normal distribution (\(mean=1\), \(SD=0.1\)) whereas that for R(t) was set to zero. The LIMI model was adopted from [9] with the following quantities:
\(\beta _{N} ,~\beta _{D} ,~\beta _{R}\): production rates of Notch, Delta, and reporter target genes, respectively.
\(k_{c}^{{  1}} ,~k_{t}^{{  1}}\): strength of the cis and transinteractions, respectively.
\(\gamma\), \(\gamma _{R}\): degradation rate of Notch and Delta or reporter target gene.
\(k_{{RS}} ,~n\): affinity and Hill coefficient, respectively, of reporter induction by Notch signaling.
\(\left\langle D_j \right\rangle_i\): average concentration of Delta in all cells, indexed by j, that are in the von Neumann neighborhood to the ith cell.
\(\left\langle N_j \right\rangle_i\): average concentration of Notch in all cells, indexed by j, that are in the von Neumann neighborhood to the ith cell.”
N(t) was assumed to be constantly produced and proportionately degraded at rates of \(\beta _{N}\) and \(\gamma\), respectively. In addition, N(t) was also assumed to be degraded in proportion to trans and cisinteractions. Therefore, a differential equation for N (t) is:
R(t) was assumed to be an increasing function of a transinteraction and proportionately degraded at the rate of \(\gamma _{R}\). Therefore, a differential equation for R(t) is:
Owing to lateral inhibition, D (t) should be a decreasing function of R(t). By modifying the production term in Eq. 1, a differential equation for D(t) is:
Because the knowledge on the production and degradation rates of Delta ligands and Notch receptors is limited, the arbitrary numbers were assumed following [9]: \(\beta _{R} = 1,000,000\), \(k_{t} = 1\), \(k_{c} = 0.1\), \(\gamma = 1\), \(\gamma _{R} = 1\) and \(k_{{RS}} = 300,000\). Sensitivity analysis is available in Additional file 1, Additional file 2. Varying numbers were used for production rates (\({\beta }_{N}\) and \({\beta }_{D}\)), the feedback strength (m) and Hill coefficient (n). For D(t) in PV cells, \(D\left(t\right)=1,000\), which would be considerably higher than D(t) in the other field cells, were used irrespective of Eq. 3 because they maintain a high Jagged1 mRNA expression level during E13.5 and E18.5 [5]. To secure the stability of the calculation, the left terms in Eqs. 1–3 were multiplied by \(dt=0.0001\) and used in a stepwise fashion in the iteration. D(t), N(t) and R(t) were set to zero if they became negative values at each iteration. The iteration was conducted until creating the equilibrium states where all the left terms in Eqs. 1–3 for every cell excluding the PV cells reached below 0.001. The values for D(t), N(t), and R(t) at the equilibrium are indicated in a gray scale except for the portal vein cells. At the equilibrium state, the average (Ave) and standard deviation (SD) was calculated for R(t) of all the cells except for the PV cell and its neighboring cells. Here we introduced a new parameter, diff, for R(t) of each neighboring cell at the equilibrium state:
A neighboring cell with \(diff > 2\) was defined as a cholangiocyte because cholangiocytes account for 3% to 5% of the liver cell population [12]. The diff values of the four neighboring cells were averaged prior to plotting the log2scaled color map for varying production rates of Delta ligands and Notch receptors. In this color map, the infinity values and negative values were indicated with crosses.
Results
A case with PV cellinduced cholangiocyte differentiation
We started from a simulation with \({\beta }_{N}=100\), \({\beta }_{D}=10\), \(m=1\) and \(n=3\) without PV cells. This condition resulted in almost homogeneous distributions in D(t), N(t) and R(t), although the distributions were finegrained patterns with very small differences among the cells (Fig. 2a).
To examine the effect of a single PV cell in cholangiocyte differentiation, a PV cell was put in the simulation field. Because PV cells express extremely high amounts of Delta ligands, \(D\left(t\right)=1,000>3\) (highest D(t) in Fig. 2a) was used for this single PV cell (Fig. 2b). Owing to large \(\left\langle D_j \right\rangle_i\) in Eq. 1, N(t) in the adjacent cells was small compared to other cells. Importantly, R(t) in the adjacent cells was exceptionally high (\(\mathrm{l}\mathrm{o}\mathrm{g}2(\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f})=19.22>1\)), suggesting cholangiocyte differentiation under \({\beta }_{N}=100\) and \({\beta }_{D}=10\).
A case without PV cellinduced cholangiocyte differentiation
Another simulation was then conducted with \({\beta }_{N}=10\), \({\beta }_{D}=100\), \(m=1\) and \(n=3\). The magnitude of each variable was considerably different from that in the simulation with \({\beta }_{N}=100\) and \({\beta }_{D}=10\) because this alteration markedly affected the system’s behavior, and the distribution pattern was heterogeneous (Fig. 2c). Although a PV cell increased or decreased the R(t) values in some cells near itself (the inlets), cholangiocyte differentiation was lacking (\(\mathrm{l}\mathrm{o}\mathrm{g}2\left(\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\right)=0.02<1\)) (Fig. 2d).
Effect of the production rates and feedback strength
Because cholangiocyte differentiation was dependent on the production rates of Delta ligands and Notch receptors (Fig. 2b, and d), we hypothesized that these parameters were deterministic in the LIMI model even under the presence of a disturbance. Therefore, we examined the diff in the neighboring cell for varying Delta or Notch production (\({\beta }_{D}\) or \({\beta }_{N}\)). In addition, because the feedback strength (m) and Hill coefficient (n) might also affect the result, we conducted this analysis in four cases with different values for m and n. Figure 3 shows the averaged diff values in a log2scaled color map for the indicated values for \({\beta }_{D}\), \({\beta }_{N}\), m and n. The averaged diff values exceeding two (exceeding one in log2scale), indicating cholangiocyte differentiation, were shown in red whereas those less than two (less than one in log2scale), indicating no cholangiocyte differentiation, were shown in blue. According to Eq. 4, infinity values resulted from small SD (homogeneity), whereas negative values resulted from large SD (heterogeneity) or small differences between R(t) values in the neighboring cells and the others. Although cholangiocyte differentiation was observed in a wide range of Delta/Notch production (\({\beta }_{D}\) or \({\beta }_{N}\)), it was lacking in some cases with high Delta production (\({\beta }_{D}\)). The cholangiocyte differentiation tended to occur in conditions where finegrained differentiation was lacking and vice versa [9]. The trend for the cholangiocyte differentiation was consistent for varying feedback strength (m) and Hill coefficient (n) values. Therefore, we concluded that the production rates of Delta ligands and Notch receptors in the LIMI model were also important in a disturbancedependent context, although the occurrence of the patterning was dependent on the presence of disturbance.
Discussion
The differentiation of cholangiocytes around portal veins is unique because this is driven by Deltarich PV cells. Here, we observed spatially restricted cholangiocyte differentiation around a PV cell using LIMI model, which is expected to yield finegrained patterning of differentiation in the absence of PV cells. In addition, this result supported a potential importance in the disturbance and production rates of Delta ligands and Notch receptors in cholangiocyte differentiation (disturbancedriven context). Although the production rates of Delta ligands and Notch receptors were important in both cholangiocyte differentiation and Drosophila bristle formation [9], the occurrence of the patterning was oppositely different between the two biological contexts. However, we speculate that these two results are consistent with each other. The parametric conditions where finegrained differentiation was lacking, stabilized the homogeneity of the R(t) distribution in the field. Therefore, the effect of a disturbance was buffered and induced cholangiocyte differentiation only in the neighboring cells. In contrast, the parametric conditions where finegrained differentiation occurred had a limited ability to stabilize the R(t) distribution in homogeneity. Therefore, the effect of a disturbance became widespread (the inlets in Figs. 2c and d) and no cholangiocyte differentiation occurred. Although it has been extensively documented how DeltaNotch signaling pathway could induce differentiation in some cell populations (divergence) [7], this study raises the potential importance of the stabilization effect of this signaling pathway towards homogeneity (convergence) in the disturbancedriven cholangiocyte differentiation.
Although the present study showed a case without cholangiocyte differentiation, this does not exclude a possibility of such conditions in the physical developing liver because some factors, other than the DeltaNotch signaling pathway, may modify the signaling. For example, the epidermal growth factor receptor (EGFR) signaling spatially narrows the effects of Notch signaling in Drosophila male embryonic gonads by antagonizing the DeltaNotch signaling pathway [13]. Although the contribution of EGF signaling in the liver remains unclear, the antagonists, if any, may adjust the signaling intensity to achieve cholangiocyte differentiation around PV cells.
Although such chemical modulators may exist, the notion that the PV cell is important for cholangiocyte differentiation is also supported clinically. For example, mutations in genes of this signaling pathway, such as Jagged1 [14, 15] and Notch2 [16] have been identified as the cause of defects in IHBD development in patients with Alagille syndrome. In addition, the number of intrahepatic bile ducts decreases in patients with loss of the intrahepatic part of the portal vein, owing to the shunt between the portal vein and inferior vena cava (Abernethy malformation) [17]. In the same literature, several cases of concomitant biliary atresia and Abernethy malformation have been reported, suggesting a relationship between PV cells and cholangiocyte differentiation.
Conclusion
The conversing nature of the DeltaNotch signaling pathway may support spatially restricted differentiation in a disturbancedriven context.
Limitations
Firstly, cell arrangement was much simpler than that in the physical liver and the spatial effect of hematopoietic cells was not considered. Secondly, cell proliferation and death were not considered. However, these aspects may be acceptable for the simulation of cholangiocyte differentiation, which occurs in direct contact with portal vein cells. Thirdly, intracellular mechanisms, such as Cdk8mediated NICD degradation [18], have not been considered because this aspect is beyond the scope.
Availability of data and material
Simulations were conducted using the C++ language in Visual Studio Community 2019 on a Windows 10 × 64based system. The datasets generated and/or analyzed during the current study are available in the GitHub repository, https://github.com/MasaharuYoshihara/cholangiocyte. The C++ codes and the results in this repository are distributed under a threeclause Berkeley Software Distribution license. The authors have not examined the codes on a Linuxbased system.
Abbreviations
 PV:

Portal vein
 NICD:

Notch intracellular domain
 LIMI:

Lateral inhibition with mutual inactivation
 SD:

Standard deviation
 IHBD:

Intrahepatic bile duct
References
 1.
Shiojiri N, Kametani H, Ota N, Akai Y, Fukuchi T, Abo T, et al. Phylogenetic analyses of the hepatic architecture in vertebrates. J Anat. 2018;232:200–13.
 2.
Gordillo M, Evans T, GouonEvans V. Orchestrating liver development Dev. 2015;142:2094–108. https://doi.org/10.1242/dev.114215.
 3.
Hofmann JJ, Zovein AC, Koh H, Radtke F, Weinmaster G, IruelaArispe ML. Jagged1 in the portal vein mesenchyme regulates intrahepatic bile duct development: Insights into Alagille syndrome. Development. 2010;137:4061–72.
 4.
Axelrod JD. Delivering the lateral inhibition punchline: It’s all about the timing. Sci Signal. 2010;3:1–4.
 5.
Tanimizu N, Miyajima A. Notch signaling controls hepatoblast differentiation by altering the expression of liverenriched transcription factors. J Cell Sci. 2004;117:3165–74.
 6.
Sprinzak D, Lakhanpal A, Lebon L, Santat LA, Fontes ME, Anderson GA, et al. Cisinteractions between Notch and Delta generate mutually exclusive signalling states. Nature. 2010;465:86–90.
 7.
Collier JR, Monk NAM, Maini PK, Lewis JH. Pattern formation by lateral inhibition with feedback: A mathematical model of deltanotch intercellular signalling. J Theor Biol. 1996;183:429–46.
 8.
Miller AC, Lyons EL, Herman TG. cisInhibition of Notch by Endogenous Delta Biases the Outcome of Lateral Inhibition. Curr Biol. 2009;19:1378–83. https://doi.org/10.1016/j.cub.2009.06.042.
 9.
Sprinzak D, Lakhanpal A, LeBon L, GarciaOjalvo J, Elowitz MB. Mutual inactivation of Notch receptors and ligands facilitates developmental patterning. PLoS Comput Biol. 2011;7(6):e1002069.
 10.
Kopinke D, Brailsford M, Shea JE, Leavitt R, Scaife CL, Murtaugh LC. Lineage tracing reveals the dynamic contribution of Hes1+ cells to the developing and adult pancreas. Development. 2011;138:431–41.
 11.
Sparks EE, Huppert KA, Brown MA, Washington MK, Huppert SS. Notch signaling regulates formation of the threedimensional architecture of intrahepatic bile ducts in mice. Hepatology. 2010;51:1391–400. https://doi.org/10.1002/hep.23431.
 12.
Masyuk AI, LaRusso NF. Aquaporins in the hepatobiliary system. Hepatology. 2006;43(2 SUPPL. 1):75–81.
 13.
Kitadate Y, Kobayashi S. Notch and Egfr signaling act antagonistically to regulate germline stem cell niche formation in Drosophila male embryonic gonads. Proc Natl Acad Sci U S A. 2010;107:14241–6.
 14.
Li L, Krantz ID, Deng Y, Genin A, Banta AB, Collins CC, et al. Alagille Syndrome Is Caused by Mutations in Human Jagged1, Which Encodes a Ligand for Notch1. Nat Genet. 1997;16:243–51.
 15.
Oda T, Elkahloun AG, Pike BL, Okajima K, Krantz ID, Genin A, et al. Mutations in the Human Jagged1 Gene Are Responsible for Alagille Syndrome. Nat Genet. 1997;16:235–42.
 16.
McDaniell R, Warthen DM, SanchezLara PA, Pai A, Krantz ID, Piccoli DA, et al. NOTCH2 mutations cause Alagille syndrome, a heterogeneous disorder of the notch signaling pathway. Am J Hum Genet. 2006;79:169–73.
 17.
Emre S, Arnon R, Cohen E, Morotti RA, Vaysman D, Shneider BL. Resolution of hepatopulmonary syndrome after auxiliary partial orthotopic liver transplantation in abernethy malformation. A case report Liver Transplant. 2007;13:1662–8. https://doi.org/10.1002/lt.21349.
 18.
Kuang Y, Golan O, Preusse K, Cain B, Christensen CJ, Salomone J, et al. Enhancer architecture sensitizes cell specific responses to Notch gene dose via a bind and discard mechanism. Elife. 2020;9:1–28.
Acknowledgements
The authors thank Prof. Satoru Kobayashi (TARA Center, University of Tsukuba) for his valuable comments on antagonistic agents and Dr. Hideyuki Câteau (University of Tsukuba) for his valuable comments on differential equations, as well as all the members of the Department of Anatomy and Embryology, Faculty of Medicine, University of Tsukuba, for their helpful discussions. We would like to thank Editage (www.editage.com) for English language editing. MY is supported by Japan Society for the Promotion of Science through Research Fellowships for Young Scientists (DC1).
Funding
This work was supported by the Japan Society for the Promotion of Science through GrantsinAid for Scientific Research (Grant Number 19H00966), KAKENHI GrantinAid for EarlyCareer Scientists (Grant Number 20K15775), and Ph.D. Program in Humanics (Doctoral Program for Worldleading Innovative and Smart Education). These funding sources had no involvement in study design, data handling, writing the manuscript, or in the decision to submit the article for publication.
Author information
Affiliations
Contributions
MY conceptualized and designed the study, including its methodology, and played a central role in developing the C++ codes, conducting the simulation, analyzing and visualizing the results, and writing the manuscript. TN helped interpretation and curation of the results and writing the manuscript. MKY helped visualization of the data. AK helped computer settings and acquired the funding. TN was a major contributor in developing the C++ codes, and helped interpretation and visualization of the results and writing the manuscript. HA helped the developing the C++ codes and interpretation of the results. ST contributed to the conception and design of the work in addition to the supervision of the study and the funding acquisition. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary Information
Additional file 1
. Supplemental methods and results.
Additional file 2
: Figure S1. The effect of the changes in each parameter while βN and βD were fixed at 100 and 10, respectively. Note that log2(diff) values exceeded one in these conditions. Figure S2. The effect of the changes in each parameter while βN and βD were fixed at 10 and 100, respectively. Note that log2(diff) values were stable at low levels except for low βR values and high kRS values.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated in a credit line to the data.
About this article
Cite this article
Yoshihara, M., Nishino, T., Yadav, M.K. et al. Mathematical analysis of the effect of portal vein cells on biliary epithelial cell differentiation through the DeltaNotch signaling pathway. BMC Res Notes 14, 243 (2021). https://doi.org/10.1186/s1310402105656y
Received:
Accepted:
Published:
Keywords
 Disturbance
 Lateral inhibition with mutual inactivation model
 Production rate
 Convergence
 Divergence
 Homogeneity
 Heterogeneity
 Stability
 Cholangiocyte