Pattern Formation in the Longevity-Related Expression of Heat Shock Protein-16.2 in Caenorhabditis elegans

Abstract

Aging in Caenorhabditis elegans is controlled, in part, by the insulin-like signaling and heat shock response pathways. Following thermal stress, expression levels of small heat shock protein-16.2 show a spatial patterning across the 20 intestinal cells that reside along the length of the worm. Here, we present a hypothesized mechanism that could lead to this patterned response and develop a mathematical model of this system to test our hypothesis. We propose that the patterned expression of heat shock protein is caused by a diffusion-driven instability within the pseudocoelom, or fluid-filled cavity, that borders the intestinal cells in C. elegans. This instability is due to the interactions between two classes of insulin-like peptides that serve antagonistic roles. We examine output from the developed model and compare it to experimental data on heat shock protein expression. Given biologically bounded parameters, the model presented is capable of producing patterns similar to what is observed experimentally and provides a first step in mathematically modeling aging-related mechanisms in C. elegans.

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Notes

  1. 1.

    ReactionDiffusionCElegans repository from MathBioCU group on GitHub (https://github.com/MathBioCU).

  2. 2.

    Values are given as average ± standard error.

  3. 3.

    Due to errors in numerical approximation, the difference between each simulation and the final stationary mode plateaus between 0.006 and 0.007.

References

  1. Altun Z, Hall D (2009) Alimentary system, intestine. In: WormAtlas. https://doi.org/10.3908/wormatlas.1.4

  2. Bai JP, Chang LL (1995) Transepithelial transport of insulin: I. Insulin degradation by insulin-degrading enzyme in small intestinal epithelium. Pharm Res 12(8):1171–1175. https://doi.org/10.1023/A:1016263926946

    Article  Google Scholar 

  3. Banse SA, Hunter CP (2012) Vampiric isolation of extracellular fluid from Caenorhabditis elegans. J Vis Exp. https://doi.org/10.3791/3647

  4. Belle A, Tanay A, Bitincka L, Shamir R, O’Shea EK (2006) Quantification of protein half-lives in the budding yeast proteome. Proc Natl Acad Sci USA 103(35):13004–13009. https://doi.org/10.1073/pnas.0605420103

    Article  Google Scholar 

  5. Cambridge SB, Gnad F, Nguyen C, Bermejo JL, Krüger M, Mann M (2011) Systems-wide proteomic analysis in mammalian cells reveals conserved, functional protein turnover. J Proteome Res 10(12):5275–5284. https://doi.org/10.1021/pr101183k

    Article  Google Scholar 

  6. Duckworth WC, Bennett RG, Hamel FG (1998) Insulin degradation: progress and potential. Endocr Rev 19(5):608–24. https://doi.org/10.1210/edrv.19.5.0349

    Article  Google Scholar 

  7. Ewbank J (2006) Signaling in the immune response. WormBook. https://doi.org/10.1895/wormbook.1.83.1

  8. Fares H, Grant B (2002) Deciphering endocytosis in Caenorhabditis elegans. Traffic (Copenhagen, Denmark) 3(1):11–19. https://doi.org/10.1034/j.1600-0854.2002.30103.x

    Article  Google Scholar 

  9. Guex N, Peitsch MC (1997) SWISS-MODEL and the Swiss-Pdb Viewer: an environment for comparative protein modeling. Electrophoresis 18(15):2714–2723. https://doi.org/10.1002/elps.1150181505

    Article  Google Scholar 

  10. Hartwig K, Heidler T, Moch J, Daniel H, Wenzel U (2009) Feeding a ROS-generator to Caenorhabditis elegans leads to increased expression of small heat shock protein HSP-16.2 and hormesis. Genes Nutr 4(1):59–67. https://doi.org/10.1007/s12263-009-0113-x

    Article  Google Scholar 

  11. Haslbeck M, Franzmann T, Weinfurtner D, Buchner J (2005) Some like it hot: the structure and function of small heat-shock proteins. Nat Struct Mol Biol 12(10):842–846. https://doi.org/10.1038/nsmb993

    Article  Google Scholar 

  12. Hasselman B (2016) nleqslv: solve systems of nonlinear equations. https://cran.r-project.org/package=nleqslv

  13. Hennig C (2015) fpc: flexible procedures for clustering. https://cran.r-project.org/package=fpc

  14. Hirose T, Nakano Y, Nagamatsu Y, Misumi T, Ohta H, Ohshima Y (2003) Cyclic GMP-dependent protein kinase EGL-4 controls body size and lifespan in C. elegans. Development (Cambridge, England) 130(6):1089–1099. https://doi.org/10.1242/dev.00330

    Article  Google Scholar 

  15. Hsu AL, Murphy CT, Kenyon C (2003) Regulation of aging and age-related disease by DAF-16 and heat-shock factor. Science (New York, NY) 300(5622):1142–1145. https://doi.org/10.1126/science.1083701

    Article  Google Scholar 

  16. Hua QX, Nakagawa SH, Wilken J, Ramos RR, Jia W, Bass J, Weiss MA (2003) A divergent INS protein in Caenorhabditis elegans structurally resembles human insulin and activates the human insulin receptor. Genes Dev 17(7):826–831. https://doi.org/10.1101/gad.1058003

    Article  Google Scholar 

  17. Kaletsky R, Lakhina V, Arey R, Williams A, Landis J, Ashraf J, Murphy CT (2016) The C. elegans adult neuronal IIS/FOXO transcriptome reveals adult phenotype regulators. Nature 529(7584):92–96. https://doi.org/10.1038/nature16483

    Article  Google Scholar 

  18. Kao G, Nordenson C, Still M, Rönnlund A, Tuck S, Naredi P (2007) ASNA-1 positively regulates insulin secretion in C. elegans and mammalian cells. Cell 128(3):577–587. https://doi.org/10.1016/j.cell.2006.12.031

    Article  Google Scholar 

  19. Kimura KD, Tissenbaum HA, Liu Y, Ruvkun G (1997) Daf-2, an insulin receptor-like gene that regulates longevity and diapause in Caenorhabditis elegans. Science 277(5328):942–946. https://doi.org/10.1126/science.277.5328.942

    Article  Google Scholar 

  20. Li GW, Burkhardt D, Gross C, Weissman JS (2014) Quantifying absolute protein synthesis rates reveals principles underlying allocation of cellular resources. Cell 157(3):624–635. https://doi.org/10.1016/j.cell.2014.02.033

    Article  Google Scholar 

  21. Lin K, Hsin H, Libina N, Kenyon C (2001) Regulation of the Caenorhabditis elegans longevity protein DAF-16 by insulin/IGF-1 and germline signaling. Nat Genet 28(2):139–145. https://doi.org/10.1038/88850

    Article  Google Scholar 

  22. Lund J, Tedesco P, Duke K, Wang J, Kim SK, Johnson TE (2002) Transcriptional profile of aging in C. elegans. Curr Biol 12(18):1566–1573. https://doi.org/10.1016/S0960-9822(02)01146-6

    Article  Google Scholar 

  23. Mendenhall AR, Tedesco PM, Sands B, Johnson TE, Brent R (2015) Single cell quantification of reporter gene expression in live adult Caenorhabditis elegans reveals reproducible cell-specific expression patterns and underlying biological variation. PLoS One 10(5):e0124289. https://doi.org/10.1371/journal.pone.0124289

    Article  Google Scholar 

  24. Mendenhall A, Crane MM, Tedesco PM, Johnson TE, Brent R (2017) Caenorhabditis elegans genes affecting interindividual variation in life-span biomarker gene expression. J Gerontol Ser A Biol Sci Med Sci 72(10):1305–1310. https://doi.org/10.1093/gerona/glw349

    Article  Google Scholar 

  25. Milo R, Phillips R (2015) Cell biology by the numbers. Garland Science. http://book.bionumbers.org/

  26. Murphy CT, McCarroll SA, Bargmann CI, Fraser A, Kamath RS, Ahringer J, Li H, Kenyon C (2003) Genes that act downstream of DAF-16 to influence the lifespan of Caenorhabditis elegans. Nature 424(6946):277–283. https://doi.org/10.1038/nature01789

    Article  Google Scholar 

  27. Murray JD (2001) Mathematical biology II: spatial models and biomedical applications, 3rd edn. Springer, Berlin

    Google Scholar 

  28. Pierce SB, Costa M, Wisotzkey R, Devadhar S, Homburger SA, Buchman AR, Ferguson KC, Heller J, Platt DM, Pasquinelli AA, Liu LX, Doberstein SK, Ruvkun G (2001) Regulation of DAF-2 receptor signaling by human insulin and ins-1, a member of the unusually large and diverse C. elegans insulin gene family. Genes Dev 15(6):672–686. https://doi.org/10.1101/gad.867301

    Article  Google Scholar 

  29. Prahlad V, Morimoto RI (2009) Integrating the stress response: lessons for neurodegenerative diseases from C. elegans. Trends Cell Biol 19(2):52–61. https://doi.org/10.1016/j.tcb.2008.11.002

    Article  Google Scholar 

  30. Prahlad V, Morimoto RI (2011) Neuronal circuitry regulates the response of Caenorhabditis elegans to misfolded proteins. Proc Natl Acad Sci USA 108(34):14204–14209. https://doi.org/10.1073/pnas.1106557108

    Article  Google Scholar 

  31. Prahlad V, Cornelius T, Morimoto RI (2008) Regulation of the cellular heat shock response in Caenorhabditis elegans by thermosensory neurons. Science (New York, NY) 320(5877):811–814. https://doi.org/10.1126/science.1156093

    Article  Google Scholar 

  32. R Core Team (2015) R: a language and environment for statistical computing. Technical report, R Foundation for Statistical Computing, Vienna. https://www.r-project.org/

  33. Rea SL, Wu D, Cypser JR, Vaupel JW, Johnson TE (2005) A stress-sensitive reporter predicts longevity in isogenic populations of Caenorhabditis elegans. Nat Genet 37(8):894–8. https://doi.org/10.1038/ng1608

    Article  Google Scholar 

  34. Roy A, Kucukural A, Zhang Y (2010) I-TASSER: a unified platform for automated protein structure and function prediction. Nat Protoc 5(4):725–738. https://doi.org/10.1038/nprot.2010.5, arXiv:1011.1669v3

    Article  Google Scholar 

  35. Seewald AK, Cypser J, Mendenhall A, Johnson T (2010) Quantifying phenotypic variation in isogenic Caenorhabditis elegans expressing Phsp-16.2::gfp by clustering 2D expression patterns. PLoS One 5(7):e11426. https://doi.org/10.1371/journal.pone.0011426

    Article  Google Scholar 

  36. Soetaert K, Meysman F (2009) Solving partial differential equations, using R package ReacTran, R package vignette. https://cran.rproject.org/package=ReacTran

  37. Subramanian K, Fee CJ, Fredericks R, Stubbs RS, Hayes MT (2013) Insulin receptor–insulin interaction kinetics using multiplex surface plasmon resonance. J Mol Recognit 26(12):643–652. https://doi.org/10.1002/jmr.2307

    Article  Google Scholar 

  38. Tepper RG, Ashraf J, Kaletsky R, Kleemann G, Murphy CT, Bussemaker HJ (2013) PQM-1 complements DAF-16 as a key transcriptional regulator of DAF-2-mediated development and longevity. Cell 154(3):676–690. https://doi.org/10.1016/j.cell.2013.07.006

    Article  Google Scholar 

  39. Walker GA, Lithgow GJ (2003) Lifespan extension in C. elegans by a molecular chaperone dependent upon insulin-like signals. Aging Cell 2(2):131–139. https://doi.org/10.1046/j.1474-9728.2003.00045.x

    Article  Google Scholar 

  40. Wolfram Research Inc (2016) Mathematica 11.0

  41. Yang J, Yan R, Roy A, Xu D, Poisson J, Zhang Y (2015) The I-TASSER suite: protein structure and function prediction. Nat Methods 12(1):7–8. https://doi.org/10.1038/nmeth.3213

    Article  Google Scholar 

  42. Zhang Y (2008) I-TASSER server for protein 3D structure prediction. BMC Bioinform. https://doi.org/10.1186/1471-2105-9-40, https://zhanglab.ccmb.med.umich.edu/papers/2008_2.pdf, arXiv:1011.1669v3

    Article  Google Scholar 

  43. Zhang Q, Bhattacharya S, Andersen ME (2013) Ultrasensitive response motifs: basic amplifiers in molecular signalling networks. Open Biol 3(4):130031. https://doi.org/10.1098/rsob.130031

    Article  Google Scholar 

Download references

Acknowledgements

JMW is supported in part by an NSF GRFP and in part by the Interdisciplinary Quantitative Biology (IQ Biology) program at the BioFrontiers Institute, University of Colorado, Boulder. IQ Biology is generously supported by NSF IGERT Grant Number 1144807. ARM is supported by the National Institute on Aging at the National Institutes of Health by Grant 4R00AG045341. The authors would also like to thank T.E. Johnson (University of Colorado, Boulder) for insightful discussions and suggestions concerning this work.

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Correspondence to D. M. Bortz.

Appendix A Alternative Mechanism

Appendix A Alternative Mechanism

Here, we explore an alternate system in which the ILP class that acts as a DAF-2 agonist, B, is repressed by DAF-16 nuclear localization. This implies that B regulates its own production through a positive feedback loop rather than through a negative feedback loop as presented in the main paper. The following system of equations describes this alternative model:

$$\begin{aligned} \frac{\partial A}{\partial t}&=F(A,B)+D_{A}\nabla ^{2}A \end{aligned}$$
(22)
$$\begin{aligned} \frac{\partial B}{\partial t}&=G(A,B)+D_{B}\nabla ^{2}B \end{aligned}$$
(23)

where

$$\begin{aligned} F(A,B)&= k_{1}-k_{2}\frac{P^{r}}{H^{r}+P^{r}}-k_{3}A \end{aligned}$$
(24)
$$\begin{aligned} G(A,B)&= k_{4}+k_{5}\frac{P}{H^{r}+P^{r}}-k_{6}B \end{aligned}$$
(25)

and

$$\begin{aligned} P=\frac{V_\mathrm{max}B}{k_{7}\left( K_{D,B}\left( 1+\frac{A}{K_{D,A}}\right) +B\right) }. \end{aligned}$$
(26)

The system was made dimensionless using the following substitutions

$$\begin{aligned} t^{*}&=D_{A}t/L^{2}&x^{*}&=x/L&d&=D_{B}/D_{A}\\ \gamma&=k_{3}L^{2}/D_{A}&u&=Ak_{3}/k_{1}&v&=Bk_{3}/k_{4}\\ p&=Pk_{7}/V_\mathrm{max}&a&=k_{2}/k_{1}&b&=k_{5}/k_{4}\\ c&=k_{6}/k_{3}&h&=H(k_{7}/V_\mathrm{max})&k_{D,A}&=K_{D,A}k_{3}/k_{1}\\ k_{D,B}&=K_{D,B}k_{3}/k_{4} \end{aligned}$$

where all the parameters must take on positive values. This leads to the following system of equations

$$\begin{aligned} \frac{\partial u}{\partial t^{*}}=\gamma f(u,v)+\nabla ^{2}u \end{aligned}$$
(27)
$$\begin{aligned} \frac{\partial v}{\partial t^{*}}=\gamma g(u,v)+d \nabla ^2 v \end{aligned}$$
(28)

where

$$\begin{aligned} f(u,v)&=1-a\frac{p(u,v)^{r}}{h^{r}+p(u,v)^{r}}-u\end{aligned}$$
(29)
$$\begin{aligned} g(u,v)&=1+b\frac{p(u,v)^{r}}{h^{r}+p(u,v)^{r}}-cv.\end{aligned}$$
(30)
$$\begin{aligned} p(u,v)&=\frac{v}{k_{D,B}\left( 1+\frac{u}{k_{D,A}}\right) +v} \end{aligned}$$
(31)

Ignoring diffusion and linearizing about the steady state (\(u_{0},v_{0}\)) leads to the following differential equation:

$$\begin{aligned} w_{t}=\gamma Aw,\quad w=\left[ \begin{array}{l} u-u_{0}\\ v-v_{0} \end{array}\right] ,\quad A=\left[ \begin{array}{ll} f_{u} &{}\quad f_{v}\\ g_{u} &{}\quad g_{v} \end{array}\right] _{(u_{0},v_{0})}. \end{aligned}$$
(32)

For diffusion-driven instability to occur, the system must be stable without diffusion, leading to the following requirements:

$$\begin{aligned} f_{u}+g_{v}&<0\end{aligned}$$
(33)
$$\begin{aligned} f_{u}g_{v}-g_{u}f_{v}&>0. \end{aligned}$$
(34)

The partial derivatives of f(uv) and g(uv) are

$$\begin{aligned} f_{u}&=\frac{ah^{r}rk_{D,B}v^{r}}{k_{D,A}Z^{2}}Y-1\end{aligned}$$
(35)
$$\begin{aligned} f_{v}&=-\frac{arv^{r-1}}{Z}+\frac{av^{r}}{Z^{2}}(rv^{r-1}+h^{r}rY)\end{aligned}$$
(36)
$$\begin{aligned} g_{u}&=-\frac{bh^{r}rk_{D,B}v^{r}}{k_{D,A}Z^{2}}Y\end{aligned}$$
(37)
$$\begin{aligned} g_{v}&=\frac{brv^{r-1}}{Z}-\frac{bv^{r}}{Z^{2}}(rv^{r-1}+h^{r}rY)-c \end{aligned}$$
(38)

where

$$\begin{aligned} Z&=v^{r}+h^{r}\left( k_{D,B}\left( 1+\frac{u}{k_{D,A}}\right) +v\right) ^{r}\end{aligned}$$
(39)
$$\begin{aligned} Y&=\left( k_{D,B}\left( 1+\frac{u}{k_{D,A}}\right) +v\right) ^{r-1}. \end{aligned}$$
(40)

Using the condition given by Eq. 34 and a series of algebraic manipulations, we obtain the following inequalities that must hold for diffusion-driven instability to occur

$$\begin{aligned}&f_{u}g_{v}-g_{u}f_{v}>0\nonumber \\&\quad \implies \left( \frac{ah^{r}rk_{D,B}v^{r}}{k_{D,A}Z^{2}}Y-1\right) \left( \frac{brv^{r-1}}{Z}-\frac{bv^{r}}{Z^{2}}(rv^{r-1}+h^{r}rY)-c\right) \nonumber \\&\qquad \qquad -\,\left( -\frac{bh^{r}rk_{D,B}v^{r}}{k_{D,A}Z^{2}}Y\right) \left( -\frac{arv^{r-1}}{Z}+\frac{av^{r}}{Z^{2}}(rv^{r-1}+h^{r}rY)\right) >0\end{aligned}$$
(41)
$$\begin{aligned}&\quad \implies -\frac{brv^{r-1}}{Z}+\frac{bv^{r}}{Z^{2}}(rv^{r-1}+h^{r}rY)-\left( \frac{ah^{r}rk_{D,B}v^{r}}{k_{D,A}Z^{2}}Y-1\right) c>0\end{aligned}$$
(42)
$$\begin{aligned}&\quad \implies -Zbrv^{r-1}+bv^{r}(rv^{r-1}+h^{r}rY)-\frac{ah^{r}rk_{D,B}cv^{r}}{k_{D,A}}Y+cZ^{2}>0\end{aligned}$$
(43)
$$\begin{aligned}&\quad \implies -\left( v^{r}+h^{r}\left( k_{D,B}\left( 1+\frac{u}{k_{D,A}}\right) +v\right) ^{r}\right) brv^{r-1}\nonumber \\&\qquad \qquad +\,bv^{r}(rv^{r-1}+h^{r}rY)-\frac{ah^{r}rk_{D,B}v^{r}c}{k_{D,A}}Y+cZ^{2}>0\end{aligned}$$
(44)
$$\begin{aligned}&\quad \implies -brv^{2r-1}-bh^{r}rv^{r-1}Y\left( k_{D,B}\left( 1+\frac{u}{k_{D,A}}\right) +v\right) \end{aligned}$$
(45)
$$\begin{aligned}&\qquad \qquad +\,brv^{2r-1}+bh^{r}rv^{r}Y-\frac{ah^{r}rk_{D,B}cv^{r}}{k_{D,A}}Y+cZ^{2}>0\end{aligned}$$
(46)
$$\begin{aligned}&\quad \implies -bh^{r}rv^{r-1}Y\left( k_{D,B}\left( 1+\frac{u}{k_{D,A}}\right) +v\right) \nonumber \\&\qquad \qquad +\,bh^{r}rv^{r}Y-\frac{ah^{r}rk_{D,B}cv^{r}}{k_{D,A}}Y+cZ^{2}>0\end{aligned}$$
(47)
$$\begin{aligned}&\quad \implies -bh^{r}rk_{D,B}v^{r-1}Y\left( 1+\frac{u}{k_{D,A}}\right) -\frac{ah^{r}rk_{D,B}cv^{r}}{k_{D,A}}Y+cZ^{2}>0 \end{aligned}$$
(48)

Furthermore, for diffusion to cause instability in the system the following relation must hold

$$\begin{aligned} df_{u}+g_{v}>0. \end{aligned}$$
(49)

Taken together with Eq. 33, this implies that \(f_{u}\) and \(g_{v}\) must have opposite signs. This leads to two possible cases. In Case 1, \(f_{u}>0\) and \(g_{v}<0\), and in Case 2, \(f_{u}<0\) and \(g_{v}>0\). For Case 1, using Eqs. 35 and 38, we derive the following inequalities:

$$\begin{aligned} f_{u}>0\implies&\frac{ah^{r}rk_{D,B}v^{r}}{k_{D,A}}\left( k_{D,B}\left( 1+\frac{u}{k_{D,A}}\right) +v\right) ^{r-1}>Z^{2}\end{aligned}$$
(50)
$$\begin{aligned} g_{v}<0\implies&cZ^{2}>Zbrv^{r-1}-bv^{r}\left( rv^{r-1}+h^{r}r\left( k_{D,B}\left( 1+\frac{u}{k_{D,A}}\right) +v\right) ^{r-1}\right) \end{aligned}$$
(51)
$$\begin{aligned} \implies&cZ^{2}>brv^{2r-1}+bh^{r}rv^{r-1}\left( k_{D,B}\left( 1+\frac{u}{k_{D,A}}\right) +v\right) ^{r}-brv^{2r-1}\nonumber \\&-bh^{r}rv^{r}\left( k_{D,B}\left( 1+\frac{u}{k_{D,A}}\right) +v\right) ^{r-1}\end{aligned}$$
(52)
$$\begin{aligned} \implies&Z^{2}>\frac{bh^{r}rv^{r-1}}{c}\cdot \nonumber \\&\left( \left( k_{D,B}\left( 1+\frac{u}{k_{D,A}}\right) +v\right) ^{r}-v\left( k_{D,B}\left( 1+\frac{u}{k_{D,A}}\right) +v\right) ^{r-1}\right) \end{aligned}$$
(53)
$$\begin{aligned} \implies&Z^{2}>\frac{bh^{r}rv^{r-1}}{c}\cdot \nonumber \\&\left( k_{D,B}\left( 1+\frac{u}{k_{D,A}}\right) +v\right) ^{r-1}\left( \left( k_{D,B}\left( 1+\frac{u}{k_{D,A}}\right) +v\right) -v\right) \end{aligned}$$
(54)
$$\begin{aligned} \implies&Z^{2}>\frac{bh^{r}rk_{D,B}v^{r-1}}{c}\left( k_{D,B}\left( 1+\frac{u}{k_{D,A}}\right) +v\right) ^{r-1}\left( 1+\frac{u}{k_{D,A}}\right) . \end{aligned}$$
(55)

In summary, for Case 1, \(Z^{2}\) must satisfy the following inequality (using Eq. 40):

$$\begin{aligned} \frac{bh^{r}rk_{D,B}v^{r-1}}{c}\left( 1+\frac{u}{k_{D,A}}\right) Y<Z^{2}<\frac{ah^{r}rk_{D,B}v^{r}}{k_{D,A}}Y. \end{aligned}$$
(56)

Thus, for Case 1, using the fact that the determinate of A must be greater than zero (Eq. 48) and the upper bound on \(Z^{2}\) given in Eq. 56 we have that

$$\begin{aligned}&\frac{ah^{r}rk_{D,B}v^{r}}{k_{D,A}}Y>Z^{2}>\frac{bh^{r}rk_{D,B}v^{r-1}}{c}\left( 1+\frac{u}{k_{D,A}}\right) Y+\frac{ah^{r}rk_{D,B}v^{r}}{k_{D,A}}Y\end{aligned}$$
(57)
$$\begin{aligned}&\implies 0>Z^{2}-\frac{ah^{r}rk_{D,B}v^{r}}{k_{D,A}}Y>\frac{bh^{r}rk_{D,B}v^{r-1}}{c}\left( 1+\frac{u}{k_{D,A}}\right) Y. \end{aligned}$$
(58)

However, this inequality is not possible since all the parameter values are greater than zero.

A similar argument holds for Case 2. Using the same steps as shown in Eqs. 5055, but reversing the equality sign, \(Z^{2}\) must satisfy the following inequality:

$$\begin{aligned} \frac{ahrk_{D,B}v^{r}}{k_{D,A}}Y<Z^{2}<\frac{bhrk_{D,B}v^{r-1}}{c}\left( 1+\frac{u}{k_{D,A}}\right) Y. \end{aligned}$$
(59)

Equations 48 and 59 imply that

$$\begin{aligned}&\frac{bhrk_{D,B}v^{r-1}}{c}\left( 1+\frac{u}{k_{D,A}}\right) Y>Z^{2}>\frac{bhrk_{D,B}v^{r-1}}{c}\left( 1+\frac{u}{k_{D,A}}\right) Y+\frac{ahrk_{D,B}v^{r}}{k_{D,A}}Y\end{aligned}$$
(60)
$$\begin{aligned}&\implies 0>Z^{2}-\frac{bhrk_{D,B}v^{r-1}}{c}\left( 1+\frac{u}{k_{D,A}}\right) Y>\frac{ahrk_{D,B}v^{r}}{k_{D,A}}Y. \end{aligned}$$
(61)

Since the parameters are positive, the final term in Eq. 61 cannot be less than zero. Thus, for this system it is not possible for the determinate of A to be greater than zero and for \(f_{u}\) and \(g_{v}\) to have opposite signs. Therefore, diffusion-driven instability cannot occur.

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Wentz, J.M., Mendenhall, A.R. & Bortz, D.M. Pattern Formation in the Longevity-Related Expression of Heat Shock Protein-16.2 in Caenorhabditis elegans. Bull Math Biol 80, 2669–2697 (2018). https://doi.org/10.1007/s11538-018-0482-7

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Keywords

  • Aging
  • Diffusion-driven instability
  • Insulin-like signaling
  • Reaction diffusion model