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


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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  1. 1.

    ReactionDiffusionCElegans repository from MathBioCU group on GitHub (

  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.


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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:

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


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


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

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

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


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

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

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

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

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

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

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


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

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

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

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

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

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:

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

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

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

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

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

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:

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

Equations 48 and 59 imply that

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

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).

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