notes added

thesis_notes.tex

@@ -5,33 +5,68 @@
 \usepackage{amssymb}
 \usepackage{amssymb}
 \usepackage{graphicx}
 \usepackage{graphicx}
 \usepackage[left=2cm,right=2cm,top=2cm,bottom=2cm]{geometry}
 \usepackage[left=2cm,right=2cm,top=2cm,bottom=2cm]{geometry}
-\author{Jigyasa Watwani}
-\title{Physics of Growth Regulation in Cells and Tissues - Notes}
 \begin{document}
 \begin{document}
-\chapter{A minimal model}
-We model a 1-D tissue as a visco-elastic medium. We write hydrodynamic equations for the evolution of a displacement field $\boldsymbol u(x,t)$, a density field $\rho(x,t)$ and a concentration field for the signaling morphogen $c(x,t)$ that tells the cells when to divide.
-THe equation for $\boldsymbol u$ is a force balance equation (divergence of the total stress = force density). We use the Kelvin Voigt model to be able to get long-time leastic behaviour:
-\begin{equation*}
-\gamma \dot u = \partial_x \sigma
-\end{equation*}
+
+\author{Jigyasa Watwani}
+\title{Physics of Growth Regulation in Cells and Tissues}
+\maketitle
+\tableofcontents
+\chapter{Size regulation in living systems}
+Evidence for size regulation: Transplantation experiments in salamander limbs and mouse spleens \cite{Bryant1984-pa}
+\chapter{Cell size regulation}
+\section{Evidence}
+Probability distributions of quantities like cell length at division, added length in a generation, generation time, cell elongation rate, etc are sharply peaked \cite{Jun2018-wz}, indicating that single cells robustly control their sizes.
+\section{Cell size control mechanisms: Sizers, Adders and Timers}
+Definitions, correlations between growth and size at birth, time taken (in generations) to achieve size homeostasis for the three mechanisms (\cite{RHIND2021R1414})
+\section{Modeling sizers, adders and timers}
+Largely following \cite{RHIND2021R1414}
+\subsection*{Division protein model}
+\subsection*{Diluted inhibitor sizer}
+\subsection*{Accumulating activator sizer}
+\chapter{Tissue size regulation}
+\section{Existing models}
+\subsection{Signaling-based models}
+\subsection{Mechanical models}
+\cite{AEGERTERWILMSEN2007318}, \cite{Shraiman2005-wl}, \cite{Hufnagel2007-nt}, \cite{PhysRevLett.129.048102}
+\chapter{A minimal mechanical model}
+We model a 1-D tissue as a visco-elastic medium. The relevant fields are a displacement field $\boldsymbol u(x,t)$, a density field $\rho(x,t)$ and a concentration field for the signaling morphogen $c(x,t)$ that tells the cells when to divide.
+\begin{figure}[h]
+\includegraphics[scale=0.3]{fields}
+\end{figure}
+
+The equation for $\boldsymbol u$ is a force balance equation $\boldsymbol \nabla \cdot \boldsymbol \sigma = \boldsymbol f$, where $\boldsymbol \sigma$ is the total stress in the tissue and $\boldsymbol f$ is the force density. We use the Kelvin Voigt model [\ref{kelvin_voigt}] to be able to get long-time eleastic behaviour. Thus, 
 \begin{equation*}
 \begin{equation*}
-\sigma = \sigma_{e} + \sigma_{v} + \sigma_{a} = K \partial_x u + \eta \partial_x \dot u + \lambda \partial_x f(c,\rho)
+\boldsymbol \sigma = \boldsymbol {\sigma_{e}} + {\boldsymbol \sigma_{v}} + {\boldsymbol \sigma_{a}}
 \end{equation*}
 \end{equation*}
-Thus, 
+The elastic and viscous stresses can be written in terms of their irreducible components as:
+\begin{equation}
+\boldsymbol{\sigma_e} = -K \frac{1}{d} (\boldsymbol \nabla \cdot \boldsymbol u)\boldsymbol g - 2\mu \left[\frac{1}{2}[\boldsymbol \nabla \boldsymbol u + (\boldsymbol \nabla \boldsymbol u)^T] - \frac{1}{d} (\boldsymbol \nabla \cdot \boldsymbol u)\boldsymbol g \right] 
+\end{equation}
+\begin{equation}
+\boldsymbol{\sigma_v} = -\zeta \frac{1}{d} (\boldsymbol \nabla \cdot \boldsymbol v)\boldsymbol g - 2\eta \left[\frac{1}{2}[\boldsymbol \nabla \boldsymbol v + (\boldsymbol \nabla \boldsymbol v)^T] - \frac{1}{d} (\boldsymbol \nabla \cdot \boldsymbol v)\boldsymbol g \right] 
+\end{equation}
+where $K, \mu, \zeta, \eta$ are the bulk modulus, shear modulus, bulk viscosity and shear viscosity of the visco-elastic material.
+We choose
+\begin{equation}
+\boldsymbol {\sigma_a} = \lambda \boldsymbol g(c, \rho, \boldsymbol u)
+\end{equation}
+The equation for the velocity field then becomes, in the overdamped limit:
 \begin{equation}
 \begin{equation}
-\gamma \dot u = K \partial_x^2 u + \eta \partial_x^2 \dot u + \lambda \partial_x f 
+\gamma \boldsymbol v = \boldsymbol \nabla \cdot (\boldsymbol{\sigma_e} + \boldsymbol{\sigma_v} + \boldsymbol{\sigma_a})
 \end{equation}
 \end{equation}
+
 The equation for the cell density is a diffusion advection equation, with terms added to account for cell birth and death:
 The equation for the cell density is a diffusion advection equation, with terms added to account for cell birth and death:
 \begin{equation}
 \begin{equation}
-\dot \rho = D_\rho \partial_x^2 \rho - \partial_x(\dot u \rho) - k_{\rho}(\rho - \rho_0) + F(u, c_i, \rho, t)
+\dot \rho = D_\rho \boldsymbol \nabla^2 \rho - \boldsymbol \nabla \cdot (\boldsymbol v \rho) - k_{\rho}(\rho - \rho_0) + F(\boldsymbol u, c_i, \rho, t)
 \end{equation} 
 \end{equation} 
 The $c-$dependence of $F$ is the sizer/adder-like behaviour that changes cell density/number based on the concentration field.
 The $c-$dependence of $F$ is the sizer/adder-like behaviour that changes cell density/number based on the concentration field.
+
 The dynamics for concentration of the $i-$th signaling morphogen is:
 The dynamics for concentration of the $i-$th signaling morphogen is:
 \begin{equation}
 \begin{equation}
-\dot c_i = D_{c_i} \partial_x^2 c_i - \partial_x(\dot {u} c_i) + G(u, c_i, \rho, t)
+\dot c_i = D_{c_i} \boldsymbol \nabla^2 c_i - \boldsymbol \nabla \cdot(\boldsymbol v c_i) + G(\boldsymbol u, c_i, \rho, t)
 \end{equation}
 \end{equation}
 \section{No signaling on fixed boundaries}
 \section{No signaling on fixed boundaries}
-Choosing $f = \frac{\rho}{\rho + \rho_s}$,
+Working in 1-D, ignoring the couplings $F, G$ and choosing $g(u,\rho,t) = \frac{\rho}{\rho + \rho_s}$,
 \begin{eqnarray}
 \begin{eqnarray}
 \dot u = v \\
 \dot u = v \\
 \gamma \dot u = K \partial_x^2 u + \eta \partial_x^2 \dot u + \lambda \partial_x \left( \frac{\rho}{\rho + \rho_s}\right)\\
 \gamma \dot u = K \partial_x^2 u + \eta \partial_x^2 \dot u + \lambda \partial_x \left( \frac{\rho}{\rho + \rho_s}\right)\\
@@ -48,6 +83,7 @@ The boundary conditions are:
 \begin{equation}
 \begin{equation}
 \rho \big|_{\partial \Omega} = 0 \qquad \sigma \big|_{\partial \Omega} = 0
 \rho \big|_{\partial \Omega} = 0 \qquad \sigma \big|_{\partial \Omega} = 0
 \end{equation}
 \end{equation}
+At every timestep, we move the boundary by $vdt$.
 \section{Controlling growth}
 \section{Controlling growth}
 A generic control problem goes as follows: Given a differential equation $\dot x = f(x,u)$ with the initial condition $x(0)=0$, find the control path $u(t)$ that minimizes a cost (which we will construct) subject to the equation $\dot x = f(x,u)$ being satisfied.
 A generic control problem goes as follows: Given a differential equation $\dot x = f(x,u)$ with the initial condition $x(0)=0$, find the control path $u(t)$ that minimizes a cost (which we will construct) subject to the equation $\dot x = f(x,u)$ being satisfied.
 The cost $J$ involves a running cost, for the path reaching a final state and a terminal cost.
 The cost $J$ involves a running cost, for the path reaching a final state and a terminal cost.
@@ -59,7 +95,16 @@ Going back to oyr problem, we tackle the control aspects in two steps:
 \item What form of active stress would minimize the cost function $J = \Phi[L(T)] + \int_0^T h(L(t)) dt$ subject to $\gamma \dot u = \partial_x \sigma$
 \item What form of active stress would minimize the cost function $J = \Phi[L(T)] + \int_0^T h(L(t)) dt$ subject to $\gamma \dot u = \partial_x \sigma$
 \item Use the optimum active stress obtained in step 1 to get appropriate reaction terms/boundary conditions on $c_i$.
 \item Use the optimum active stress obtained in step 1 to get appropriate reaction terms/boundary conditions on $c_i$.
 \end{enumerate}
 \end{enumerate}
-\chapter{Model on fixed bouundaries}
+\chapter{Model on fixed boundaries}
+On fixed boundaries, our model can explain pattern formation in the desnity field of cells and/or signaling morphogens. 
+
+In the absence of the signaling morphogen, we first look at patterns in the density field. We obtained homogeneous, patterned and oscillatory states in different regions of the phase space:
+\begin{figure}[h]
+\includegraphics[scale=0.2]{phaseplot}
+\end{figure}
+\begin{figure}
+\includegraphics[scale=0.3]{patterns}
+\end{figure}
 \chapter{Model on moving boundaries}
 \chapter{Model on moving boundaries}
 Note:
 Note:
 \section{Diffusion on fixed boundaries becomes diffusion-advection on moving boundaries}
 \section{Diffusion on fixed boundaries becomes diffusion-advection on moving boundaries}
@@ -235,7 +280,8 @@ The final solution is the inverse fourier transform of this, viz,
 c(x,t) = \exp(-\alpha t_c) \left[ 1 + 0.2 \cos\left(\frac{m \pi x}{L}\right) \exp \left( -\frac{D m^2 \pi^2 (t - t_c)}{L^2} - \frac{D m^2 \pi^2 (1 - e^{-2 \alpha t_c})}{2 \alpha L_0^2}\right) \right]
 c(x,t) = \exp(-\alpha t_c) \left[ 1 + 0.2 \cos\left(\frac{m \pi x}{L}\right) \exp \left( -\frac{D m^2 \pi^2 (t - t_c)}{L^2} - \frac{D m^2 \pi^2 (1 - e^{-2 \alpha t_c})}{2 \alpha L_0^2}\right) \right]
 \end{equation}
 \end{equation}
 where again $L = L_0 \exp(\alpha t_c)$
 where again $L = L_0 \exp(\alpha t_c)$
-\chapter*{Appendix: Visco-elastic models}
+\appendix
+\chapter{Visco-elastic models}
 As the name suggests, viscoelastic materials have an elastic component and a viscous component. For the elastic component, the stress $\sigma_e$ is related to the strain $\epsilon_e$ as:
 As the name suggests, viscoelastic materials have an elastic component and a viscous component. For the elastic component, the stress $\sigma_e$ is related to the strain $\epsilon_e$ as:
 \begin{equation}
 \begin{equation}
 \sigma_e = E \epsilon_e
 \sigma_e = E \epsilon_e
@@ -270,7 +316,7 @@ As $t \to \infty$,
 \sigma(t) = \eta \dot \epsilon
 \sigma(t) = \eta \dot \epsilon
 \end{equation}
 \end{equation}
 That is, at long times, a Maxwell viscoelastic material behaves like a fluid.
 That is, at long times, a Maxwell viscoelastic material behaves like a fluid.
-\subsection{Kelvin-Voigt model}
+\subsection{Kelvin-Voigt model} \label{kelvin_voigt}
 In this model, the elastic and viscous components(spring and dashpot, respectively) are in parallel with each other. We subject both of them to equal strain $\epsilon = \epsilon_e = \epsilon_v$. The total stress of the combination is 
 In this model, the elastic and viscous components(spring and dashpot, respectively) are in parallel with each other. We subject both of them to equal strain $\epsilon = \epsilon_e = \epsilon_v$. The total stress of the combination is 
 \begin{equation}
 \begin{equation}
 \sigma = \sigma_e + \sigma_v
 \sigma = \sigma_e + \sigma_v
@@ -287,7 +333,7 @@ As $t \to \infty$,
 \epsilon(t) = \frac{\sigma}{\epsilon}
 \epsilon(t) = \frac{\sigma}{\epsilon}
 \end{equation}
 \end{equation}
 That is, at long times, a Kelvin-Voigt viscoelastic material behaves like a solid.
 That is, at long times, a Kelvin-Voigt viscoelastic material behaves like a solid.
-\chapter*{Appendix: Proof of the Reynolds Transport Theorem}
+\chapter{Proof of the Reynolds Transport Theorem}
 From Leibniz theorem,
 From Leibniz theorem,
 \begin{equation}
 \begin{equation}
 \begin{split}
 \begin{split}
@@ -303,5 +349,5 @@ Thus,
 \begin{equation}
 \begin{equation}
 \frac{d}{dt} \int_{\alpha(t)}^{\beta(t)} c(x,t) dx = \int_{\alpha(t)}^{\beta(t)} \left[ \frac{\partial c}{\partial t} + \frac{\partial}{\partial x}(cv) \right]dx
 \frac{d}{dt} \int_{\alpha(t)}^{\beta(t)} c(x,t) dx = \int_{\alpha(t)}^{\beta(t)} \left[ \frac{\partial c}{\partial t} + \frac{\partial}{\partial x}(cv) \right]dx
 \end{equation}
 \end{equation}
-\bibliography{ref}
+\bibliography{references}
 \end{document}
 \end{document}
\ No newline at end of file