notes added

thesis_notes.tex

@@ -5,33 +5,68 @@
 \usepackage{amssymb}
 \usepackage{graphicx}
 \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}
-\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*}
-\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*}
-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}
-\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}
+
 The equation for the cell density is a diffusion advection equation, with terms added to account for cell birth and death:
 \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} 
 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:
 \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}
 \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}
 \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)\\
@@ -48,6 +83,7 @@ The boundary conditions are:
 \begin{equation}
 \rho \big|_{\partial \Omega} = 0 \qquad \sigma \big|_{\partial \Omega} = 0
 \end{equation}
+At every timestep, we move the boundary by $vdt$.
 \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.
 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 Use the optimum active stress obtained in step 1 to get appropriate reaction terms/boundary conditions on $c_i$.
 \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}
 Note:
 \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]
 \end{equation}
 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:
 \begin{equation}
 \sigma_e = E \epsilon_e
@@ -270,7 +316,7 @@ As $t \to \infty$,
 \sigma(t) = \eta \dot \epsilon
 \end{equation}
 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 
 \begin{equation}
 \sigma = \sigma_e + \sigma_v
@@ -287,7 +333,7 @@ As $t \to \infty$,
 \epsilon(t) = \frac{\sigma}{\epsilon}
 \end{equation}
 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,
 \begin{equation}
 \begin{split}
@@ -303,5 +349,5 @@ Thus,
 \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
 \end{equation}
-\bibliography{ref}
+\bibliography{references}
 \end{document}
\ No newline at end of file