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- .gitattributes +66 -0
- chapters/collecting-trajectories.log +0 -0
- chapters/collecting-trajectories.tex +294 -0
- chapters/langevin-equation.log +0 -0
- chapters/langevin-equation.tex +367 -0
- chapters/preamble.tex +108 -0
- chapters/xenopus-equilibrium.tex +71 -0
- images/Pulse1-Fourier.pdf +3 -0
- images/Pulse1.pdf +3 -0
- images/Pulse2-Fourier.pdf +3 -0
- images/Pulse2.pdf +3 -0
- images/Pulse3-Fourier.pdf +3 -0
- images/Pulse3.pdf +3 -0
- images/a-logo.png +0 -0
- images/coke.jpg +3 -0
- images/fourier-1.pdf +3 -0
- images/fourier-2.pdf +3 -0
- images/fourier-3.pdf +3 -0
- images/fourier-4.pdf +3 -0
- images/fourier-5.pdf +3 -0
- mtrack.png +3 -0
- presentation.aux +12 -75
- presentation.fdb_latexmk +13 -51
- presentation.fls +2 -190
- presentation.log +21 -282
- presentation.nav +11 -58
- presentation.out +2 -6
- presentation.pdf +2 -2
- presentation.snm +0 -6
- presentation.synctex.gz +2 -2
- presentation.tex +1 -1
- presentation.toc +0 -4
- presentation_images/1DCnn.png +0 -0
- presentation_images/Derivative_Plots_of_ALL_PCs_Across_Cell_Types_all_PCs_Fifth_density_stats.png +3 -0
- presentation_images/Derivative_Plots_of_ALL_PCs_Across_Cell_Types_all_PCs_Second_density_stats.png +3 -0
- presentation_images/Derivative_Plots_of_ALL_PCs_Across_Cell_Types_all_PCs_Second_membrane_density_stats.png +3 -0
- presentation_images/Derivative_Plots_of_ALL_PCs_Across_Cell_Types_all_PCs_Sixth_density_stats.png +3 -0
- presentation_images/Derivative_Plots_of_ALL_PCs_Across_Cell_Types_all_PCs_Sixth_membrane_density_stats.png +3 -0
- presentation_images/Derivative_Plots_of_PC1_Across_Cell_Types_PC1_Fifth_density_stats.png +3 -0
- presentation_images/Derivative_Plots_of_PC1_Across_Cell_Types_PC1_Second_density_stats.png +3 -0
- presentation_images/Derivative_Plots_of_PC1_Across_Cell_Types_PC1_Second_membrane_density_stats.png +3 -0
- presentation_images/Derivative_Plots_of_PC1_Across_Cell_Types_PC1_Sixth_density_stats.png +3 -0
- presentation_images/Derivative_Plots_of_PC1_Across_Cell_Types_PC1_Sixth_membrane_density_stats.png +3 -0
- presentation_images/Diffusion_Constant_Plots_of_ALL_PCs_Across_Cell_Types_all_PCs_Fifth_density_stats.png +3 -0
- presentation_images/Diffusion_Constant_Plots_of_ALL_PCs_Across_Cell_Types_all_PCs_Second_density_stats.png +3 -0
- presentation_images/Diffusion_Constant_Plots_of_ALL_PCs_Across_Cell_Types_all_PCs_Second_membrane_density_stats.png +3 -0
- presentation_images/Diffusion_Constant_Plots_of_ALL_PCs_Across_Cell_Types_all_PCs_Sixth_density_stats.png +3 -0
- presentation_images/Diffusion_Constant_Plots_of_ALL_PCs_Across_Cell_Types_all_PCs_Sixth_membrane_density_stats.png +3 -0
- presentation_images/Diffusion_Constant_Plots_of_PC1_Across_Cell_Types_PC1_Fifth_density_stats.png +3 -0
- presentation_images/Diffusion_Constant_Plots_of_PC1_Across_Cell_Types_PC1_Second_density_stats.png +3 -0
.gitattributes
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chapters/collecting-trajectories.log
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|
|
|
|
|
|
|
|
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|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
|
|
|
| 1 |
+
\section{Collecting Trajectories}
|
| 2 |
+
|
| 3 |
+
\subsection{Nuclei Segmentation: VollSeg}
|
| 4 |
+
\frame{\sectionpage}
|
| 5 |
+
|
| 6 |
+
\begin{frame}{VollSeg}
|
| 7 |
+
\begin{itemize}[<+->]
|
| 8 |
+
|
| 9 |
+
\item VollSeg is a versatile tool built over StarDist with modular segmentation methodologies.
|
| 10 |
+
|
| 11 |
+
\item Xenopus: Early stage of development occupies limited space, $\rightarrow$ lots of background signal $\rightarrow$ Normalization issues.
|
| 12 |
+
|
| 13 |
+
\item Train a ROI and StarDist nuclei model $\rightarrow$ nuclei segmentation in the region of signal.
|
| 14 |
+
|
| 15 |
+
\end{itemize}
|
| 16 |
+
|
| 17 |
+
\end{frame}
|
| 18 |
+
|
| 19 |
+
\begin{frame}{VollSeg}
|
| 20 |
+
\href{run:presentation_movies/sixth_nuclei_seg.mp4}{
|
| 21 |
+
\fcolorbox{blue!80!black}{blue!10}{% Border and background color
|
| 22 |
+
\parbox[c][1.5em][c]{0.6\textwidth}{% Box size
|
| 23 |
+
\centering\textbf{\textcolor{blue!90!black}{Nuclei Segmentation (Click)}}% Text styling
|
| 24 |
+
}%
|
| 25 |
+
}%
|
| 26 |
+
}
|
| 27 |
+
|
| 28 |
+
\end{frame}
|
| 29 |
+
|
| 30 |
+
\subsection{Mitosis Action Classification: Oneat}
|
| 31 |
+
\frame{\sectionpage}
|
| 32 |
+
\begin{frame}{Mitosis Action Classification}
|
| 33 |
+
|
| 34 |
+
\begin{itemize}[<+->]
|
| 35 |
+
|
| 36 |
+
\item Mitosis is a process that completes in $\delta T$ time-frame $\rightarrow$ action event
|
| 37 |
+
|
| 38 |
+
\item Oneat: (Osis network for event as action topology) is a network that is jointly trained to find the plane of focus and space-time location of mitotic cells (arbitrary shaped images).
|
| 39 |
+
|
| 40 |
+
\item Leveraging the segmentation of nuclei $\rightarrow$ apply trained network to obtain TZYX location of mitotic cells.
|
| 41 |
+
\item Oneat is organism and action event type independent, extensible to Apoptosis, static cell type classification.
|
| 42 |
+
|
| 43 |
+
\end{itemize}
|
| 44 |
+
\end{frame}
|
| 45 |
+
\begin{frame}{Mitosis Action Classification}
|
| 46 |
+
\href{run:presentation_movies/all_timepoints_mitosis.mp4}{
|
| 47 |
+
\fcolorbox{blue!80!black}{blue!10}{% Border and background color
|
| 48 |
+
\parbox[c][1.5em][c]{0.6\textwidth}{% Box size
|
| 49 |
+
\centering\textbf{\textcolor{blue!90!black}{Action Classification Video (Click)}}% Text styling
|
| 50 |
+
}%
|
| 51 |
+
}%
|
| 52 |
+
}
|
| 53 |
+
|
| 54 |
+
|
| 55 |
+
\end{frame}
|
| 56 |
+
|
| 57 |
+
\subsection{Lineage tree correction: TrackMate-Oneat}
|
| 58 |
+
|
| 59 |
+
\frame{\sectionpage}
|
| 60 |
+
\begin{frame}{TrackMate-Oneat}
|
| 61 |
+
\begin{itemize}[<+->]
|
| 62 |
+
|
| 63 |
+
\item Mitosis Locations $\rightarrow$ a strong prior for track linkage
|
| 64 |
+
\begin{alertblock}{Correcting Branch Bifurcations}
|
| 65 |
+
Leveraging prior information to correct the mitotic tracks/lineage tree branch bifurcation
|
| 66 |
+
\end{alertblock}
|
| 67 |
+
|
| 68 |
+
|
| 69 |
+
\item TrackMate extension $\rightarrow$ TrackMate-Oneat $\rightarrow$ track without branch linking, input Mitosis locations csv $\rightarrow$ updates the tracks xml file
|
| 70 |
+
|
| 71 |
+
\end{itemize}
|
| 72 |
+
\end{frame}
|
| 73 |
+
|
| 74 |
+
\begin{frame}{TrackMate-Oneat}
|
| 75 |
+
\begin{figure}
|
| 76 |
+
\centering
|
| 77 |
+
% Adjust the width and spacing as needed
|
| 78 |
+
\includegraphics[width=1\textwidth]{presentation_images/trackmate-panes-1536x495.png}
|
| 79 |
+
|
| 80 |
+
\end{figure}
|
| 81 |
+
\end{frame}
|
| 82 |
+
|
| 83 |
+
\begin{frame}{TrackMate-Oneat}
|
| 84 |
+
\begin{figure}
|
| 85 |
+
\centering
|
| 86 |
+
% Adjust the width and spacing as needed
|
| 87 |
+
|
| 88 |
+
\includegraphics[width=0.8\textwidth]{presentation_images/tm-oneat-panel}
|
| 89 |
+
\end{figure}
|
| 90 |
+
\end{frame}
|
| 91 |
+
|
| 92 |
+
\begin{frame}{TrackMate-Oneat}
|
| 93 |
+
\href{run:presentation_movies/fifth_data_tracks.mp4}{
|
| 94 |
+
\fcolorbox{blue!80!black}{blue!10}{% Border and background color
|
| 95 |
+
\parbox[c][1.5em][c]{0.6\textwidth}{% Box size
|
| 96 |
+
\centering\textbf{\textcolor{blue!90!black}{Tracks Example1 (Click)}}% Text styling
|
| 97 |
+
}%
|
| 98 |
+
}%
|
| 99 |
+
}
|
| 100 |
+
|
| 101 |
+
\href{run:presentation_movies/fifth_data_tracks_second.mp4}{
|
| 102 |
+
\fcolorbox{blue!80!black}{blue!10}{% Border and background color
|
| 103 |
+
\parbox[c][1.5em][c]{0.6\textwidth}{% Box size
|
| 104 |
+
\centering\textbf{\textcolor{blue!90!black}{Tracks Example2 (Click)}}% Text styling
|
| 105 |
+
}%
|
| 106 |
+
}%
|
| 107 |
+
}
|
| 108 |
+
\end{frame}
|
| 109 |
+
|
| 110 |
+
\subsection{Membrane Segmentation: VollSeg + CellPose}
|
| 111 |
+
\frame{\sectionpage}
|
| 112 |
+
\begin{frame}{Membrane Segmentation}
|
| 113 |
+
|
| 114 |
+
\begin{itemize}[<+->]
|
| 115 |
+
|
| 116 |
+
\item CellPose: Slice by Slice 3D segmentation, object shape stretched, 3D features washed out.
|
| 117 |
+
|
| 118 |
+
\item Mode VollSeg: Autoencoder to enhance membrane edges $\rightarrow$ Nuclei seeds perform 3D watershed.
|
| 119 |
+
|
| 120 |
+
\item Computer vision methods to remove 'overflows', cellpose regions as mask.
|
| 121 |
+
\href{run:presentation_movies/fifth_data_membrane_segmentation.mp4}{
|
| 122 |
+
\fcolorbox{blue!80!black}{blue!10}{% Border and background color
|
| 123 |
+
\parbox[c][1.5em][c]{0.6\textwidth}{% Box size
|
| 124 |
+
\centering\textbf{\textcolor{blue!90!black}{Membrane Segmentation (Click)}}% Text styling
|
| 125 |
+
}%
|
| 126 |
+
}%
|
| 127 |
+
}
|
| 128 |
+
\begin{alertblock}{3D membrane leveraged segmentation}
|
| 129 |
+
Leveraging nuclei seeds on membrane enhanced signal for 3D shape preserving segmentation
|
| 130 |
+
\end{alertblock}
|
| 131 |
+
|
| 132 |
+
\end{itemize}
|
| 133 |
+
|
| 134 |
+
\end{frame}
|
| 135 |
+
\subsection{Tracks: Nuclei $\rightarrow$ Membrane: NapaTrackMater}
|
| 136 |
+
\frame{\sectionpage}
|
| 137 |
+
\begin{frame}{Transfer Tracks}
|
| 138 |
+
\begin{itemize}[<+->]
|
| 139 |
+
|
| 140 |
+
\item Nuclei channel in 3D $\rightarrow$ easy to segment-track-auto-manual correct.
|
| 141 |
+
\item membrane channel in 3D $\rightarrow$ difficult to segment-track, impossible to manually correct.
|
| 142 |
+
\begin{alertblock}{Simply transfer the tracks}
|
| 143 |
+
Using nuclei tracks, membrane segmentation transfer the links, recompute morphological features
|
| 144 |
+
\end{alertblock}
|
| 145 |
+
\item If membrane seg label not found, skip timepoint, mark morphological features as -1 (lost track) but continue.
|
| 146 |
+
\end{itemize}
|
| 147 |
+
\end{frame}
|
| 148 |
+
\begin{frame}{Transfer Tracks}
|
| 149 |
+
\href{run:presentation_movies/fifth_data_tracks_membrane.mp4}{
|
| 150 |
+
\fcolorbox{blue!80!black}{blue!10}{% Border and background color
|
| 151 |
+
\parbox[c][1.5em][c]{0.6\textwidth}{% Box size
|
| 152 |
+
\centering\textbf{\textcolor{blue!90!black}{Tracks Example1 (Click)}}% Text styling
|
| 153 |
+
}%
|
| 154 |
+
}%
|
| 155 |
+
}
|
| 156 |
+
|
| 157 |
+
\href{run:presentation_movies/fifth_data_tracks_second_membrane.mp4}{
|
| 158 |
+
\fcolorbox{blue!80!black}{blue!10}{% Border and background color
|
| 159 |
+
\parbox[c][1.5em][c]{0.6\textwidth}{% Box size
|
| 160 |
+
\centering\textbf{\textcolor{blue!90!black}{Tracks Example2 (Click)}}% Text styling
|
| 161 |
+
}%
|
| 162 |
+
}%
|
| 163 |
+
}
|
| 164 |
+
\end{frame}
|
| 165 |
+
\subsection{Point Cloud based morphodynamic features: NapaTrackMater}
|
| 166 |
+
\frame{\sectionpage}
|
| 167 |
+
\begin{frame}{Point Cloud based morphodynamic features}
|
| 168 |
+
\begin{itemize}[<+->]
|
| 169 |
+
|
| 170 |
+
\item Simple TrackMate XML files of nuclei and membrane tracks.
|
| 171 |
+
|
| 172 |
+
\item Input: NapaTrackMater $\rightarrow$ obtain point clouds from segmentation image, compute accurate morphodynamic properties $\rightarrow$ re-create a Master XML file $\rightarrow$ visualize tracks in 3D in Napari: vollseg-napari-trackmate
|
| 173 |
+
\item Shape features: Eccentricity components, surface area, radius, Volume
|
| 174 |
+
\item Dynamic features: Speed, Acceleration, Distance to tissue border, {Radial, Cell-Axis, Motion} angles wrt image co-ordinate frame
|
| 175 |
+
|
| 176 |
+
|
| 177 |
+
|
| 178 |
+
\end{itemize}
|
| 179 |
+
\end{frame}
|
| 180 |
+
|
| 181 |
+
\begin{frame}{Point Cloud based morphodynamic features: NapaTrackMater}
|
| 182 |
+
\href{run:presentation_movies/fifth_data_features.mp4}{
|
| 183 |
+
\fcolorbox{blue!80!black}{blue!10}{% Border and background color
|
| 184 |
+
\parbox[c][1.5em][c]{0.6\textwidth}{% Box size
|
| 185 |
+
\centering\textbf{\textcolor{blue!90!black}{Tracks Example1 (Click)}}% Text styling
|
| 186 |
+
}%
|
| 187 |
+
}%
|
| 188 |
+
}
|
| 189 |
+
\end{frame}
|
| 190 |
+
\subsection{Trajectory Analysis: NapaTrackMater}
|
| 191 |
+
\frame{\sectionpage}
|
| 192 |
+
\begin{frame}{Trajectory Dataframe}
|
| 193 |
+
\begin{itemize}[<+->]
|
| 194 |
+
\item Representation of tracks as feature matrices $F \cross T$.
|
| 195 |
+
\item Dataframe with columns 'Dividing', 'Number Dividing', 't,z,y,x, shape-dynamic features'
|
| 196 |
+
\item Such representation more amenable to Machine Learning based approaches.
|
| 197 |
+
\item Knowing ground truth cell type append column of cell type (Goblet, Basal, Radial) to the dataframe.
|
| 198 |
+
\end{itemize}
|
| 199 |
+
\end{frame}
|
| 200 |
+
|
| 201 |
+
\begin{frame}{Inception Model Prediction}
|
| 202 |
+
\begin{itemize}[<+->]
|
| 203 |
+
\item A totally theoretical physics based network, on ideas of non-equilibrium statistical physics (later slides).
|
| 204 |
+
\item Cell type $\rightarrow$ Track label, Motion type $\rightarrow$ Another Track label
|
| 205 |
+
\item $F$ features along channel dimension, T as other dimension: Input Data for Neural Network(s)
|
| 206 |
+
\item 1D densenet as the model for training a) Cell type/Track type b) Motion type
|
| 207 |
+
\item Not whole track in training, cut out tracklets. Feature not Vision based neural network.
|
| 208 |
+
\begin{alertblock}{Model can Predict!}
|
| 209 |
+
Given a cut-out of tracklet it can reliably predict a) \& b)
|
| 210 |
+
\end{alertblock}
|
| 211 |
+
\end{itemize}
|
| 212 |
+
\end{frame}
|
| 213 |
+
|
| 214 |
+
|
| 215 |
+
\subsection{Time Cross Correlations}
|
| 216 |
+
|
| 217 |
+
\frame{\sectionpage}
|
| 218 |
+
\begin{frame}{Time Cross Correlations: Motivation}
|
| 219 |
+
\begin{itemize}[<+->]
|
| 220 |
+
|
| 221 |
+
\item Equilibrium statistical mechanics $\rightarrow$ thermodynamic properties: partition function of the ensemble
|
| 222 |
+
\item Nonequilibrium systems $\rightarrow$ no unique partition function
|
| 223 |
+
\item System properties obtained via time cross-correlations
|
| 224 |
+
\end{itemize}
|
| 225 |
+
\uncover<+->{\begin{equation*}
|
| 226 |
+
\langle v(t) v(t^\prime) \rangle_\mathrm{time} = \frac{kT}{m}\exp^{-\gamma(t-t^\prime)/m} \label{time-cross}
|
| 227 |
+
\end{equation*}}
|
| 228 |
+
|
| 229 |
+
\uncover<+->{
|
| 230 |
+
\begin{alertblock}{Inception Model}
|
| 231 |
+
Can we design a neural network to learn time-cross correlations $\rightarrow$ predict cell-fate (Basal, Goblet, Radial)/action (Mitosis, Apoptosis) classfication.
|
| 232 |
+
\end{alertblock}
|
| 233 |
+
}
|
| 234 |
+
|
| 235 |
+
\end{frame}
|
| 236 |
+
|
| 237 |
+
\begin{frame}{Time Cross Correlations}
|
| 238 |
+
|
| 239 |
+
|
| 240 |
+
|
| 241 |
+
\uncover<+->{
|
| 242 |
+
Consider the vector of morphodynamic properties $\vec{A(t)}$, let us compute time average of this vector
|
| 243 |
+
\begin{equation*}
|
| 244 |
+
\langle \vec{A} \rangle = \frac{1}{\tau}\int_0^\tau dt \vec{A(t)}
|
| 245 |
+
\end{equation*}}
|
| 246 |
+
|
| 247 |
+
|
| 248 |
+
\uncover<+->{
|
| 249 |
+
Subtract the average to get the fluctuation $\delta \vec{A}$
|
| 250 |
+
\begin{equation*}
|
| 251 |
+
\delta \vec{A(t)} = \vec{A(t)} - \langle \vec{A} \rangle
|
| 252 |
+
\end{equation*}}
|
| 253 |
+
|
| 254 |
+
|
| 255 |
+
\uncover<+->{
|
| 256 |
+
The time-averaged product of two fluctuations
|
| 257 |
+
\begin{equation*}
|
| 258 |
+
C(t) = \frac{1}{\tau}\int_0^\tau ds \delta \vec{A}(s)\delta \vec{A}(t + s) \label{cross-corr}
|
| 259 |
+
\end{equation*}
|
| 260 |
+
$C(t)$: cross-correlation of the morphodynamic feature vector.
|
| 261 |
+
}
|
| 262 |
+
|
| 263 |
+
|
| 264 |
+
|
| 265 |
+
\end{frame}
|
| 266 |
+
|
| 267 |
+
\begin{frame}{Neural Network: Learning cross-correlations}
|
| 268 |
+
\begin{itemize}[<+->]
|
| 269 |
+
\item Construct a $(B,F,T)$ dimensional tensor, B: batch dimension, F: channels/features and T: time interval, take all the tracks belonging to a certain cell type $S$.
|
| 270 |
+
\item Inception Model: Customized $1D$ Densenet model with soft attention over time
|
| 271 |
+
\item Convolution operations in Densenet $=$ computing the cross correlations as 1D convolutions are mathematically the same.
|
| 272 |
+
\item Training data: several tracks of $25$ timepoints with $18$ features for each cell type $S$
|
| 273 |
+
\item Data is $Z$ scored to have zero mean and are normalized by the standard deviation of the feature vector (component wise) with $S$ being the ground truth label
|
| 274 |
+
\item Categorical Cross entropy as objective to train the network to classify trajectories as belonging to certain cell type.
|
| 275 |
+
\end{itemize}
|
| 276 |
+
\end{frame}
|
| 277 |
+
\begin{frame}{Neural Network: Learning cross-correlations}
|
| 278 |
+
\begin{columns} % Use columns for better layout
|
| 279 |
+
\begin{column}{0.5\textwidth} % Adjust width as needed
|
| 280 |
+
\includegraphics[width=0.8\linewidth, angle=-90]{presentation_images/1DCNN.png} % Use \linewidth to fit within the column
|
| 281 |
+
\end{column}
|
| 282 |
+
|
| 283 |
+
\begin{column}{0.5\textwidth}
|
| 284 |
+
|
| 285 |
+
\begin{itemize}[<+->]
|
| 286 |
+
\item Input 1D data with features as channels $\rightarrow$ 1D Densenet: a) Convolutional, b) Dense, c) Transition, d) Attention block.
|
| 287 |
+
\item Convolutions compute temporal correlations b/w features
|
| 288 |
+
\item Network output class: cell type, $S$ (Goblet, Basal, Radial)/ action class, $A$ (Mitosis, Non-Mitosis)
|
| 289 |
+
\end{itemize}
|
| 290 |
+
\end{column}
|
| 291 |
+
\end{columns}
|
| 292 |
+
\end{frame}
|
| 293 |
+
|
| 294 |
+
|
chapters/langevin-equation.log
ADDED
|
The diff for this file is too large to render.
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|
|
|
chapters/langevin-equation.tex
ADDED
|
@@ -0,0 +1,367 @@
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| 1 |
+
\section{Langevin Equation for macroscopic properties}
|
| 2 |
+
|
| 3 |
+
\subsection{Brownian Motion}
|
| 4 |
+
\frame{\sectionpage}
|
| 5 |
+
|
| 6 |
+
\begin{frame}{Brownian Particle as real Particle}
|
| 7 |
+
\uncover<+->{\begin{equation*}
|
| 8 |
+
\frac{d}{dt}\left( \frac{\partial L}{ \partial \dot{q}} \right) = \frac{\partial L}{ \partial q} - \frac{\partial R}{ \partial \dot{q}} + \sqrt{2B}\Gamma(t) \label{Lagrange-eqs}
|
| 9 |
+
\end{equation*}}
|
| 10 |
+
|
| 11 |
+
\uncover<+->{\begin{equation*}
|
| 12 |
+
L = \frac{1}{2}m\dot{q}^2 - V(q) \label{Lagrangian-real}
|
| 13 |
+
\end{equation*}}
|
| 14 |
+
|
| 15 |
+
\uncover<+->{\begin{equation*}m \frac{d\dot{q}}{dt} = -\frac{\partial V}{\partial q} -\gamma\dot{q} + \sqrt{2B}\Gamma(t)
|
| 16 |
+
\end{equation*}}
|
| 17 |
+
\uncover<+->{\begin{equation*}\Phi(q,t) = -\frac{\partial V}{\partial q} -\gamma\dot{q}
|
| 18 |
+
\end{equation*}}
|
| 19 |
+
\uncover<+->
|
| 20 |
+
{\begin{equation*}
|
| 21 |
+
\frac{dv}{dt} = \frac{\Phi(q,t)}{m} + \frac{\sqrt{2B}}{m} \Gamma(t),
|
| 22 |
+
\end{equation*}}
|
| 23 |
+
\end{frame}
|
| 24 |
+
\begin{frame}{Obtaining deterministic and random force}
|
| 25 |
+
|
| 26 |
+
\uncover<+->
|
| 27 |
+
{\begin{equation*}
|
| 28 |
+
v(t+dt) - v(t) = \int_{t}^{t+dt} \frac{\Phi(q,t)}{m}dt + \int_{t}^{t+dt} \frac{\sqrt{2B}}{m} \Gamma(t) dt
|
| 29 |
+
\end{equation*}}
|
| 30 |
+
|
| 31 |
+
\uncover<+->
|
| 32 |
+
{\begin{equation*}
|
| 33 |
+
\langle \Gamma(t) \rangle = 0, \quad \langle \Gamma(t) \Gamma(t_1) \rangle = \delta(t - t_1).
|
| 34 |
+
\end{equation*}}
|
| 35 |
+
|
| 36 |
+
|
| 37 |
+
\uncover<+->
|
| 38 |
+
{\begin{equation*}
|
| 39 |
+
\Phi(q,t) = m \frac{ \langle v(t+dt) -v(t) \rangle}{dt}
|
| 40 |
+
\end{equation*}}
|
| 41 |
+
|
| 42 |
+
|
| 43 |
+
\uncover<+->
|
| 44 |
+
{\begin{equation*}
|
| 45 |
+
B = \frac{m^2}{2} \frac{\langle (v(t+dt) - v(t))^2 \rangle}{dt} - \frac{\Phi(q,t) ^2}{2} dt
|
| 46 |
+
\end{equation*}}
|
| 47 |
+
|
| 48 |
+
\end{frame}
|
| 49 |
+
|
| 50 |
+
\begin{frame}{Macroscopic properties as Brownian Particles}
|
| 51 |
+
\uncover<+->{\begin{equation*}
|
| 52 |
+
\vec{a} = ( a_1, \cdots, a_n )
|
| 53 |
+
\end{equation*}}
|
| 54 |
+
\uncover<+->{
|
| 55 |
+
\begin{equation*}
|
| 56 |
+
L = \frac{1}{2}\dot{a}^2 - V(a) \label{Lagrangian-cell}
|
| 57 |
+
\end{equation*}}
|
| 58 |
+
\uncover<+->{
|
| 59 |
+
\begin{equation*}
|
| 60 |
+
\frac{d\dot{a}}{dt} = -\frac{\partial V}{\partial a} -\gamma\dot{a} + \sqrt{2B}\Gamma(t)
|
| 61 |
+
\end{equation*}}
|
| 62 |
+
\uncover<+->{
|
| 63 |
+
\begin{equation*}
|
| 64 |
+
B_i(a,t) = \frac{\langle (a_i(t+dt) - a_i(t))^2 \rangle}{2dt} - \frac{\Phi_i(a,t) ^2}{2} dt \label{diffusion-constant}
|
| 65 |
+
\end{equation*}}
|
| 66 |
+
|
| 67 |
+
\uncover<+->{
|
| 68 |
+
\begin{equation*}
|
| 69 |
+
\Phi_i(a,t) = \frac{ \langle a_i(t+dt) -a_i(t) \rangle}{dt} \label{force-constant}
|
| 70 |
+
\end{equation*}}
|
| 71 |
+
|
| 72 |
+
|
| 73 |
+
|
| 74 |
+
\end{frame}
|
| 75 |
+
|
| 76 |
+
|
| 77 |
+
\begin{frame}{Obtaining deterministic and random force}
|
| 78 |
+
\begin{figure}
|
| 79 |
+
\centering
|
| 80 |
+
% Adjust the width and spacing as needed
|
| 81 |
+
\includegraphics[width=0.3\textwidth]{sixth_phase/nuclei_predicted_morphospaces/Derivative_Plots_of_ALL_PCs_Across_Cell_Types_all_PCs_Sixth_nuclei_density_stats.png}
|
| 82 |
+
\hspace{0.5cm}
|
| 83 |
+
\includegraphics[width=0.3\textwidth]{sixth_phase/nuclei_predicted_morphospaces/Diffusion_Constant_Plots_of_ALL_PCs_Across_Cell_Types_all_PCs_Sixth_nuclei_density_stats.png}
|
| 84 |
+
|
| 85 |
+
\end{figure}
|
| 86 |
+
\textbf{Description:} \\
|
| 87 |
+
Mean and Variance for Deterministic and random force, nuclei channel, Sixth Dataset (all principal components).
|
| 88 |
+
\begin{alertblock}{State Transition}
|
| 89 |
+
For each cell type, the deterministic and random force strength decreases with time, system transitions to be less 'fluidic'
|
| 90 |
+
\end{alertblock}
|
| 91 |
+
\end{frame}
|
| 92 |
+
|
| 93 |
+
|
| 94 |
+
|
| 95 |
+
\begin{frame}{Obtaining deterministic and random force}
|
| 96 |
+
\begin{figure}
|
| 97 |
+
\centering
|
| 98 |
+
% Adjust the width and spacing as needed
|
| 99 |
+
\includegraphics[width=0.3\textwidth]{sixth_phase/membrane_predicted_morphospaces/Derivative_Plots_of_ALL_PCs_Across_Cell_Types_all_PCs_Sixth_membrane_density_stats.png}
|
| 100 |
+
\hspace{0.5cm}
|
| 101 |
+
\includegraphics[width=0.3\textwidth]{sixth_phase/membrane_predicted_morphospaces/Diffusion_Constant_Plots_of_ALL_PCs_Across_Cell_Types_all_PCs_Sixth_membrane_density_stats.png}
|
| 102 |
+
|
| 103 |
+
\end{figure}
|
| 104 |
+
\textbf{Description:} \\
|
| 105 |
+
Mean and Variance for Deterministic and random force, membrane channel, Sixth Dataset (all principal components).
|
| 106 |
+
\begin{alertblock}{State Transition}
|
| 107 |
+
For each cell type, the deterministic and random force strength decreases with time, system transitions to be less 'fluidic'
|
| 108 |
+
\end{alertblock}
|
| 109 |
+
\end{frame}
|
| 110 |
+
|
| 111 |
+
|
| 112 |
+
|
| 113 |
+
\subsection{Ergodic Test: Quantifying $t_{eq}$}
|
| 114 |
+
|
| 115 |
+
\frame{\sectionpage}
|
| 116 |
+
|
| 117 |
+
|
| 118 |
+
\begin{frame}{Ergodic}
|
| 119 |
+
\begin{itemize}[<+->]
|
| 120 |
+
|
| 121 |
+
\uncover<+->{
|
| 122 |
+
\item Time Average:
|
| 123 |
+
\[
|
| 124 |
+
\langle A \rangle_T = \lim_{T \to \infty} \frac{1}{T} \int_0^T A(t) \, dt
|
| 125 |
+
\]}
|
| 126 |
+
|
| 127 |
+
\uncover<+->{
|
| 128 |
+
\item Ensemble Average:
|
| 129 |
+
\[
|
| 130 |
+
\langle A \rangle_E = \frac{1}{N} \sum_{i=1}^{N} A_i
|
| 131 |
+
\]
|
| 132 |
+
where \( A_i \) is the value of the observable for the \( i \)-th system.}
|
| 133 |
+
|
| 134 |
+
|
| 135 |
+
\item
|
| 136 |
+
\uncover<+->{\[
|
| 137 |
+
\text{Ergodicity: } \langle A \rangle_T = \langle A \rangle_E \quad \text{for all relevant observables } A
|
| 138 |
+
\]}
|
| 139 |
+
\end{itemize}
|
| 140 |
+
|
| 141 |
+
\end{frame}
|
| 142 |
+
|
| 143 |
+
\begin{frame}{Ergodicity for cell types: Quantifying $t_{eq}$}
|
| 144 |
+
|
| 145 |
+
\begin{itemize}
|
| 146 |
+
\item \uncover<+->{For each cell track of Basal, Radial, Goblet type.}
|
| 147 |
+
\item \uncover<+->{Use $f_i(t)$: feature values for track i at time t and $T$ in the interval $\Delta t$.}
|
| 148 |
+
\item
|
| 149 |
+
\uncover<+->{
|
| 150 |
+
Ensemble Average
|
| 151 |
+
\begin{equation*}
|
| 152 |
+
\langle f \rangle_{\mathrm{ensemble}, t} = \frac{1}{N}\sum_{i=1}^N f_i(t)
|
| 153 |
+
\end{equation*}
|
| 154 |
+
}
|
| 155 |
+
\item
|
| 156 |
+
\uncover<+->{
|
| 157 |
+
Time Average
|
| 158 |
+
\begin{equation*}
|
| 159 |
+
\langle f \rangle_{\mathrm{time}, i} = \frac{1}{\Delta t} \sum_{t\in T} f_i(t)
|
| 160 |
+
\end{equation*}
|
| 161 |
+
}
|
| 162 |
+
\item
|
| 163 |
+
\uncover<+->{
|
| 164 |
+
Ergodic Test:
|
| 165 |
+
\begin{equation*}
|
| 166 |
+
\Delta f = \arrowvert\sum_i \langle f\rangle_{\mathrm{time}, i} - \sum_t \langle f \rangle_{\mathrm{ensemble},t} \arrowvert \label{ergodic_test}
|
| 167 |
+
\end{equation*}
|
| 168 |
+
}
|
| 169 |
+
\end{itemize}
|
| 170 |
+
|
| 171 |
+
|
| 172 |
+
|
| 173 |
+
|
| 174 |
+
\end{frame}
|
| 175 |
+
|
| 176 |
+
\begin{frame}{Ergodicity for cell types: Quantifying $t_{eq}$}
|
| 177 |
+
\begin{itemize}
|
| 178 |
+
\item \uncover<+->{
|
| 179 |
+
For numerical computation take T = 20, compute $\Delta f$ for each cell type.
|
| 180 |
+
\item $\Delta f(t)$ for $[20,40,\cdots T]$, each cell type $\rightarrow$ discrete plot time and ensemble average.
|
| 181 |
+
\item Initially system for each cell type is in non-equilibrium $\rightarrow$ non-zero gap b/w time-ensemble average.
|
| 182 |
+
\item As system becomes ergodic the gap reduces, $\Delta f_s < \epsilon$, $s$ being cell type.
|
| 183 |
+
\item $t_{eq}$: Critical time when gap reduces to $\epsilon$ for $s$, $t_{eq}(s)$.
|
| 184 |
+
}
|
| 185 |
+
\end{itemize}
|
| 186 |
+
|
| 187 |
+
\end{frame}
|
| 188 |
+
|
| 189 |
+
\begin{frame}{Ergodicity for cell types: Quantifying $t_{eq}$}
|
| 190 |
+
|
| 191 |
+
\begin{figure}
|
| 192 |
+
\centering
|
| 193 |
+
% Adjust the width and spacing as needed
|
| 194 |
+
\includegraphics[width=0.45\textwidth]{sixth_phase/nuclei_predicted_phasespaces/ergodicity_differences_Basal.png}
|
| 195 |
+
\hspace{0.5cm}
|
| 196 |
+
\includegraphics[width=0.45\textwidth]{sixth_phase/nuclei_predicted_phasespaces/ergodicity_differences_Radial.png}
|
| 197 |
+
\hspace{0.5cm}
|
| 198 |
+
\includegraphics[width=0.45\textwidth]{sixth_phase/nuclei_predicted_phasespaces/ergodicity_differences_Goblet.png}
|
| 199 |
+
\end{figure}
|
| 200 |
+
|
| 201 |
+
\vspace{0.5cm} % Add some vertical space for readability
|
| 202 |
+
|
| 203 |
+
\textbf{Description:} \\
|
| 204 |
+
Time and Ensemble average difference plots, Sixth Dataset.
|
| 205 |
+
\end{frame}
|
| 206 |
+
|
| 207 |
+
\subsection{Phase Space}
|
| 208 |
+
|
| 209 |
+
\frame{\sectionpage}
|
| 210 |
+
\begin{frame}{Trajectories in Phase Space}
|
| 211 |
+
\begin{itemize}
|
| 212 |
+
|
| 213 |
+
\item \uncover<+->{The force controls evolution of morphodynamic features}
|
| 214 |
+
\item \uncover<+->{How do we determine the potential of force for each cell type?}
|
| 215 |
+
\item \uncover<+->{Visualizing the evolution of trajectories in phase space would offer the clue.}
|
| 216 |
+
\item \uncover<+->{Phase space is described by each generalized co-ordinate and it's conjugate momentum.
|
| 217 |
+
\begin{equation*}
|
| 218 |
+
\vec{a} = ( a_1, \cdots, a_n )
|
| 219 |
+
\end{equation*}
|
| 220 |
+
\begin{equation*}
|
| 221 |
+
L = \frac{1}{2}\dot{a}^2 - V(a)
|
| 222 |
+
\end{equation*}
|
| 223 |
+
\begin{equation*}
|
| 224 |
+
\vec{p} = \left (\frac{\partial L}{\partial a_1},\cdots,\frac{\partial L}{\partial a_n}\right )
|
| 225 |
+
\end{equation*}
|
| 226 |
+
}
|
| 227 |
+
\end{itemize}
|
| 228 |
+
\end{frame}
|
| 229 |
+
|
| 230 |
+
\begin{frame}{Trajectories in Phase Space: Nuclei}
|
| 231 |
+
|
| 232 |
+
\begin{figure}
|
| 233 |
+
\centering
|
| 234 |
+
% Adjust the width and spacing as needed
|
| 235 |
+
\includegraphics[width=0.2\textwidth]{sixth_phase/nuclei_predicted_phasespaces/phasespace_Goblet_Eccentricity_Comp_First_dataset_phasespace.png}
|
| 236 |
+
\hspace{0.5cm}
|
| 237 |
+
\includegraphics[width=0.2\textwidth]{sixth_phase/nuclei_predicted_phasespaces/phasespace_Goblet_Eccentricity_Comp_Third_dataset_phasespace.png}
|
| 238 |
+
\hspace{0.5cm}
|
| 239 |
+
\includegraphics[width=0.2\textwidth]{sixth_phase/nuclei_predicted_phasespaces/phasespace_Goblet_Speed_dataset_phasespace.png}
|
| 240 |
+
\hspace{0.5cm}
|
| 241 |
+
\includegraphics[width=0.2\textwidth]{sixth_phase/nuclei_predicted_phasespaces/phasespace_Goblet_Surface_Area_dataset_phasespace.png}
|
| 242 |
+
\end{figure}
|
| 243 |
+
|
| 244 |
+
\vspace{0.5cm} % Add some vertical space for readability
|
| 245 |
+
|
| 246 |
+
\textbf{Description:} \\
|
| 247 |
+
Phase space plots Goblet Cells, nuclei channel, Sixth Dataset.
|
| 248 |
+
\begin{alertblock}{Potential of Force}
|
| 249 |
+
The potential corresponding to deterministic force corresponds to a Harmonic potential ($\frac{1}{2}ka_i^ 2$)
|
| 250 |
+
\end{alertblock}
|
| 251 |
+
\end{frame}
|
| 252 |
+
|
| 253 |
+
\begin{frame}{Trajectories in Phase Space: Nuclei}
|
| 254 |
+
|
| 255 |
+
\begin{figure}
|
| 256 |
+
\centering
|
| 257 |
+
% Adjust the width and spacing as needed
|
| 258 |
+
\includegraphics[width=0.2\textwidth]{sixth_phase/nuclei_predicted_phasespaces/phasespace_Radial_Eccentricity_Comp_First_dataset_phasespace.png}
|
| 259 |
+
\hspace{0.5cm}
|
| 260 |
+
\includegraphics[width=0.2\textwidth]{sixth_phase/nuclei_predicted_phasespaces/phasespace_Radial_Eccentricity_Comp_Third_dataset_phasespace.png}
|
| 261 |
+
\hspace{0.5cm}
|
| 262 |
+
\includegraphics[width=0.2\textwidth]{sixth_phase/nuclei_predicted_phasespaces/phasespace_Radial_Speed_dataset_phasespace.png}
|
| 263 |
+
\hspace{0.5cm}
|
| 264 |
+
\includegraphics[width=0.2\textwidth]{sixth_phase/nuclei_predicted_phasespaces/phasespace_Radial_Surface_Area_dataset_phasespace.png}
|
| 265 |
+
\end{figure}
|
| 266 |
+
|
| 267 |
+
\vspace{0.5cm} % Add some vertical space for readability
|
| 268 |
+
|
| 269 |
+
\textbf{Description:} \\
|
| 270 |
+
Phase space plots Radial Cells, nuclei channel, Sixth Dataset.
|
| 271 |
+
\begin{alertblock}{Potential of Force}
|
| 272 |
+
The potential corresponding to deterministic force corresponds to a Harmonic potential ($\frac{1}{2}ka_i^ 2$)
|
| 273 |
+
\end{alertblock}
|
| 274 |
+
\end{frame}
|
| 275 |
+
|
| 276 |
+
\begin{frame}{Trajectories in Phase Space: Nuclei}
|
| 277 |
+
|
| 278 |
+
\begin{figure}
|
| 279 |
+
\centering
|
| 280 |
+
% Adjust the width and spacing as needed
|
| 281 |
+
\includegraphics[width=0.2\textwidth]{sixth_phase/nuclei_predicted_phasespaces/phasespace_Basal_Eccentricity_Comp_First_dataset_phasespace.png}
|
| 282 |
+
\hspace{0.5cm}
|
| 283 |
+
\includegraphics[width=0.2\textwidth]{sixth_phase/nuclei_predicted_phasespaces/phasespace_Basal_Eccentricity_Comp_Third_dataset_phasespace.png}
|
| 284 |
+
\hspace{0.5cm}
|
| 285 |
+
\includegraphics[width=0.2\textwidth]{sixth_phase/nuclei_predicted_phasespaces/phasespace_Basal_Speed_dataset_phasespace.png}
|
| 286 |
+
\hspace{0.5cm}
|
| 287 |
+
\includegraphics[width=0.2\textwidth]{sixth_phase/nuclei_predicted_phasespaces/phasespace_Basal_Surface_Area_dataset_phasespace.png}
|
| 288 |
+
\end{figure}
|
| 289 |
+
|
| 290 |
+
\vspace{0.5cm} % Add some vertical space for readability
|
| 291 |
+
|
| 292 |
+
\textbf{Description:} \\
|
| 293 |
+
Phase space plots Basal Cells, nuclei channel, Sixth Dataset.
|
| 294 |
+
\begin{alertblock}{Potential of Force}
|
| 295 |
+
The potential corresponding to deterministic force corresponds to a Harmonic potential ($\frac{1}{2}ka_i^ 2$)
|
| 296 |
+
\end{alertblock}
|
| 297 |
+
\end{frame}
|
| 298 |
+
|
| 299 |
+
|
| 300 |
+
\begin{frame}{Trajectories in Phase Space: Membrane}
|
| 301 |
+
|
| 302 |
+
\begin{figure}
|
| 303 |
+
\centering
|
| 304 |
+
% Adjust the width and spacing as needed
|
| 305 |
+
\includegraphics[width=0.2\textwidth]{sixth_phase/membrane_predicted_phasespaces/phasespace_Goblet_Eccentricity_Comp_First_dataset_phasespace.png}
|
| 306 |
+
\hspace{0.5cm}
|
| 307 |
+
\includegraphics[width=0.2\textwidth]{sixth_phase/membrane_predicted_phasespaces/phasespace_Goblet_Eccentricity_Comp_Third_dataset_phasespace.png}
|
| 308 |
+
\hspace{0.5cm}
|
| 309 |
+
\includegraphics[width=0.2\textwidth]{sixth_phase/membrane_predicted_phasespaces/phasespace_Goblet_Speed_dataset_phasespace.png}
|
| 310 |
+
\hspace{0.5cm}
|
| 311 |
+
\includegraphics[width=0.2\textwidth]{sixth_phase/membrane_predicted_phasespaces/phasespace_Goblet_Surface_Area_dataset_phasespace.png}
|
| 312 |
+
\end{figure}
|
| 313 |
+
|
| 314 |
+
\vspace{0.5cm} % Add some vertical space for readability
|
| 315 |
+
|
| 316 |
+
\textbf{Description:} \\
|
| 317 |
+
Phase space plots Goblet Cells, membrane channel, Sixth Dataset.
|
| 318 |
+
\begin{alertblock}{Potential of Force}
|
| 319 |
+
The potential corresponding to deterministic force corresponds to a Harmonic potential of higher order, e.g ($\frac{1}{2}k(a_i^ 2 - b)^ 2$), such a potential has minima at b
|
| 320 |
+
\end{alertblock}
|
| 321 |
+
\end{frame}
|
| 322 |
+
|
| 323 |
+
\begin{frame}{Trajectories in Phase Space: Membrane}
|
| 324 |
+
|
| 325 |
+
\begin{figure}
|
| 326 |
+
\centering
|
| 327 |
+
% Adjust the width and spacing as needed
|
| 328 |
+
\includegraphics[width=0.2\textwidth]{sixth_phase/membrane_predicted_phasespaces/phasespace_Radial_Eccentricity_Comp_First_dataset_phasespace.png}
|
| 329 |
+
\hspace{0.5cm}
|
| 330 |
+
\includegraphics[width=0.2\textwidth]{sixth_phase/membrane_predicted_phasespaces/phasespace_Radial_Eccentricity_Comp_Third_dataset_phasespace.png}
|
| 331 |
+
\hspace{0.5cm}
|
| 332 |
+
\includegraphics[width=0.2\textwidth]{sixth_phase/membrane_predicted_phasespaces/phasespace_Radial_Speed_dataset_phasespace.png}
|
| 333 |
+
\hspace{0.5cm}
|
| 334 |
+
\includegraphics[width=0.2\textwidth]{sixth_phase/membrane_predicted_phasespaces/phasespace_Radial_Surface_Area_dataset_phasespace.png}
|
| 335 |
+
\end{figure}
|
| 336 |
+
|
| 337 |
+
\vspace{0.5cm} % Add some vertical space for readability
|
| 338 |
+
|
| 339 |
+
\textbf{Description:} \\
|
| 340 |
+
Phase space plots Radial Cells, membrane channel, Sixth Dataset.
|
| 341 |
+
\begin{alertblock}{Potential of Force}
|
| 342 |
+
The potential corresponding to deterministic force corresponds to a Harmonic potential of higher order, e.g ($\frac{1}{2}k(a_i^ 2 - b)^ 2$), such a potential has minima at b
|
| 343 |
+
\end{alertblock}
|
| 344 |
+
\end{frame}
|
| 345 |
+
|
| 346 |
+
\begin{frame}{Trajectories in Phase Space: Membrane}
|
| 347 |
+
|
| 348 |
+
\begin{figure}
|
| 349 |
+
\centering
|
| 350 |
+
% Adjust the width and spacing as needed
|
| 351 |
+
\includegraphics[width=0.2\textwidth]{sixth_phase/membrane_predicted_phasespaces/phasespace_Basal_Eccentricity_Comp_First_dataset_phasespace.png}
|
| 352 |
+
\hspace{0.5cm}
|
| 353 |
+
\includegraphics[width=0.2\textwidth]{sixth_phase/membrane_predicted_phasespaces/phasespace_Basal_Eccentricity_Comp_Third_dataset_phasespace.png}
|
| 354 |
+
\hspace{0.5cm}
|
| 355 |
+
\includegraphics[width=0.2\textwidth]{sixth_phase/membrane_predicted_phasespaces/phasespace_Basal_Speed_dataset_phasespace.png}
|
| 356 |
+
\hspace{0.5cm}
|
| 357 |
+
\includegraphics[width=0.2\textwidth]{sixth_phase/membrane_predicted_phasespaces/phasespace_Basal_Surface_Area_dataset_phasespace.png}
|
| 358 |
+
\end{figure}
|
| 359 |
+
|
| 360 |
+
\vspace{0.5cm} % Add some vertical space for readability
|
| 361 |
+
|
| 362 |
+
\textbf{Description:} \\
|
| 363 |
+
Phase space plots Basal Cells, membrane channel, Sixth Dataset.
|
| 364 |
+
\begin{alertblock}{Potential of Force}
|
| 365 |
+
The potential corresponding to deterministic force corresponds to a Harmonic potential of higher order, e.g ($\frac{1}{2}k(a_i^ 2 - b)^ 2$), such a potential has minima at b
|
| 366 |
+
\end{alertblock}
|
| 367 |
+
\end{frame}
|
chapters/preamble.tex
ADDED
|
@@ -0,0 +1,108 @@
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|
| 1 |
+
\usepackage{amsthm}
|
| 2 |
+
\usepackage{mathtools}
|
| 3 |
+
\usepackage{physics}
|
| 4 |
+
\usepackage{calligra}
|
| 5 |
+
\usepackage{csquotes}
|
| 6 |
+
\usepackage{tensor}
|
| 7 |
+
\usepackage[thicklines]{cancel}
|
| 8 |
+
\usepackage{tcolorbox}
|
| 9 |
+
\usepackage{pstricks}
|
| 10 |
+
|
| 11 |
+
|
| 12 |
+
\DeclareMathAlphabet{\mathcalligra}{T1}{calligra}{m}{n}
|
| 13 |
+
\DeclareFontShape{T1}{calligra}{m}{n}{<->s*[2.2]callig15}{}
|
| 14 |
+
\newcommand{\scriptr}{\mathcalligra{r}\,}
|
| 15 |
+
\newcommand{\boldscriptr}{\pmb{\mathcalligra{r}}\,}
|
| 16 |
+
\def\rc{\scriptr}
|
| 17 |
+
\def\brc{\boldscriptr}
|
| 18 |
+
\def\hrc{\hat\brc}
|
| 19 |
+
\newcommand{\ie}{\emph{i.e.}} %id est
|
| 20 |
+
\newcommand{\eg}{\emph{e.g.}} %exempli gratia
|
| 21 |
+
\newcommand{\rtd}[1]{\ensuremath{\left\lfloor #1 \right\rfloor}}
|
| 22 |
+
\newcommand{\dirac}[1]{\ensuremath{\delta \left( #1 \right)}}
|
| 23 |
+
\newcommand{\diract}[1]{\ensuremath{\delta^3 \left( #1 \right)}}
|
| 24 |
+
\newcommand{\e}{\ensuremath{\epsilon_0}}
|
| 25 |
+
\newcommand{\m}{\ensuremath{\mu_0}}
|
| 26 |
+
\newcommand{\V}{\ensuremath{\mathcal{V}}}
|
| 27 |
+
\newcommand{\prnt}[1]{\ensuremath{\left(#1\right)}} %parentheses
|
| 28 |
+
\newcommand{\colch}[1]{\ensuremath{\left[#1\right]}} %square brackets
|
| 29 |
+
\newcommand{\chave}[1]{\ensuremath{\left\{#1\right\}}} %curly brackets
|
| 30 |
+
|
| 31 |
+
\useoutertheme{infolines}
|
| 32 |
+
\useinnertheme{rectangles}
|
| 33 |
+
\usefonttheme{professionalfonts}
|
| 34 |
+
|
| 35 |
+
|
| 36 |
+
\definecolor{orange}{HTML}{f28165}
|
| 37 |
+
\definecolor{gray}{HTML}{303030}
|
| 38 |
+
\definecolor{yellow}{HTML}{f0be52}
|
| 39 |
+
\definecolor{lightorange}{HTML}{f19e58}
|
| 40 |
+
|
| 41 |
+
\renewcommand{\CancelColor}{\color{orange}}
|
| 42 |
+
|
| 43 |
+
\makeatletter
|
| 44 |
+
\newcommand{\mybox}[1]{%
|
| 45 |
+
\setbox0=\hbox{#1}%
|
| 46 |
+
\setlength{\@tempdima}{\dimexpr\wd0+13pt}%
|
| 47 |
+
\begin{tcolorbox}[colback=orange,colframe=orange,boxrule=0.5pt,arc=4pt,
|
| 48 |
+
left=6pt,right=6pt,top=6pt,bottom=6pt,boxsep=0pt,width=\@tempdima]
|
| 49 |
+
\textcolor{white}{#1}
|
| 50 |
+
\end{tcolorbox}
|
| 51 |
+
}
|
| 52 |
+
\makeatother
|
| 53 |
+
|
| 54 |
+
\usecolortheme[named=orange]{structure}
|
| 55 |
+
\usecolortheme{sidebartab}
|
| 56 |
+
\usecolortheme{orchid}
|
| 57 |
+
\usecolortheme{whale}
|
| 58 |
+
\setbeamercolor{alerted text}{fg=yellow}
|
| 59 |
+
\setbeamercolor{block title alerted}{bg=alerted text.fg!90!black}
|
| 60 |
+
\setbeamercolor{block title example}{bg=lightorange!60!black}
|
| 61 |
+
\setbeamercolor{background canvas}{bg=gray}
|
| 62 |
+
\setbeamercolor{normal text}{bg=gray,fg=white}
|
| 63 |
+
|
| 64 |
+
\setbeamertemplate{footline}
|
| 65 |
+
{
|
| 66 |
+
\leavevmode%
|
| 67 |
+
\hbox{%
|
| 68 |
+
\begin{beamercolorbox}[wd=.333333\paperwidth,ht=2.25ex,dp=1ex,center]{author in head/foot}%
|
| 69 |
+
\usebeamerfont{author in head/foot}\insertshortauthor~~(\insertshortinstitute)
|
| 70 |
+
\end{beamercolorbox}%
|
| 71 |
+
\begin{beamercolorbox}[wd=.333333\paperwidth,ht=2.25ex,dp=1ex,center]{title in head/foot}%
|
| 72 |
+
\usebeamerfont{title in head/foot}\insertshorttitle
|
| 73 |
+
\end{beamercolorbox}%
|
| 74 |
+
\begin{beamercolorbox}[wd=.333333\paperwidth,ht=2.25ex,dp=1ex,center]{date in head/foot}%
|
| 75 |
+
\usebeamerfont{date in head/foot}\insertshortdate{}%\hspace*{2em}
|
| 76 |
+
|
| 77 |
+
%#turning the next line into a comment, erases the frame numbers
|
| 78 |
+
%\insertframenumber{} / \inserttotalframenumber\hspace*{2ex}
|
| 79 |
+
|
| 80 |
+
\end{beamercolorbox}}%
|
| 81 |
+
\vskip0pt%
|
| 82 |
+
}
|
| 83 |
+
|
| 84 |
+
|
| 85 |
+
\setbeamertemplate{blocks}[rectangle]
|
| 86 |
+
\setbeamercovered{dynamic}
|
| 87 |
+
|
| 88 |
+
\setbeamertemplate{section page}
|
| 89 |
+
{
|
| 90 |
+
\begin{centering}
|
| 91 |
+
\begin{beamercolorbox}[sep=27pt,center]{part title}
|
| 92 |
+
\usebeamerfont{section title}\insertsection\par
|
| 93 |
+
\usebeamerfont{subsection title}\insertsubsection\par
|
| 94 |
+
\end{beamercolorbox}
|
| 95 |
+
\end{centering}
|
| 96 |
+
}
|
| 97 |
+
|
| 98 |
+
%\setbeamertemplate{subsection page}
|
| 99 |
+
%{
|
| 100 |
+
% \begin{centering}
|
| 101 |
+
% \begin{beamercolorbox}[sep=12pt,center]{part title}
|
| 102 |
+
% \usebeamerfont{subsection title}\insertsubsection\par
|
| 103 |
+
% \end{beamercolorbox}
|
| 104 |
+
% \end{centering}
|
| 105 |
+
%}
|
| 106 |
+
|
| 107 |
+
\newcommand{\hlight}[1]{\colorbox{violet!50}{#1}}
|
| 108 |
+
\newcommand{\hlighta}[1]{\colorbox{red!50}{#1}}
|
chapters/xenopus-equilibrium.tex
ADDED
|
@@ -0,0 +1,71 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
|
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|
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|
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|
|
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|
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|
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|
|
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|
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|
|
|
|
|
|
|
|
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|
|
|
|
|
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|
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|
|
|
|
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|
|
|
|
|
|
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|
|
|
|
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|
|
|
|
|
|
|
|
|
|
|
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|
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|
|
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|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
\section{Experimental Results}
|
| 2 |
+
|
| 3 |
+
\frame{\sectionpage}
|
| 4 |
+
|
| 5 |
+
\begin{frame}{Nuclei Channel: Forces}
|
| 6 |
+
\only<-3>{\uncover<+->{\begin{equation*}
|
| 7 |
+
f(t) = \cos(\omega_0 t)e^{-\pi t^2}
|
| 8 |
+
\end{equation*}}
|
| 9 |
+
\uncover<+->{\begin{equation*}
|
| 10 |
+
\widehat{f}(\omega) = \frac{e^{-\frac{(\omega - \omega_0)^2}{4\pi}} + e^{-\frac{(\omega + \omega_0)^2}{4\pi}}}{2\sqrt{2\pi}}
|
| 11 |
+
\end{equation*}}
|
| 12 |
+
\uncover<+->{\begin{equation*}
|
| 13 |
+
\omega = 2\pi\nu
|
| 14 |
+
\end{equation*}}}
|
| 15 |
+
\only<4>{\begin{equation*}
|
| 16 |
+
f(t) = \cos(2\pi\nu_0 t)e^{-\pi t^2}
|
| 17 |
+
\end{equation*}
|
| 18 |
+
\begin{equation*}
|
| 19 |
+
\widehat{f}(\nu) = \frac{e^{-\pi(\nu - \nu_0)^2} + e^{-\pi(\nu + \nu_0)^2}}{2\sqrt{2\pi}}
|
| 20 |
+
\end{equation*}}
|
| 21 |
+
\end{frame}
|
| 22 |
+
|
| 23 |
+
\begin{frame}{Nuclei Channel: $T_{eq}$}
|
| 24 |
+
\begin{equation*}
|
| 25 |
+
f(t) = \cos(2\pi\nu_0 t)e^{-\pi t^2}
|
| 26 |
+
\end{equation*}
|
| 27 |
+
\centering
|
| 28 |
+
|
| 29 |
+
\includegraphics[height = 0.7 \textheight]{images/Pulse1.pdf}
|
| 30 |
+
\end{frame}
|
| 31 |
+
|
| 32 |
+
\begin{frame}{Nuclei Channel: Phase Space}
|
| 33 |
+
\begin{equation*}
|
| 34 |
+
\widehat{f}(\nu) = \frac{e^{-\pi(\nu - \nu_0)^2} + e^{-\pi(\nu + \nu_0)^2}}{2\sqrt{2\pi}}
|
| 35 |
+
\end{equation*}
|
| 36 |
+
\centering
|
| 37 |
+
|
| 38 |
+
\includegraphics[height = 0.65 \textheight]{images/Pulse1-Fourier.pdf}
|
| 39 |
+
\end{frame}
|
| 40 |
+
|
| 41 |
+
\begin{frame}{Membrane Channel: Forces}
|
| 42 |
+
\uncover<+->{\begin{equation*}
|
| 43 |
+
f(t) = e^{i\omega_0 t} = \cos(\omega_0 t) + i \sin(\omega_0 t)
|
| 44 |
+
\end{equation*}}
|
| 45 |
+
\uncover<+->{\begin{equation*}
|
| 46 |
+
\widehat{f}(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} e^{i\omega_0 t} e^{-i \omega t} \dd{t}
|
| 47 |
+
\end{equation*}}
|
| 48 |
+
\end{frame}
|
| 49 |
+
|
| 50 |
+
\begin{frame}{Membrane Channel: $T_{eq}$}
|
| 51 |
+
\uncover<+->{\begin{equation*}
|
| 52 |
+
f(t) = e^{i\omega_0 t}
|
| 53 |
+
\end{equation*}}
|
| 54 |
+
\uncover<+->{\begin{equation*}
|
| 55 |
+
e^{i\omega_0 t} = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} \widehat{f}(\omega) e^{i \omega t} \dd{\omega}
|
| 56 |
+
\end{equation*}}
|
| 57 |
+
\uncover<+->{\begin{equation*}
|
| 58 |
+
\widehat{f}(\omega) = \sqrt{2\pi} \dirac{\omega - \omega_0}
|
| 59 |
+
\end{equation*}}
|
| 60 |
+
\end{frame}
|
| 61 |
+
|
| 62 |
+
\begin{frame}{Membrane Channel: Phase Space}
|
| 63 |
+
\begin{equation*}
|
| 64 |
+
f(t) = \cos(\omega_0 t) = \frac{e^{i\omega_0t} + e^{-i\omega_0t}}{2}
|
| 65 |
+
\end{equation*}
|
| 66 |
+
|
| 67 |
+
\centering
|
| 68 |
+
\includegraphics[height = 0.65 \textheight]{images/Pulse2.pdf}
|
| 69 |
+
\end{frame}
|
| 70 |
+
|
| 71 |
+
|
images/Pulse1-Fourier.pdf
ADDED
|
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presentation.pdf
CHANGED
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| 1 |
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presentation.snm
CHANGED
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@@ -1,8 +1,2 @@
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| 1 |
\beamer@slide {time-cross}{50}
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| 2 |
\beamer@slide {cross-corr}{55}
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| 4 |
-
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| 5 |
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| 6 |
-
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| 7 |
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| 8 |
-
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| 1 |
\beamer@slide {time-cross}{50}
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| 2 |
\beamer@slide {cross-corr}{55}
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presentation.synctex.gz
CHANGED
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@@ -1,3 +1,3 @@
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| 1 |
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presentation.tex
CHANGED
|
@@ -22,7 +22,7 @@
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|
| 22 |
\end{frame}
|
| 23 |
|
| 24 |
\input{chapters/collecting-trajectories.tex}
|
| 25 |
-
|
| 26 |
\section*{Acknowledgments} %You can remove this if you do not want to use it
|
| 27 |
\begin{frame}{Acknowledgments}
|
| 28 |
The author is extremely thankful to Jean-Zay supercomputing support for this project.
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|
|
|
| 22 |
\end{frame}
|
| 23 |
|
| 24 |
\input{chapters/collecting-trajectories.tex}
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| 25 |
+
|
| 26 |
\section*{Acknowledgments} %You can remove this if you do not want to use it
|
| 27 |
\begin{frame}{Acknowledgments}
|
| 28 |
The author is extremely thankful to Jean-Zay supercomputing support for this project.
|
presentation.toc
CHANGED
|
@@ -8,7 +8,3 @@
|
|
| 8 |
\beamer@subsectionintoc {1}{6}{Point Cloud based morphodynamic features: NapaTrackMater}{32}{0}{1}
|
| 9 |
\beamer@subsectionintoc {1}{7}{Trajectory Analysis: NapaTrackMater}{38}{0}{1}
|
| 10 |
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|
| 11 |
-
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|
| 12 |
-
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| 13 |
-
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|
| 14 |
-
\beamer@subsectionintoc {2}{3}{Phase Space}{98}{0}{2}
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|
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| 8 |
\beamer@subsectionintoc {1}{6}{Point Cloud based morphodynamic features: NapaTrackMater}{32}{0}{1}
|
| 9 |
\beamer@subsectionintoc {1}{7}{Trajectory Analysis: NapaTrackMater}{38}{0}{1}
|
| 10 |
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presentation_images/1DCnn.png
ADDED
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presentation_images/Derivative_Plots_of_ALL_PCs_Across_Cell_Types_all_PCs_Fifth_density_stats.png
ADDED
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Git LFS Details
|
presentation_images/Derivative_Plots_of_ALL_PCs_Across_Cell_Types_all_PCs_Second_density_stats.png
ADDED
|
Git LFS Details
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presentation_images/Derivative_Plots_of_ALL_PCs_Across_Cell_Types_all_PCs_Second_membrane_density_stats.png
ADDED
|
Git LFS Details
|
presentation_images/Derivative_Plots_of_ALL_PCs_Across_Cell_Types_all_PCs_Sixth_density_stats.png
ADDED
|
Git LFS Details
|
presentation_images/Derivative_Plots_of_ALL_PCs_Across_Cell_Types_all_PCs_Sixth_membrane_density_stats.png
ADDED
|
Git LFS Details
|
presentation_images/Derivative_Plots_of_PC1_Across_Cell_Types_PC1_Fifth_density_stats.png
ADDED
|
Git LFS Details
|
presentation_images/Derivative_Plots_of_PC1_Across_Cell_Types_PC1_Second_density_stats.png
ADDED
|
Git LFS Details
|
presentation_images/Derivative_Plots_of_PC1_Across_Cell_Types_PC1_Second_membrane_density_stats.png
ADDED
|
Git LFS Details
|
presentation_images/Derivative_Plots_of_PC1_Across_Cell_Types_PC1_Sixth_density_stats.png
ADDED
|
Git LFS Details
|
presentation_images/Derivative_Plots_of_PC1_Across_Cell_Types_PC1_Sixth_membrane_density_stats.png
ADDED
|
Git LFS Details
|
presentation_images/Diffusion_Constant_Plots_of_ALL_PCs_Across_Cell_Types_all_PCs_Fifth_density_stats.png
ADDED
|
Git LFS Details
|
presentation_images/Diffusion_Constant_Plots_of_ALL_PCs_Across_Cell_Types_all_PCs_Second_density_stats.png
ADDED
|
Git LFS Details
|
presentation_images/Diffusion_Constant_Plots_of_ALL_PCs_Across_Cell_Types_all_PCs_Second_membrane_density_stats.png
ADDED
|
Git LFS Details
|
presentation_images/Diffusion_Constant_Plots_of_ALL_PCs_Across_Cell_Types_all_PCs_Sixth_density_stats.png
ADDED
|
Git LFS Details
|
presentation_images/Diffusion_Constant_Plots_of_ALL_PCs_Across_Cell_Types_all_PCs_Sixth_membrane_density_stats.png
ADDED
|
Git LFS Details
|
presentation_images/Diffusion_Constant_Plots_of_PC1_Across_Cell_Types_PC1_Fifth_density_stats.png
ADDED
|
Git LFS Details
|
presentation_images/Diffusion_Constant_Plots_of_PC1_Across_Cell_Types_PC1_Second_density_stats.png
ADDED
|
Git LFS Details
|