.center[ .vertical-center[ # Variational stress inference: A robust method to measure tissue mechanics Nicholas Noll Kavli Institute for Theoretical Physics, Santa Barbara, USA ] ] ??? hello everyone, I'm Nicholas Noll with the KITP of Santa Barbara. I'm excited to talk to you today about a new way to think of cellular stress in epithelia. --- # Cellular mechanics drives morphogenesis .center[ ![:scale 1000](/figs/devbio/morpho/flow1.svg) ] ??? in recent years, a central dogma of morphogenesis has emerged --- count: false # Cellular mechanics drives morphogenesis .center[ ![:scale 1000](/figs/devbio/morpho/flow2.svg) ] ??? regulated subcellular chemistry controls cellular mechanics patterns a mesoscopic tissue stress --- count: false # Cellular mechanics drives morphogenesis .center[ ![:scale 1000](/figs/devbio/morpho/flow3.svg) ] ??? in turn drives global cellular rearrangements and directs cellular flow --- count: false # Cellular mechanics drives morphogenesis .center[ ![:scale 1000](/figs/devbio/morpho/flow4.svg) ] ??? precise quantitative understanding of the relation of these scales is in part impeded by the difficulty of measuring cell mechanics in-vivo --- # In-silico allows non-invasive cell-scale measurement .left-col50[ .center[ ![:scale 400](/figs/devbio/morpho/gbe_tissue_bw.png) ] ] ??? 1. computational inference is commonly used to tackle this challenge as it is non-invasive and can measure cell-scale stress in a high-throughput manner 2. traditional pipeline involves acquiring high-quality live-image datasets of flourescent labelled cell membranes --- count:false # In-silico allows non-invasive cell-scale measurement .left-col50[ .center[ ![:scale 400](/figs/devbio/morpho/gbe_tissue.png) ] ] ??? 3. cell geometry is then segmented and then commonly abstracted as a vertex model --- count:false # In-silico allows non-invasive cell-scale measurement .left-col50[ .center[ ![:scale 350](/figs/devbio/morpho/vertex_model_cartoon.png) ] #### Vertex model * vertex labelled by $\{i,j,k\}$ * cells labelled by $\{\alpha,\beta,\gamma\}$ * edges are ![:emph](straight lines) ] --- count:false # In-silico allows non-invasive cell-scale measurement .left-col50[ .center[ ![:scale 350](/figs/devbio/morpho/vertex_model_cartoon_edge.png) ] #### Vertex model * vertex labelled by $\{i,j,k\}$ * cells labelled by $\{\alpha,\beta,\gamma\}$ * edges are ![:emph](straight lines) ] #### Mechanical state * interfacial .green[tension] $T_{ij}$ on edge $\langle i,j \rangle$ --- count:false # In-silico allows non-invasive cell-scale measurement .left-col50[ .center[ ![:scale 350](/figs/devbio/morpho/vertex_model_cartoon_cell.png) ] #### Vertex model * vertex labelled by $\{i,j,k\}$ * cells labelled by $\{\alpha,\beta,\gamma\}$ * edges are ![:emph](straight lines) ] #### Mechanical state * interfacial .green[tension] $T_{ij}$ on edge $\langle i,j \rangle$ * intercellular .red["pressure"] $p\_\alpha$ in cell $\alpha$ -- #### Mechanical equilibrium $$ \vec{0} = \sum\limits\_{j\in \mathcal{N}\_i} T\_{ij} \hat{r}\_{ij} + \Delta p\_{ij}\left(\hat{z} \wedge \vec{r}\_{ij}\right)$$ -- #### Challenges for wide-spread use 1. Inversion sensitive to segmentation noise 2. Have to know boundary conditions (global) --- class: middle, center # Can we reformulate the relationship between apical cell geometry and mechanics to remedy problems? --- # Revisit vertex model parameterization .left-col50[ .center[ ![:scale 425](/figs/devbio/morpho/mem52_box.svg) ] ] --- count:false # Revisit vertex model parameterization .left-col50[ .center[ ![:scale 425](/figs/devbio/morpho/mem52_box_vertex.svg) ] ] -- #### Straight edge insufficient * Differential cell pressure + interfacial tension -- * Edges are ![:emph](circular arcs) to ensure force balance --- count:false # Revisit vertex model parameterization .left-col50[ .center[ ![:scale 425](/figs/devbio/morpho/mem52_box_edge.svg) ] ] #### Straight edge insufficient * Differential cell pressure + interfacial tension * Edges are ![:emph](circular arcs) to ensure force balance * Radius obeys Young-Laplace $$ \Delta p\_{\alpha\beta} R\_{\alpha\beta} = T\_{\alpha\beta}$$ -- #### Reparameterize epithelial geometry by 1. Edge arc centroids $\vec{\rho}\_{\alpha\beta}$ 2. Edge arc radii $R\_{\alpha\beta}$ --- # Variational segmentation of epithelial geometry .left-col50[ $$ E(\vec{\rho},R) = \sum\_{\langle \alpha,\beta \rangle} \sum\_n (|\vec{r}\_{\alpha\beta}(n) - \vec{\rho}\_{\alpha\beta}| - R_{\alpha\beta})^2 $$ .center[ ![:scale 200](/figs/devbio/morpho/optimization.png) ] ] -- .center[ ![:scale 450](/figs/devbio/morpho/CAPfit.png) ] -- #### As posed, has problems 1. Can't vary $\rho$ and $R$ independently! Have to ensure intersection at vertex -- 2. No obvious relation to mechanics! Ultimately want to infer $T$ and $p$ --- # Epithelial tissues as circular arc polygonal tilings .left-col50[ .center[ ![:scale 350](/figs/devbio/morpho/CAPnetwork.png) ] Tension acts tangent to the circular arc ] -- .right-col50[ .center[ ![:scale 300](/figs/devbio/morpho/CAPedge.png) ] ] #### Equilibrium constraints on CAP networks * Young-Laplace condition $ R\_{\alpha\beta} \Delta p\_{\alpha\beta} = T_{\alpha\beta} $ * Force on $i$, from edge $\langle i,j \rangle$ or $\langle \alpha,\beta\rangle$ is $$ \vec{T}\_{i,j} = \hat{z} \wedge \frac{T\_{\alpha\beta}}{R\_{\alpha\beta}}\left(\vec{r}\_i - \vec{\rho}\_{\alpha\beta} \right) $$ --- count:false # Epithelial tissues as circular arc polygonal tilings .left-col50[ .center[ ![:scale 350](/figs/devbio/morpho/CAPnetwork.png) ] Tension acts tangent to the circular arc ] .right-col50[ .center[ ![:scale 300](/figs/devbio/morpho/CAPedge.png) ] ] #### Equilibrium constraints on CAP networks * Young-Laplace condition $ R\_{\alpha\beta} \Delta p\_{\alpha\beta} = T_{\alpha\beta} $ * Force on $i$, from edge $\langle i,j \rangle$ or $\langle \alpha,\beta\rangle$ is $$ \vec{T}\_{i,j} = \hat{z} \wedge \Delta p\_{\alpha\beta}\left(\vec{r}\_i - \vec{\rho}\_{\alpha\beta} \right) $$ --- # Mechanical equilibrium imposes non-trivial constraints .left-col50[ .center[ ![:scale 250](/figs/devbio/morpho/CAPvertex.png) ] #### Force balance at vertex $i$ $$ \vec{0} = \Delta p\_{\beta\alpha} \, \vec{\rho}\_{\beta\alpha} + \Delta p\_{\alpha\gamma} \, \vec{\rho}\_{\alpha\gamma} + \Delta p\_{\gamma\beta} \, \vec{\rho}\_{\gamma\beta} $$ ] -- .right-col50[ .center[ ![:scale 425](/figs/devbio/morpho/CAPcollinear.png) ] #### Geometric constraint Centroids $\vec{\rho}$ that meet at vertex are collinear ] -- Implies general solution of \{$p$, $\vec{q}$\}: $\quad \vec{\rho}\_{\beta\alpha} = \frac{p\_{\beta} \vec{q}\_\beta - p\_{\alpha}\vec{q}\_{\alpha}}{p\_\beta - p\_\alpha}$ --- # Unconstrained parameterization of equilibrium CAP We parameterize an equilibrium epithelial tissue by ![:emph](independent) degrees of freedom -- .left-col50[ ![:scale 450](/figs/devbio/morpho/dual_cartoon.png) ] * $4C$ d.o.f \{$\vec{q}$, $p$, $z$\} $\leftrightarrow$ $4C$ mechanics \{$p$, $T$\} -- * Parameterize ![:emph](mechanics): $p\_\alpha =$ pressure, $$ T\_{\alpha\beta}^2 = p\_\alpha p\_\beta |\vec{q}\_\alpha - \vec{q}\_\beta|^2 - \Delta p\_{\alpha\beta} \Delta z\_{\alpha\beta} $$ -- * Parameterize ![:emph](geometry): $\Delta p\_{\alpha\beta} R\_{\alpha\beta} = T\_{\alpha\beta}$, $$ \vec{\rho}\_{\alpha\beta} = \frac{p\_{\alpha}\vec{q}\_{\alpha} - p\_{\beta}\vec{q}\_{\beta}}{p\_{\alpha} - p\_{\beta}} $$ -- * Any intuitive interpretation of these parameters? --- # Epithelia generalize Voronoi tessellations $\vec{q}\_\alpha$ denotes generating points of Voronoi. Distance from $\alpha$ is $d\_\alpha^2(\vec{x}) = |\vec{x} - \vec{q}\_\alpha|^2$ .center[ ![:scale 550](/figs/devbio/morpho/voronoi1.png) ] --- count:false # Epithelia generalize Voronoi tessellations $z\_\alpha$ allows for transverse movement. Distance from $\alpha$ is $d\_\alpha^2(\vec{x}) = |\vec{x} - \vec{q}\_\alpha|^2 - z^2\_\alpha$ .center[ ![:scale 550](/figs/devbio/morpho/voronoi2.png) ] --- count:false # Epithelia generalize Voronoi tessellations $p\_\alpha$ weights space. Distance from $\alpha$ is $d\_\alpha^2(\vec{x}) = p\_\alpha|\vec{x} - \vec{q}\_\alpha|^2 - z^2\_\alpha$ .center[ ![:scale 550](/figs/devbio/morpho/voronoi3.png) ] --- # Formulation of variational inference Utilize "dual" parameterization of mechanics and geometry. Find "closest" equilibrium geometry. -- .left-col50[ $$ E(\vec{q},z,p) = \sum\_{\langle \alpha,\beta \rangle} \sum\_n (|\vec{r}\_{\alpha\beta}(n) - \vec{\rho}\_{\alpha\beta}| - R_{\alpha\beta})^2 $$ .center[ ![:scale 200](/figs/devbio/morpho/optimization.png) ] ] -- .center[ ![:scale 450](/figs/devbio/morpho/CAPfit.png) ] -- #### Important features * Insensitive to noise in input geometry * Overdetermined (local) --- # Improves reconstruction of synthetic data .center[ ![:scale 1000](/figs/devbio/morpho/inverse_sensitivity.svg) ] --- # Predicts interfacial myosin from cell geometry .left-col50[ #### Lateral ectoderm during GBE .center[ ![:scale 580](/figs/devbio/morpho/gbe.png) ] ] .right-col50[ .center[ ![:scale 580](/figs/devbio/morpho/myotension.png) ] ] --- # Stress axis correlates with mitotic spindle #### Divisions during late GBE .center[ ![:scale 1000](/figs/devbio/morpho/divisionaxis.svg) ] --- # Future outlook .center[ ![:scale 900](/figs/devbio/morpho/future1.svg) ] --- count:false # Future outlook .center[ ![:scale 900](/figs/devbio/morpho/future2.svg) ] --- count:false # Future outlook .center[ ![:scale 900](/figs/devbio/morpho/future3.svg) ] --- # Acknowledgements .center[ ![:scale 700](/figs/devbio/morpho/acknowledgements.svg) ] --- count:false # Measuring mechanics in-vivo is difficult .center[ ![:scale 1100](/figs/devbio/morpho/invivo_measurements_1.svg) ] --- count:false # Measuring mechanics in-vivo is difficult .center[ ![:scale 1100](/figs/devbio/morpho/invivo_measurements_2.svg) ] --- count:false # Measuring mechanics in-vivo is difficult .center[ ![:scale 1100](/figs/devbio/morpho/invivo_measurements_3.svg) ] --- count:false # Radii are constrained by centroids .left-col50[ .center[ ![:scale 405](/figs/devbio/morpho/CAPradii.png) ] ] #### Must intersect at vertex $$ \Delta p\_\{\alpha\beta\}(R\_{\alpha\beta}^2 - \vec{\rho}^2\_{\alpha\beta}) + [...]\_{\beta\gamma} + [...]\_{\gamma\alpha} = 0 $$ -- #### Defines family of solutions $$ R\_{\alpha\beta}^2(z\_\alpha, z\_\beta) = \vec{\rho}\_{\alpha\beta}^2 + \frac{p\_\alpha z^2\_\alpha - p\_\beta z^2\_\beta}{p\_\alpha - p\_\beta} $$ -- We can fully parameterize an equilibrium epithelial tissue by {$\{\vec{q}\_\alpha, p\_\alpha, z\_\alpha\}$} * \# dof = $4 c$ = $e + c$ = \# of parameters -- * cell autonomous parameterization -- * directly encodes mechanics * <sub><sup>$p\_\alpha$ is apical pressure </sup></sub> * <sub><sup>$T\_{\alpha\beta}$ given by Young-Laplace and above radial eqn</sup></sub> --- count:false # Tracking divisions .center[ ![:scale 800](/figs/devbio/morpho/celldivisions.svg) ]