Calculation of the force field required for nucleus deformation during cell migration through constrictions

During cell migration in confinement, the nucleus has to deform for a cell to pass through small constrictions. Such nuclear deformations require significant forces. A direct experimental measure of the deformation force field is extremely challenging. However, experimental images of nuclear shape are relatively easy to obtain. Therefore, here we present a method to calculate predictions of the deformation force field based purely on analysis of experimental images of nuclei before and after deformation. Such an inverse calculation is technically non-trivial and relies on a mechanical model for the nucleus. Here we compare two simple continuum elastic models of a cell nucleus undergoing deformation. In the first, we treat the nucleus as a homogeneous elastic solid and, in the second, as an elastic shell. For each of these models we calculate the force field required to produce the deformation given by experimental images of nuclei in dendritic cells migrating in microchannels with constrictions of controlled dimensions. These microfabricated channels provide a simplified confined environment mimicking that experienced by cells in tissues. Our calculations predict the forces felt by a deforming nucleus as a migrating cell encounters a constriction. Since a direct experimental measure of the deformation force field is very challenging and has not yet been achieved, our numerical approaches can make important predictions motivating further experiments, even though all the parameters are not yet available. We demonstrate the power of our method by showing how it predicts lateral forces corresponding to actin polymerisation around the nucleus, providing evidence for actin generated forces squeezing the sides of the nucleus as it enters a constriction. In addition, the algorithm we have developed could be adapted to analyse experimental images of deformation in other situations.

We then calculate the mean value and standard deviation of the fluorescence of the pixels. Then we decrease the intensity of each image by an amount given by I new = I old −Ĩ − aσ, where I new is the intensity with the background removed, I old is the intensity with background,Ĩ is the mean intensity of all pixels, σ is the standard deviation of the intensities and a is a value that is dependent on the image. The value of a is increased in increments of 0.1 from 0, until the number of fluorescing pixels matches that of the area of the curves around the nuclei, which we had previously drawn using the threshold, spline fitting and interpolation tools within imageJ, for each given image and frame.
The change in DNA intensity between the nuclear region inside and outside the constriction of 60 − 70% is consistent with the change in height of the channel from 5µm outside of the constriction to 3.4µm within the constriction, i.e. the constriction is approximately 70% of the height of the channel. The mean area of each nucleus in µm 2 before the nucleus enters the constriction, while any part of the nucleus is between the constriction entry and exit, and after the nucleus has exited the constriction. The mean area of each nucleus is calculated by the mean area of that nucleus at all available time frames during which the nucleus is before/in/after the constriction. This is a mean of a few time frames since typically there are 10-20 time frames in total for each cell. Each point on the graph represents an individual nucleus with an identifying colour. The colour scale blue to red to orange is the area of the nucleus before the constriction from small to large. Nuclei in the later positions keep their colour as defined by their area before the constriction. The black horizontal lines show the mean area of nuclei at that position and the vertical black lines indicate the standard deviations.
The change in area shown in Fig A.1 is also consistent with the change in height of the channel from 5µm outside of the constriction to 3.4µm within the constriction.
Multiplying this area by the height gives us the volume shown in Fig 2A in the main text. We therefore conclude that there is no significant volume change of the nuclei as they travel through the constrictions.

B Numerical calculation of the strain tensor
In order to accommodate the boundary conditions given by solid model assumption of a deformation field decreasing lineally along the radial direction, we calculate the derivatives in this model using polar coordinates in the 2D plane seen in images.
However, the coordinates of the pixels in the experimental images are known in Cartesian coordinates. Therefore a transformation between these two coordinate systems is necessary.
The polar form of the strain and Eq (2) in the main text can be written using where we can use the standard Polar-Cartesian relations to evaluate the following terms; Substituting these into Eq (1) gives the polar form of the strain tensor.
Numerically, we calculate these derivatives using finite central difference forms of numerical derivatives on a mesh. The input pixels give the mesh points along the perimeter. To define the radial mesh, we draw concentric rings with shapes similar to the perimeter but of decreasing size at pixel sized intervals along the radii. We denote these mesh points using the indices n,m to label the points along the perimeter and radii respectively. The finite central difference numerical derivatives are given by, ∂u x (x n,m , y n,m ) ∂r = u x (x n,m+1 , y n,m+1 ) − u x (x n,m−1 , y n,m−1 ) r n,m+1 − r n,m−1 , ∂u y (x n,m , y n,m ) ∂r = u x (x n,m+1 , y n,m+1 ) − u x (x n,m−1 , y n,m−1 ) r n,m+1 − r n,m−1 , ∂u x (x n,m , y n,m ) ∂θ = R n+1 u x (x n+1,m , y n+1,m )∆θ n − R n−1 u x (x n−1,m , y n−1,m )∆θ n+1 + u x (x n,m , y n,m )((∆θ n+1 ) 2 − (∆θ n ) 2 ) / ∆θ n+1 ∆θ n (∆θ n + ∆θ n+1 ) , ∂u y (x n,m , y n,m ) ∂θ = R n+1 u y (x n+1,m , y n+1,m )∆θ n − R n−1 u y (x n−1,m , y n−1,m )∆θ n+1 where R n+1 is the ratio of the radius at perimeter point n + 1 to the radius at perimeter point n and ∆θ n+1 is the angle between these points. Similar expressions are used for the innermost and outermost shapes, but replaced with forward/backward finite difference methods respectively. The factors of R n in the final two equations are included to scale for small variations in the radius between points. These equations, together with the standard relations between Cartesian and polar coordinates allow the strains to be numerically evaluated from Eq (2) in the main text in the Cartesian coordinate basis directly.

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Nucleus deformation To calculate derivatives of the deformation field in the solid model we need to define the deformation at the inner mesh points along the concentric rings inside the perimeter. where |r inner n,m | is the distance from the origin to the inner ring point n, m corresponding to point n, M on the perimeter. To calculate the derivatives of these deformations numerically at the outer limits we use a backwards difference method.

C Differential geometry of surfaces
In order to generally describe the more complicated unknown surface in the general tangent and normal coordinate basis, and calculate values along the surface, we first describe the general form of the derivatives using differential geometry. The general forms involve the curvatures of the surface and the Christoffel symbols of the surface, and an analytic method to calculate the derivatives is given below.
Briefly, a surface X(n, s 1 , s 2 ) described by two tangential directions s 1 , s 2 and the normal direction n has an associated metric tensor given by the derivatives of the surface along each of the directions at each point, where x i represents the tangent directions along the surface. For a positively oriented surface, where by definition when travelling along the curve describing the surface, the interior of the curve is on the left, the outwards normal to the surface is then given by The Christoffel symbols are written in terms of the metric tensor as Similarly, the curvature of the surface, measured as the rate of change of the normal 4/11 Nucleus deformation direction along the surface can be expressed as a tensor, C ij .
A thin shell surface can be written as a function of only the two tangent directions, X(n, s 1 , s 2 ), and so the metric tensor is a 2x2 matrix, with the components s 1 and s 2 representing two tangent directions along the surface g = e s1 · e s1 e s1 · e s2 e s1 · e s2 e s2 · e s2 .
The derivatives along the surface of the basis vectors are then given in terms of the curvature and metric as ∂e Using these relations, we can describe any thin shell surface, for instance those seen in images of the nucleus. Because the images considered are only in two dimensions, an analytical approach is used to describe the out of plane direction s 2 , while the in plane images provide s 1 from the outline. The normal direction is assumed outwards and in the XY plane seen in images, and so can be determined purely from the s 1 tangent vector. As the surface is flat in the out of plane direction, the vector in the out of plane direction is easily defined as a unit length vector parallel to the z axis. As such the metric tensor is the identity and the Christoffel symbols are all zero, leaving only the curvature terms in the shell model of this particular out of plane direction shape.
However, we include the full differential geometry in the code to allow for use with other, more complicated, shapes beyond the scope of this paper.

D Alignment of initial and target images
The deformation free point is likely to lie somewhere between the two limiting cases The reason that the centroid alignment is not necessarily the correct alignment is due to the fact that we calculate the centroid of the 2D images rather than the 3D volume and therefore we need a correction to this alignment. However the limits on both the spatial and temporal resolution of the nuclei in the images prevent the point of zero deformation from being identified directly from any given series of images.
In order to estimate the location where the deformed nucleus should be placed relative to the undeformed nucleus, we measured how the position of the front and rear of the nucleus changed between frames. Measurements of the change in the front, ∆F , and rear, ∆R, position of each nucleus were taken between frames at each of the five positions (before the constriction, entering in to the constriction, inside the constriction, leaving the constriction and after the constriction).

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Nucleus deformation The distance that we shift the entire target shape along x is proportional to the relative change between the front and rear position (i.e. the deformation) compared to the sum of the changes to the rear and the front in the respective directions. The proportion of the centroid aligned changes in rear/front position, |∆r| or |∆f |, that the target is shifted to align with the estimated translation free position is, where ∆F and ∆R and changes in the front/rear between the undeformed and deformed shapes and ∆f and ∆r are the changes in the front/rear between the undeformed and deformed shapes when aligned in the centroid frames respectively.

E Strain energy
We plot the free energy density per unit volume, Equation (3)  and force (right) fields for A. the solid model and B. the shell model for an example nucleus that fails to pass through the constriction. The initial shape is before and the target shape is when the nucleus is attempting to enter the constriction. After this time point the cell changes direction and moves back along the channel in the negative x direction. As in Fig 4 in the main text, the axes show pixel numbers where each pixel is is 0.215µm. The black outline is the initial shape and the green outline is the target deformed shape. Blue arrows represent the final deformation field found between the images and red arrows represent the traction force direction and magnitude, with each arrow scaled such that one unit of length on the axes represents a traction force of 250 Pa. The traction force is calculated using a Young's modulus of E = 5 kPa and assuming that the nucleus behaves as an incompressible elastic solid with a Poisson ratio ν = 0.5. C. Average actin (LifeAct-GFP fluorescence) intensity (green + points) at the time point when the nucleus is attempting to enter into the constriction (right hand y-axis). The actin intensity is the mean intensity over the width of the channel at each pixel position. This is then renormalised by the average intensity for the cell and aligned with the start of the constriction at x = 0. The red × points are the absolute values of the y components of the traction force from the solid model A. and the purple + × points are the absolute values of the y components of the traction force from shell model B. The vertical dotted line indicates the start of the constriction. D. Experimental image showing the actin intensity.