Spin controlled atom-ion inelastic collisions

The control of the ultracold collisions between neutral atoms is an extensive and successful field of study. The tools developed allow for ultracold chemical reactions to be managed using magnetic fields, light fields and spin-state manipulation of the colliding particles among other methods. The control of chemical reactions in ultracold atom-ion collisions is a young and growing field of research. Recently, the collision energy and the ion electronic state were used to control atom-ion interactions. Here, we demonstrate spin-controlled atom-ion inelastic processes. In our experiment, both spin-exchange and charge-exchange reactions are controlled in an ultracold Rb-Sr$^+$ mixture by the atomic spin state. We prepare a cloud of atoms in a single hyperfine spin-state. Spin-exchange collisions between atoms and ion subsequently polarize the ion spin. Electron transfer is only allowed for (RbSr)$^+$ colliding in the singlet manifold. Initializing the atoms in various spin states affects the overlap of the collision wavefunction with the singlet molecular manifold and therefore also the reaction rate. We experimentally show that by preparing the atoms in different spin states one can vary the charge-exchange rate in agreement with theoretical predictions

Supplementary Information from A dynamically diluted alignment model reveals the impact of cell turnover on the plasticity of tissue polarity patterns

Supplementary Information to Hoffmann, Voss-Böhme, Rink and Brusch 2017: “A Dynamically Diluted Alignment Model Reveals the Impact of Cell Turnover on the Plasticity of Tissue Polarity Patterns” with detailed derivations of equations that are used in the main text, a table summarising all variable names and notations and additional results in 7 supplementary figures: Fig.S1, Lattice size effects. Fig.S2, Effects of de-novo polarisation rate beta on the time of minimal order. Fig.S3, Statistical robustness of time of minimal order. Fig.S4, Bistability of mean-field ODE system for very high neighbour coupling strength. Fig.S5, Time of minimal order for vanishing neighbour coupling strength. Fig.S6, Time of minimal order collapses to linear function of the effective neighbour coupling strength. Fig.S7, Effects of coupling strength to global signal epsilon_s on the time of minimal order

Original research data: time of minimal order in numerical solutions of the mean-field model from A dynamically diluted alignment model reveals the impact of cell turnover on the plasticity of tissue polarity patterns

Time of minimal order data measured from numerical solutions of the ODE system of the mean-field model (main text eq. (19)), using Runge-Kutta scheme with time step 10^(-6). Columns are ID of the ODE, de-novo polarisation rate beta, cell replacement rate delta, neighbour coupling strength epsilon_n, effective neighbour coupling strength epsilon_n_eff (which is a function of beta, delta and epsilon_n), coupling strength to global signal epsilon_s, components s_x and s_y of the global signal vector, end time of the simulation, time interval of simulation logging, and the following observables: minimal order, first time at which minimal order is attained, last time at which minimal order is attained, flag whether tissue polarity inverted direction (false if first and last time of minimal order differ), corrected time of minimal order (the observable, NaN if tissue polarity did not invert direction)

Original research data: time of minimal order in sampled trajectories of the dynamically diluted alignment model from A dynamically diluted alignment model reveals the impact of cell turnover on the plasticity of tissue polarity patterns

Time of minimal order data measured from sampled trajectories of the dynamically diluted alignment model (main text eqs. (4), (6) and (7)), using a variant of the stochastic simulation algorithm that is adapted to Interacting Particle Systems [Klauß and Voss-Böhme (2008)]. Columns are simulation ID, de-novo polarisation rate beta, cell replacement rate delta, neighbour coupling strength epsilon_n, effective neighbour coupling strength epsilon_n_eff (which is a function of beta, delta and epsilon_n), coupling strength to global signal epsilon_s, lattice size in x- and y- direction, boundary condition in x- and y-direction, components s_x and s_y of the global signal vector, end time of the simulation, time interval of simulation logging, time of minimal order (the observable). There are data of 25 simulations with different random number series for each parameter combination

Distinguishing head morphologies of four different flatworm species.

<p><b>A.</b> Application of our method to parametrize head morphology of four different flatworm species. For each species, time-lapse sequences of different worms were recorded as two independent runs of duration frames. The head is defined as most anterior of the worm body. Radial distances are computed with respect to the midpoint of the head (red dot at of the worm length from the tip of the head). <b>B.</b> By applying PCA to this multi-species data set, we obtain two shape modes, which together account for of the shape variability. Deformations of the mean shape with respect to the the two modes are shown (black: mean shape, red: superposition of mean shape and first mode with and second mode with , respectively). We represent head morphology of the four species in a combined shape space of these two modes. Average head shapes for each species are indicated by crosses, with ellipses of variance including (dark color) and (light color) of motility-associated shape variability, respectively.</p

Supplementary Movie from A dynamically diluted alignment model reveals the impact of cell turnover on the plasticity of tissue polarity patterns

Typical simulation run of the IPS model on a 100-by-100 lattice with periodic boundary conditions. The movie covers 1 time unit in the non-dimensionalised model time; time resolution of visualisation is 0.001. See main text figure 2C for colour code. The bottom left detail of the same simulation is shown in main text figure 3A, subpanels a-d, for times 0.05, 0.2, 0.5, 0.8, and its analysis is shown in main text figure 3B. Parameters are neighbour coupling strength epsilon_n=4.5, cell replacement rate delta=0.2, de-novo polarisation rate beta=1 and coupling strength to global signal epsilon_s=1, the global signal s=(1,0) points to the right

Principal component analysis is used across different disciplines, giving rise to a diverse terminology, which is summarized here.

<p>Principal component analysis is used across different disciplines, giving rise to a diverse terminology, which is summarized here.</p

Principal shape modes of sperm flagellar beating.

<p><b>A.</b> High-precision tracking of planar flagellar centerline shapes (, red) are characterized by their tangent angle as a function of arc-length along the flagellum. <b>B.</b> The time-evolution of this flagellar tangent angle is shown as a kymograph. The periodicity of the flagellar beat is reflected by the regular stripe patterns in this kymograph; the slope of these stripes is related to the propagation of bending waves along the flagellum from base to tip. By averaging over the time-dimension, we define a mean flagellar shape characterized by a tangent angle profile . For illustration, this mean flagellar shape is shown in black superimposed to tracked flagellar shapes (grey). <b>C.</b> We define a feature-feature covariance matrix from the centered tangent angle data matrix as explained in the text. The negative correlation at arc-length distance reflects the half-wavelength of the flagellar bending waves. <b>D.</b> The normalized eigenvalue spectrum of the covariance matrix sharply drops after the second eigenvalue, implying that the eigenvectors corresponding to the first two eigenvalues together account for 97% of the observed variance in the tangent angle data. <b>E.</b> Using principal component analysis, we define two principal shape modes (blue, red), which correspond precisely to the two maximal eigenvalues of the covariance matrix in panel C. The lower plot shows the reconstruction of a tracked flagellar shape (black) by a superposition of the mean flagellar shape and these two principal shape modes (magenta). In addition to tangent angle profiles, respective flagellar shapes are shown on the right. <b>F.</b> Each tracked flagellar shape can now be assigned a pair of shape scores and , indicating the relative weight of the two principal shape modes in reconstruction this shape. This defines a two-dimensional abstract shape space. A sequence of shapes corresponds to a point cloud in this shape space. We find that these point form a closed loop, reflecting the periodicity of the flagellar beat. We can define a shape limit cycle by fitting a curve to the point cloud. By projecting the shape points on this shape limit cycle, we can assign a unique flagellar phase modulo to each shape. This procedure amounts to a binning of flagellar shapes according to shape similarity. <b>G.</b> By requiring that the phase variable should change continuously, we obtain a representation of the beating flagellum as a phase oscillator. The flagellar phase increases at a rate equal to the frequency of the flagellar beat and rectifies the progression through subsequent beat cycles by increasing by . <b>H.</b> Amplitude fluctuations of flagellar beating as a function of flagellar phase. An instantaneous amplitude of the flagellar beat is defined as the radial distance of a point in the -shape space, normalized by the radial distance of the corresponding point on the limit cycle of same phase. A phase-dependent standard deviation was fitted to the data (black solid line). Also shown are fits for additional cells (gray; the position of was defined using a common set of shape modes). <b>J.</b> Swimming path of the head center during one beat cycle computed for the flagellar wave given by the shape limit cycle (panel F) using resistive force theory <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0113083#pone.0113083-Gray1" target="_blank">[18]</a> as described previously <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0113083#pone.0113083-Friedrich1" target="_blank">[19]</a>. The path is characterized by a wiggling motion of the head superimposed to net propulsion. For a ‘standing wave’ beat pattern characterized by the oscillation of only one shape mode, net propulsion vanishes.</p

Three shape modes characterize projected flatworm body shape dynamics.

<p><b>A.</b> Our custom-made MATLAB software tracks worms in movies and extracts worm boundary outline (red) and centerline (blue). <b>B.</b> The radial distance between the boundary points and midpoint of the centerline (, red dot) is calculated as a parameterization of worm shape. We normalize the radial distance profile of each worm by the mean radius . <b>C.</b> The second symmetry axis (dotted line) of the covariance matrix corresponds to statistically symmetric behavior of the worm with respect to its midline. <b>D.</b> The three shape modes with the largest eigenvalues account for 94% of the shape variations. The first shape mode characterizes bending of the worm and alone accounts for 61% of the observed shape variance. On the top, we show its normalized radial profile on the left as well as the boundary outline corresponding to the superposition of the mean worm shape and this first shape mode (solid red: , dashed red: , black: mean shape with ). The second shape mode describe lateral thinning (), while the third shape mode corresponds unlike deformations of head and tail (), giving the worm a wedge-shaped appearance. <b>E.</b> The first shape mode with score describing worm bending strongly correlates with the instantaneous turning rate of worm midpoint trajectories. <b>F.</b> We manually selected 30 movies where worms clearly show inch-worming and 50 movies with no inch-worming behavior. The variance of score and increases for the inch-worming worms. <b>G.</b> The autocorrelation of mode and the crosscorrelation between mode and mode reveals an inch-worming frequency of approximately , hinting at generic behavioral patterns.</p

Illustration of principal component analysis.

<p><b>A.</b> As a minimal example, we consider a hypothetical data set of length and height measurements for a collection of individuals, <i>i.e.</i> there are just geometric features measured here. <b>B.</b> In this example, length and height are assumed to be strongly correlated, thus mimicking the partial redundancy of geometrical features commonly observed in real data. Principal component analysis now defines a change of coordinate system from the original (length,height)-axes (shown in a black) to a new set of axes (blue) that represent the principal axes of the feature-feature covariance matrix of the data. Briefly, the first new axis points in the direction of maximal data variability, while the second new axis points in the direction of minimal data variability. The change of coordinate system is indicated by a rotation around the center of the point cloud representing the data. By projecting the data on those axes that correspond to maximal feature-feature covariance, in this example the first axis, one can reduce the dimensionality of the data space, while retaining most of the variability of the data. In the context of morphology analysis, we will refer to these new axes as ‘shape modes’ , which represent specific combinations of features. The new coordinates are referred to as ‘shape scores’ .</p