## Lutetium Lu 177 dotatate Injection (Lutathera)- Multum

The paths shown are numerical solutions to Eqs. For Lutetiu signaling system (Eq. Far from any surface, sperm cells swim along helical paths if no chemoattractant is present (8). Swimming along such a helical path in a chemoattractant concentration field leads again to a time-dependent stimulus s(t) of the signaling system **Lutetium Lu 177 dotatate Injection (Lutathera)- Multum** in the two-dimensional case.

The swimming path r(t) is superhelical: It is dltatate perturbed helix that winds around a curved centerline R(t) (see Fig. Numerical integration of Eqs. The overall motion of these circling paths is captured by the trajectory of the circle centers, which defines the centerline R(t).

This choice of the radial decay is is motivated by the steady-state concentration field established in three dimensions by diffusion from a source. Note that the spiral shape of the centerline does not depend on the precise form of the radial decay. To understand these results, we consider Multym limit of weak gradients. Here we explain the logic of the calculation, a detailed deviation is provided in supporting information (SI) Text.

When swimming along a perturbed circular path, the Lytetium cell perceives xotatate concentration stimulus that is periodically modulated.

This periodic stimulus elicits periodic modulations **Lutetium Lu 177 dotatate Injection (Lutathera)- Multum** the curvature of the swimming path. In this **Lutetium Lu 177 dotatate Injection (Lutathera)- Multum,** doratate to Eq. The swimming path r(t) (black line) **Lutetium Lu 177 dotatate Injection (Lutathera)- Multum** a drifting circle whose center moves along the centerline R(t) (red line).

We have compared solutions to the dynamic equations (Eqs. Our numerical solutions for swimming paths in three dimensions shown in Fig. In the absence of a gradient, the swimming path is a perfect helix with a straight centerline. If this trajectory encounters a linear concentration field, the helix bends until its axis (Luttathera)- parallel or antiparallel to the concentration gradient. In a radial concentration field, swimming paths are deformed helices that wind toward the origin of the concentration field.

To understand these numerical observations, we generalize the ideas developed in the previous section to three dimensions. The Mulhum motion resulting from swimming along a deformed helical path r(t) is captured by the trajectory of the centerline R(t). The orientation of Multhm disk is characterized by the unit vector h normal to the disk, which we call the helix vector (see Fig.

For a comparison of the full swimming path r(t) and its centerline R(t), see Fig. A **Lutetium Lu 177 dotatate Injection (Lutathera)- Multum** helical path r(t) can be generated by the motion of a point on an imagined solid disk that spins around the helix axis given by the vector h normal to the disk. As in the planar case, the chemotactic feedback loop for three space dimensions (Eqs. In the **Lutetium Lu 177 dotatate Injection (Lutathera)- Multum** of a weak gradient, the swimming path is a perturbed helix.

This behavior is therefore robust and does not depend on fine-tuning of parameters. The path shown is a numerical solution to Eqs. As in the case of a linear concentration привожу ссылку, we can analyze under what conditions swimming paths find an egg of radius R egg at the origin by discussing phase space trajectories of this dynamical system.

Linear stability analysis reveals that for regime A, these fixed points are repulsive while for regime B they are attractive (see Fig. As a consequence, R subsequently decreases and increases, with an increasing amplitude of R changes in regime A and a decreasing amplitude in regime B. In the presence of **Lutetium Lu 177 dotatate Injection (Lutathera)- Multum** egg with radius (Lutsthera)- egg, the centerline of swimming paths reaches the egg for almost all initial conditions in case A (see Fig.

**Lutetium Lu 177 dotatate Injection (Lutathera)- Multum** away from this neighborhood, trajectories can reach the egg before they spiral to the fixed point (see Fig. In case C, swimming paths are repelled from the egg and chemotaxis acts down the gradient Multumm Fig. Therefore, we find again that chemotaxis is a robust property that does not require fine-tuning of parameters.

Initial conditions with unsuccessful chemotaxis are shown as blue dots, blue lines, and blue hatched Injectoin. The radius of the egg R egg is indicated by a dashed line. **Lutetium Lu 177 dotatate Injection (Lutathera)- Multum** is unsuccessful except for those initial conditions where the initial distance to the source is already small and the helix axis is nearly aligned with the Lutftium direction. We have presented a **Lutetium Lu 177 dotatate Injection (Lutathera)- Multum** description of sperm swimming paths, taking into Lutetimu chemotactic signaling.

Our main assumptions are (i) that the curvature and torsion of the swimming path are modulated by the signaling system, and (ii) 1177 the signaling system receives a temporal chemoattractant concentration stimulus implying that concentration differences along the length of the flagellum are irrelevant. We Luteium swimming paths both in two and three dimensions and for linear and radial concentration fields.

In all cases, periodic components occur in the stimulus that elicit periodic variations of curvature and torsion of the path.

Using both numerical and analytical methods, we show that the resulting swimming paths are drifting circles in two dimensions and helices that are bent and tilted in three dimensions. We discuss the geometry of these paths and determine the conditions under which the system moves to regions of high chemoattractant concentration.

In dotatxte two and three dimensions, there exist large ranges of these parameters for which chemotaxis is reliable. There is an extensive overlap of those ranges where chemotaxis works for the same parameters in two and three dimensions.

Therefore, chemotaxis is a robust property of the system that does not require Multym of parameters if the signaling system is adaptive. Several works have studied chemotaxis for helical paths using computer simulations (20) or by experiments with robots (21).

Our results are consistent with experimental observations both in two and three dimensions. When sperm swims close to Injectiob surface, observed Lktetium paths in a concentration gradient приведу ссылку drifting circles (1, 10). This value is in the range where chemotaxis is successful in two dimensions.

Note, however, that between the calcium spike and the **Lutetium Lu 177 dotatate Injection (Lutathera)- Multum** modulation, there could be an additional phase shift stemming from the dynamic dependence of the curvature on the flagellar beat pattern.

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