Optimal decision making for sperm chemotaxis in the presence of noise

For navigation, microscopic agents such as biological cells rely on noisy sensory input. In cells performing chemotaxis, such noise arises from the stochastic binding of signaling molecules at low concentrations. Using chemotaxis of sperm cells as application example, we address the classic problem of chemotaxis towards a single target. We reveal a fundamental relationship between the speed of chemotactic steering and the strength of directional fluctuations that result from the amplification of noise in the chemical input signal. This relation implies a trade-off between slow, but reliable, and fast, but less reliable, steering. By formulating the problem of optimal navigation in the presence of noise as a Markov decision process, we show that dynamic switching between reliable and fast steering substantially increases the probability to find a target, such as the egg. Intriguingly, this decision making would provide no benefit in the absence of noise. Instead, decision making is most beneficial, if chemical signals are above detection threshold, yet signal-to-noise ratios of gradient measurements are low. This situation generically arises at intermediate distances from a target, where signaling molecules emitted by the target are diluted, thus defining a `noise zone' that cells have to cross. Our work addresses the intermediate case between well-studied perfect chemotaxis at high signal-to-noise ratios close to a target, and random search strategies in the absence of navigation cues, e.g. far away from a target. Our specific results provide a rational for the surprising observation of decision making in recent experiments on sea urchin sperm chemotaxis. The general theory demonstrates how decision making enables chemotactic agents to cope with high levels of noise in gradient measurements by dynamically adjusting the persistence length of a biased persistent random walk.Comment: 9 pages, 5 figure

Sperm navigation mapped on a Markov decision process.

<p>(A,B) Binning of (<i>R</i>, Ψ)-phase space and sketch of trajectories for ‘low-gain’ (white) and ‘high-gain’ (black) steering. (C) Illustration of a single decision: Starting in a state 1, the player first chooses between two actions, i.e. ‘low-gain’ steering or ‘high-gain’ steering. This choice determines the transition probabilities <i>L</i>_<i>ij</i> for jumping to a different state, here labelled 2 and 3. (D) Illustration of a memoryless decision strategy, assigning a choice of action to each state. The figure shows coarse bins for sake of illustration.</p

Chemotactic success with decision making.

<p>Success probability <i>P</i>(<i>R</i>₀) for the optimal decision strategy, resulting from switching between ‘low-gain’ and ‘high-gain’ steering, as function of initial distance <i>R</i>₀ to the egg for the case of noise-free concentration measurements (A), and physiological levels of sensing noise (B) (red squares). For comparison, success probabilities for strategies without decision making are shown (circles). (C,D) Optimal decision strategies for the cases shown in panel A and B. Greyscale represents prediction frequency of ‘high-gain’ steering, using a cohort of MDPs obtained by bootstrapping, see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006109#pcbi.1006109.s001" target="_blank">S1 Appendix</a> for details. Arrows and dashed lines indicate zone boundaries as introduced in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006109#pcbi.1006109.g002" target="_blank">Fig 2</a>. (E,F) Spatial sensitivity analysis of optimal strategies: Shown is the change in chemotactic range as function of cut-off distance <i>R</i>_<i>c</i> for hybrid strategies that employ the optimal strategy for <i>R</i> < <i>R</i>_<i>c</i>, and either ‘low-gain’ steering (white circles) or ‘high-gain’ steering (black circles) else. Positive values indicate a benefit of decision making at the respective distance to the egg. Parameters, see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006109#pcbi.1006109.s001" target="_blank">S1 Appendix</a>.</p

Decision making in chemotaxis of sea urchin sperm.

<p>(A) Helical swimming path of a sea urchin sperm cell (black) with helix centreline (red), while navigating in a concentration field of the chemoattractant resact [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006109#pcbi.1006109.ref016" target="_blank">16</a>]. The concentration field is cylindrically symmetric with symmetry axis parallel to the <i>z</i>-axis (indicated in blue). (B) Projection of the same swimming path on the <i>xy</i>-plane. Dots mark the beginning (black) and peak (red) of ‘high-gain’ steering phases (or off-responses [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006109#pcbi.1006109.ref016" target="_blank">16</a>]). The concentration field is indicated by blue circles. (C) From the swimming path and the local gradient direction, we can determine a time-dependent rate <i>γ</i>(<i>t</i>) of helix bending towards the gradient [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006109#pcbi.1006109.ref016" target="_blank">16</a>]. The beginning of a ‘high-gain’ steering phase is defined as the level-crossing of <i>γ</i>(<i>t</i>) above its median as is indicated by black dots. Peaks of <i>γ</i>(<i>t</i>) are indicated by red dots. (D) Scatter plot of the orientation angle Ψ and local concentration <i>c</i> at the beginning of ‘high-gain’ steering phases (<i>n</i> = 9 cells). ‘High-gain’ steering is predominantly initiated for Ψ > <i>π</i>/2 (grey shading).</p

Decision making improves sperm chemotaxis in the presence of noise

<div><p>To navigate their surroundings, cells rely on sensory input that is corrupted by noise. In cells performing chemotaxis, such noise arises from the stochastic binding of signalling molecules at low chemoattractant concentrations. We reveal a fundamental relationship between the speed of chemotactic steering and the strength of directional fluctuations that result from the amplification of noise in a chemical input signal. This relation implies a trade-off between steering that is slow and reliable, and steering that is fast but less reliable. We show that dynamic switching between these two modes of steering can substantially increase the probability to find a target, such as an egg to be found by sperm cells. This decision making confers no advantage in the absence of noise, but is beneficial when chemical signals are detectable, yet characterized by low signal-to-noise ratios. The latter applies at intermediate distances from a target, where signalling molecules are diluted, thus defining a ‘noise zone’ that cells have to cross. Our results explain decision making observed in recent experiments on sea urchin sperm chemotaxis. More generally, our theory demonstrates how decision making enables chemotactic agents to cope with high levels of noise in gradient sensing by dynamically adjusting the persistence length of a biased random walk.</p></div

Simple implementation of optimal decision making.

<p>(A) Signalling variables <i>p</i> and <i>q</i> contain information about the helix orientation angle Ψ and distance <i>R</i> to the target. Contour levels for conditional probability densities (red) and (black) (corresponding to 1%, 10%, 50%, 90% percentiles; <i>R</i> = 1.5mm). (B) Relative frequency of ‘high-gain’ steering predicted by the optimal decision strategy, for given combination of (<i>p</i>, <i>q</i>). We define a decision boundary Θ(<i>p</i>) (yellow) by a piecewise linear fit to the 50%-contour line (up to <i>p</i> = 5ms, corresponding to a limit of sufficiently reliable state estimation, see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006109#pcbi.1006109.s001" target="_blank">S1 Appendix</a>). (C) Simulated swimming path using this decision rule with dynamic switching between ‘high-gain’ steering (red) and ‘low-gain’ steering (black); projected on <i>xy</i>-plane. The chemoattractant concentration in this plane is shown (blue gradient), together with the boundary of the noise zone. (D) Success probability <i>P</i>(<i>R</i>₀) for full simulations with simple decision making (red) as a function of initial distance <i>R</i>₀ to the egg. For comparison, success probabilities for ‘low-gain’ steering (white) and ‘high-gain’ steering (black) are shown. (E) The effective chemotactic range with decision making (red) is larger than for an optimal constant gain factor (black). Parameters, see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006109#pcbi.1006109.s001" target="_blank">S1 Appendix</a>.</p

Helical chemotaxis in the presence of sensing noise.

<p>(A) Chemoattractant molecules bind to receptors on the cell membrane. (B) The sequence of binding events defines a stochastic input signal <i>s</i>(<i>t</i>) with rate <i>b</i>(<i>t</i>), <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006109#pcbi.1006109.e001" target="_blank">Eq 1</a>. (C) A sperm cell swims along a helical swimming path (black), whose centreline (red) can bend in the direction of a concentration gradient (blue). (D) Helical swimming in a concentration gradient causes a periodic modulation of the rate <i>b</i>(<i>t</i>) of binding events (red). Representative realization of input signal <i>s</i>(<i>t</i>) (black, low-pass filtered for visualization). This signal dynamically regulates the path curvature <i>κ</i>(<i>t</i>), here shown in the absence of sensing noise (red) and for stochastic input signal (black). (E) Example swimming paths with and without sensing noise for two values of the gain factor (‘low-gain’ steering <i>ρ</i>_low = 1, ‘high-gain steering’ <i>ρ</i>_high = 10). Egg cell (yellow disk). (F) Signal-to-noise ratio (SNR) as a function of distance <i>R</i> from the egg. The SNR defines a ‘noise zone’ spanning intermediate distances <i>R</i>, bounded by a noise zone boundary , where SNR = 1, and a spatial limit of chemosensation , where <i>c</i>(<i>R</i>) = (λ<i>T</i>)⁻¹. (G) Probability to find the egg as a function of gain factor <i>ρ</i> for initial distance <i>R</i>₀ = 3 mm to the egg (and random initial orientation). Without sensing noise, the success probability increases monotonically with <i>ρ</i>, while in the presence of noise, this probability displays a maximum at an optimal <i>ρ</i>. Maximum search time 300s. Error bars smaller than symbols. Parameters chosen to match experiment, see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006109#pcbi.1006109.s001" target="_blank">S1 Appendix</a>.</p