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

Charakterisierung erkennbarer Baumreihen über starken Bimonoiden durch gewichtete MSO-Logik

Endliche Wortautomaten ermöglichen es, reguläre Wortsprachen sowohl zu erkennen als auch zu erzeugen. Julius Richard Büchi gelang es, diese erkennbaren Wortsprachen mithilfe der monadischen Logik zweiter Stufe, kurz MSO, zu charakterisieren [7, 19]. Dieses Ergebnis wurde dann auf erkennbare Baumsprachen, das heißt Mengen von geordneten Bäumen, die durch einenAufwärtsbaumautomaten erkannt werden, erweitert [11, 28]. Anstelle der <-Relation auf den Positionen eines Wortes tritt dabei die Kindrelation edgei(x; y) für die Positionen eines Baumes. Die erkennbaren Wort- und Baumsprachen haben breite Anwendung in der Informatik gefunden. Zu den bekanntesten gehören beispielsweise reguläre Ausdrücke und Syntaxbäume vieler Programmiersprachen. Im Zusammenspiel mit XML ist die Schemasprache RelaxNG zur Dokumentvalidierung [9, 29], im Gegensatz zu XML-Schema, durch die reiche Theorie erkennbarer Baumsprachen fundiert

The Orchestration Stack: The Impossible Task of Designing Software for Unknown Future Post-CMOS Hardware

Future systems based on post-CMOS technologies will be wildly heterogeneous, with properties largely unknown today. This paper presents our design of a new hardware/software stack to address the challenge of preparing software development for such systems. It combines well-understood technologies from different areas, e.g., network-on-chips, capability operating systems, flexible programming models and model checking. We describe our approach and provide details on key technologies

Model Checking Techniques for Design and Analysis of Future Hardware and Software Systems

Computer hardware and software laid the foundation for fundamental innovations in science, technology, economics and society. Novel application areas generate an ever-increasing demand for computation power and storage capacities. Classic CMOS-based hardware and the von Neumann architecture are approaching their limits in miniaturization, power density and communication speed. To meet future demands, researchers work on new device technologies and architecture approaches which in turn require new algorithms and a hardware/software co-design to exploit their capabilities. Since the overall system heterogeneity and complexity increases, the challenge is to build systems with these technologies that are both correct and performant by design. Formal methods in general and model checking in particular are established verification methods in hardware design, and have been successfully applied to many hardware, software and integrated hardware/software systems. In many systems, probabilistic effects arise naturally, e.g., from input patterns, production variations or the occurrence of faults. Probabilistic model checking facilitates the quantitative analysis of performance and reliability measures in stochastic models that formalize this probabilism. The interdisciplinary research project Center for Advancing Electronics Dresden, cfaed for short, aims to explore hardware and software technologies for future information processing systems. It joins the research efforts of different groups working on technologies for all system layers ranging from transistor device research over system architecture up to the application layer. The collaborations among the groups showed a demand for new formal methods and enhanced tools to assist the design and analysis of technologies at all system layers and their cross-layer integration. Addressing these needs is the goal of this thesis. This work contributes to probabilistic model checking for Markovian models with new methods to compute two essential measures in the analysis of hardware/software systems and a method to tackle the state-space explosion problem: 1) Conditional probabilities are well known in stochastic theory and statistics, but efficient methods did not exist to compute conditional expectations in Markov chains and extremal conditional probabilities in Markov decision processes. This thesis develops new polynomial-time algorithms, and it provides a mature implementation for the probabilistic model checker PRISM. 2) Relativized long-run and relativized conditional long-run averages are proposed in this work to reason about probabilities and expectations in Markov chains on the long run when zooming into sets of states or paths. Both types of long-run averages are implemented for PRISM. 3) Symmetry reduction is an effective abstraction technique to tame the state-space explosion problem. However, state-of-the-art probabilistic model checkers apply it only after building the full model and offer no support for specifying non-trivial symmetric components. This thesis fills this gap with a modeling language based on symmetric program graphs that facilitates symmetry reduction on the source level. The new language can be integrated seamlessly into the PRISM modeling language. This work contributes to the research on future hardware/software systems in cfaed with three practical studies that are enabled by the developed methods and their implementations. 1) To confirm relevance of the new methods in practice and to validate the results, the first study analyzes a well-understood synchronization protocol, a test-and-test-and-set spinlock. Beyond this confirmation, the analysis demonstrates the capability to compute properties that are hardly accessible to measurements. 2) Probabilistic write-copy/select is an alternative protocol to overcome the scalability issues of classic resource-locking mechanisms. A quantitative analysis verifies the protocol's principle of operation and evaluates the performance trade-offs to guide future implementations of the protocol. 3) The impact of a new device technology is hard to estimate since circuit-level simulations are not available in the early stages of research. This thesis proposes a formal framework to model and analyze circuit designs for novel transistor technologies. It encompasses an operational model of electrical circuits, a functional model of polarity-controllable transistor devices and algorithms for design space exploration in order to find optimal circuit designs using probabilistic model checking. A practical study assesses the model accuracy for a lab-device based on germanium nanowires and performs an automated exploration and performance analysis of the design space of a given switching function. The experiments demonstrate how the framework enables an early systematic design space exploration and performance evaluation of circuits for experimental transistor devices.:1. Introduction 1.1 Related Work 2. Preliminaries 3. Conditional Probabilities in Markovian Models 3.1 Methods for Discrete- and Continuous-Time Markov Chains 3.2 Reset Method for Markov Decision Processes 3.3 Implementation 3.4 Evaluation and Comparative Studies 3.5 Conclusion 4. Long-Run Averages in Markov Chains 4.1 Relativized Long-Run Average 4.2 Conditional State Evolution 4.3 Implementation 4.4 Conclusion 5. Language-Support for Immediate Symmetry Reduction 5.1 Probabilistic Program Graphs 5.2 Symmetric Probabilistic Program Graphs 5.3 Implementation 5.4 Conclusion 6. Practical Applications of the Developed Techniques 6.1 Test-and-Test-and-Set Spinlock: Quantitative Analysis of an Established Protocol 6.2 Probabilistic Write/Copy-Select: Quantitative Analysis as Design Guide for a Novel Protocol 6.3 Circuit Design for Future Transistor Technologies: Evaluating Polarity-Controllable Multiple-Gate FETs 7. Conclusion Bibliography Appendices A. Conditional Probabilities and Expectations A.1 Selection of Benchmark Models A.2 Additional Benchmark Results A.3 Comparison PRISM vs. Storm B. Language-Support for Immediate Symmetry Reduction B.1 Syntax of the PRISM Modeling Language B.2 Multi-Core Example C. Practical Applications of the Developed Techniques C.1 Test-and-Test-and-Set Spinlock C.2 Probabilistic Write/Copy-Select C.3 Circuit Design for Future Transistor Technologie

Charakterisierung erkennbarer Baumreihen über starken Bimonoiden durch gewichtete MSO-Logik

Endliche Wortautomaten ermöglichen es, reguläre Wortsprachen sowohl zu erkennen als auch zu erzeugen. Julius Richard Büchi gelang es, diese erkennbaren Wortsprachen mithilfe der monadischen Logik zweiter Stufe, kurz MSO, zu charakterisieren [7, 19]. Dieses Ergebnis wurde dann auf erkennbare Baumsprachen, das heißt Mengen von geordneten Bäumen, die durch einenAufwärtsbaumautomaten erkannt werden, erweitert [11, 28]. Anstelle der <-Relation auf den Positionen eines Wortes tritt dabei die Kindrelation edgei(x; y) für die Positionen eines Baumes. Die erkennbaren Wort- und Baumsprachen haben breite Anwendung in der Informatik gefunden. Zu den bekanntesten gehören beispielsweise reguläre Ausdrücke und Syntaxbäume vieler Programmiersprachen. Im Zusammenspiel mit XML ist die Schemasprache RelaxNG zur Dokumentvalidierung [9, 29], im Gegensatz zu XML-Schema, durch die reiche Theorie erkennbarer Baumsprachen fundiert

Charakterisierung erkennbarer Baumreihen über starken Bimonoiden durch gewichtete MSO-Logik

Endliche Wortautomaten ermöglichen es, reguläre Wortsprachen sowohl zu erkennen als auch zu erzeugen. Julius Richard Büchi gelang es, diese erkennbaren Wortsprachen mithilfe der monadischen Logik zweiter Stufe, kurz MSO, zu charakterisieren [7, 19]. Dieses Ergebnis wurde dann auf erkennbare Baumsprachen, das heißt Mengen von geordneten Bäumen, die durch einenAufwärtsbaumautomaten erkannt werden, erweitert [11, 28]. Anstelle der <-Relation auf den Positionen eines Wortes tritt dabei die Kindrelation edgei(x; y) für die Positionen eines Baumes. Die erkennbaren Wort- und Baumsprachen haben breite Anwendung in der Informatik gefunden. Zu den bekanntesten gehören beispielsweise reguläre Ausdrücke und Syntaxbäume vieler Programmiersprachen. Im Zusammenspiel mit XML ist die Schemasprache RelaxNG zur Dokumentvalidierung [9, 29], im Gegensatz zu XML-Schema, durch die reiche Theorie erkennbarer Baumsprachen fundiert

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