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The energy-speed-accuracy trade-off in sensory adaptation

Adaptation is the essential process by which an organism becomes better suited to its environment. The benefits of adaptation are well documented, but the cost it incurs remains poorly understood. Here, by analysing a stochastic model of a minimum feedback network underlying many sensory adaptation systems, we show that adaptive processes are necessarily dissipative, and continuous energy consumption is required to stabilize the adapted state. Our study reveals a general relation among energy dissipation rate, adaptation speed and the maximum adaptation accuracy. This energy-speed-accuracy relation is tested in the Escherichia coli chemosensory system, which exhibits near-perfect chemoreceptor adaptation. We identify key requirements for the underlying biochemical network to achieve accurate adaptation with a given energy budget. Moreover, direct measurements confirm the prediction that adaptation slows down as cells gradually de-energize in a nutrient-poor medium without compromising adaptation accuracy. Our work provides a general framework to study cost-performance tradeoffs for cellular regulatory functions and information processing.

Thermodynamic Costs of Information Processing in Sensory Adaptation

Biological sensory systems react to changes in their surroundings. They are characterized by fast response and slow adaptation to varying environmental cues. Insofar as sensory adaptive systems map environmental changes to changes of their internal degrees of freedom, they can be regarded as computational devices manipulating information. Landauer established that information is ultimately physical, and its manipulation subject to the entropic and energetic bounds of thermodynamics. Thus the fundamental costs of biological sensory adaptation can be elucidated by tracking how the information the system has about its environment is altered. These bounds are particularly relevant for small organisms, which unlike everyday computers, operate at very low energies. In this paper, we establish a general framework for the thermodynamics of information processing in sensing. With it, we quantify how during sensory adaptation information about the past is erased, while information about the present is gathered. This process produces entropy larger than the amount of old information erased and has an energetic cost bounded by the amount of new information written to memory. We apply these principles to the E. colis chemotaxis pathway during binary ligand concentration changes. In this regime, we quantify the amount of information stored by each methyl group and show that receptors consume energy in the range of the information-theoretic minimum. Our work provides a basis for further inquiries into more complex phenomena, such as gradient sensing and frequency response.

Kinetic versus energetic discrimination in biological copying.

We study stochastic copying schemes in which discrimination between a right and a wrong match is achieved via different kinetic barriers or different binding energies of the two matches. We demonstrate that, in single-step reactions, the two discrimination mechanisms are strictly alternative and cannot be mixed to further reduce the error fraction. Close to the lowest error limit, kinetic discrimination results in a diverging copying velocity and dissipation per copied bit. On the other hand, energetic discrimination reaches its lowest error limit in an adiabatic regime where dissipation and velocity vanish. By analyzing experimentally measured kinetic rates of two DNA polymerases, T7 and Polγ, we argue that one of them operates in the kinetic and the other in the energetic regime. Finally, we show how the two mechanisms can be combined in copying schemes implementing error correction through a proofreading pathway.

Dynamic curvature regulation accounts for the symmetric and asymmetric beats of Chlamydomonas flagella

Cilia and flagella are model systems for studying how mechanical forces control morphology. The periodic bending motion of cilia and flagella is thought to arise from mechanical feedback: dynein motors generate sliding forces that bend the flagellum, and bending leads to deformations and stresses, which feed back and regulate the motors. Three alternative feedback mechanisms have been proposed: regulation by the sliding forces, regulation by the curvature of the flagellum, and regulation by the normal forces that deform the cross-section of the flagellum. In this work, we combined theoretical and experimental approaches to show that the curvature control mechanism is the one that accords best with the bending waveforms of Chlamydomonas flagella. We make the surprising prediction that the motors respond to the time derivative of curvature, rather than curvature itself, hinting at an adaptation mechanism controlling the flagellar beat. DOI: http://dx.doi.org/10.7554/eLife.13258.001

Thermodynamics of error correction

Information processing at the molecular scale is limited by thermal fluctuations. This can cause undesired consequences in copying information since thermal noise can lead to errors that can compromise the functionality of the copy. For example, a high error rate during DNA duplication can lead to cell death. Given the importance of accurate copying at the molecular scale, it is fundamental to understand its thermodynamic features. In this paper, we derive a universal expression for the copy error as a function of entropy production and {\cred work dissipated by the system during wrong incorporations}. Its derivation is based on the second law of thermodynamics, hence its validity is independent of the details of the molecular machinery, be it any polymerase or artificial copying device. Using this expression, we find that information can be copied in three different regimes. In two of them, work is dissipated to either increase or decrease the error. In the third regime, the protocol extracts work while correcting errors, reminiscent of a Maxwell demon. As a case study, we apply our framework to study a copy protocol assisted by kinetic proofreading, and show that it can operate in any of these three regimes. We finally show that, for any effective proofreading scheme, error reduction is limited by the chemical driving of the proofreading reaction.

Motor Regulation Results in Distal Forces that Bend Partially Disintegrated Chlamydomonas Axonemes into Circular Arcs

The bending of cilia and flagella is driven by forces generated by dynein motor proteins. These forces slide adjacent microtubule doublets within the axoneme, the motile cytoskeletal structure. To create regular, oscillatory beating patterns, the activities of the axonemal dyneins must be coordinated both spatially and temporally. It is thought that coordination is mediated by stresses or strains, which build up within the moving axoneme, and somehow regulate dynein activity. During experimentation with axonemes subjected to mild proteolysis, we observed pairs of doublets associating with each other and forming bends with almost constant curvature. By modeling the statics of a pair of filaments, we show that the activity of the motors concentrates at the distal tips of the doublets. Furthermore, we show that this distribution of motor activity accords with models in which curvature, or curvature-induced normal forces, regulates the activity of the motors. These observations, together with our theoretical analysis, provide evidence that dynein activity can be regulated by curvature or normal forces, which may, therefore, play a role in coordinating the beating of cilia and flagella.

Free energy cost of reducing noise while maintaining a high sensitivity.

Living systems need to be highly responsive, and also to keep fluctuations low. These goals are incompatible in equilibrium systems due to the fluctuation dissipation theorem (FDT). Here, we show that biological sensory systems, driven far from equilibrium by free energy consumption, can reduce their intrinsic fluctuations while maintaining high responsiveness. By developing a continuum theory of the E. coli chemotaxis pathway, we demonstrate that adaptation can be understood as a nonequilibrium phase transition controlled by free energy dissipation, and it is characterized by a breaking of the FDT. We show that the maximum response at short time is enhanced by free energy dissipation. At the same time, the low frequency fluctuations and the adaptation error decrease with the free energy dissipation algebraically and exponentially, respectively.

Curvature regulation of the ciliary beat through axonemal twist.

Cilia and flagella are hairlike organelles that propel cells through fluid. The active motion of the axoneme, the motile structure inside cilia and flagella, is powered by molecular motors of the dynein family. These motors generate forces and torques that slide and bend the microtubule doublets within the axoneme. To create regular waveforms the activities of the dyneins must be coordinated. It is thought that coordination is mediated by stresses due to radial, transverse, or sliding deformations, that build up within the moving axoneme. However, which particular component of the stress regulates the motors to produce the observed flagellar waveforms remains an open question. To address this question, we describe the axoneme as a three-dimensional bundle of filaments and characterize its mechanics. We show that regulation of the motors by radial and transverse stresses can lead to a coordinated flagellar motion only in the presence of twist. By comparison, regulation by shear stress is possible without twist. We calculate emergent beating patterns in twisted axonemes resulting from regulation by transverse stresses. The waveforms are similar to those observed in flagella of Chlamydomonas and sperm. Due to the twist, the waveform has non-planar components, which result in swimming trajectories such as twisted ribbons and helices that agree with observations.

Protocols for Copying and Proofreading in Template-Assisted Polymerization

We discuss how information encoded in a template polymer can be stochastically copied into a copy polymer. We consider four different stochastic copy protocols of increasing complexity, inspired by building blocks of the mRNA translation pathway. In the first protocol, monomer incorporation occurs in a single stochastic transition. We then move to a more elaborate protocol in which an intermediate step can be used for error correction. Finally, we discuss the operating regimes of two kinetic proofreading protocols: one in which proofreading acts from the final copying step, and one in which it acts from an intermediate step. We review known results for these models and, in some cases, extend them to analyze all possible combinations of energetic and kinetic discrimination. We show that, in each of these protocols, only a limited number of these combinations leads to an improvement of the overall copying accuracy.

The Energy Cost of Sensory Adaptation

Biological sensory systems adapt to prolonged stimuli in order to maintain high sensitivity in different environments. Sensory adaptations are carried out by various molecular feedback mechanisms. Here, we show that all adaptation dynamics are dissipative and feedback control consumes energy to achieve high adaptation accuracy against intrinsic fluctuations in the underlying molecular signaling pathways. A universal relation among energy dissipation rate, adaptation time, and the optimum adaptation accuracy is established in a general continuum model and for the specific case of adaptation in E. coli chemotaxis. Our study finds that sensory adaptations are fueled by high-energy biomolecules (e.g., ATP), which provide the energy necessary in stabilizing the adapted state. For E. coli chemotaxis, hydrolysis of S-adenosylmethionine (SAM) drives the chemo-receptor adaptation, and the high energy content in SAM is crucial in maintaining the near perfect adaptation of the system. Finally, we point out that the energy-accuracy relation found here has deep connections with the energy dissipation required for molecular level error-correction and information processing.View Large Image | View Hi-Res Image | Download PowerPoint Slide