Stability is overrated. A cadaver occupies an energetically stable state. Stable systems cannot adapt to changes in their environment. For a living creature, it is more useful to be finely balanced. Like a coin on its edge, the slightest perturbation unleashes enormous changes its state. So it is with hair-cell bundles.

The hair bundle is a phylogenetically ancient sensory structure. It can be found in the mammalian hearing and vestibular systems, and in the lateral line systems of fish and amphibians. In each case the hair bundles detect mechanical disturbances in an ambient fluid, however, the nature of these disturbances does vary from system to system and from organism to organism. The hair bundles of the ears are tuned to oscillatory displacements, those of the vestibular system detect bodily accelerations, and those in the lateral line are sensitive to pulsatile changes in water pressure. These systems simultaneously boast impressive sensitivity and an extreme dynamic range. It is implausible that a linear system, with outputs directly proportional to inputs, could accommodate the millionfold range of disturbance amplitudes that hair bundles routinely process. Thus, these electromechanical structures must be actively applying non-linear amplification to their displacements. The mechanisms of this amplification, however, are poorly understood.

In order to address this mystery, Hudspeth and colleagues first built a dynamical model capable of replicating the sensory performance and variety of states hair bundles are observed to achieve [1]. While I apologize to the reader for including an equation in a blog post, Hudspeth’s relatively simple model consisted of a system of just two differential equations and two variable parameters:

eq1

in which x is the bundle’s displacement, f is the force owing to adaptation, a is a negative stiffness owing to gating of the transduction channel, τ is the timescale of adaptation, b is a compliance coupling bundle displacement to adaptation, K is the sum of the bundle’s load stiffness and pivot-spring stiffness, Fc is the sum of the constant force intrinsic to the hair bundle and that owing to the load, and F is any time-dependent force applied to the bundle. ηx and ηf are noise terms. Parameters a, b, and τ were held constant, and the state space of the system was explored by varying K and Fc; the load stiffness and constant force, respectively. The resulting state-space of these parameter manipulations is displayed in the figure below.

fig1ABC-1

In the white quiescent regions, the hair bundles demonstrate no movement at all. In the green bistable regions, the structure is mostly quiescent, but will occasionally lurch from one configuration to another. In the orange region the hair bundles spontaneously oscillate between configurations. Furthermore, regular gradients of oscillation amplitude and frequency exist within this regime. This is an elegant mathematical tale, but (A) does it exist it biology? And (B) what does this model have to do with amplification? Both these questions are addressed in Hudspeth and colleagues’  experimental paper on the matter [2].

To probe their in vitro state space, a novel apparatus was devised to mechanically clamp bullfrog hair bundles in a manner analogous to the venerable patch clamp. Mechanical force was applied to the hair bundle via a flexible glass filament. The mechanical load, once set, was maintained continuously by optically measuring the hair bundles’ displacement and adjusting the force applied with a piezoelectric actuator. Thus, the state-space mapped out theoretically could be experimentally tested by adjusting the load stiffness and constant force parameters.

In general, the bullfrog hair bundles behaved in a manner consistent with the model’s predictions. In small- and medium-sized hair cell bundles there exists a contiguous region of state-space in which spontaneous oscillations occur. In addition, the amplitude and frequency tunings of these oscillations follow the predicted gradient. Thus, the dynamical model appears to be a good representation of the biological system.

As shown in the figure above, the theoretical state-space contains a special region along the edge of the oscillatory regime. This narrow space, labeled as the line of Hopf bifurcations, is where the dynamical system is least stable: precariously balanced between periodic and quiescent behaviors. This instability allows small perturbations to effect large state changes, or amplification. Indeed, when hair bundles were mechanically clamped to states near the edge of the oscillatory regime and then stimulated with periodic displacements appropriate to their state-position, the degree of amplitude non-linearity in their resonant response was maximized. Furthermore frequency tuning was most narrow in this border region, perhaps indicative of the fragility of dynamical system.

Dynamical systems are common in nature, doubly so in the complex substrates of cellular biology. Force multiplication is not a concept limited to sensory systems. A minuscule concentration of signaling molecule can trigger macroscale developmental patterning, or the release very few hormones can begin complex cascades. The leveraging of unstable dynamical systems to achieve large outputs from small inputs may one day be regarded as a general principle of biology.

Learn more when Dr. Jim Hudspeth discusses  “Making an effort to listen: mechanical amplification by myosin molecules and ion channels in hair cells of the inner ear” at 4 pm this Tuesday, March 29th in the CNBC Marilyn Farquar Seminar Room.

Sources

  1. Maoiléidigh, D.Ó., Nicola, E.M. and Hudspeth, A.J., 2012. The diverse effects of mechanical loading on active hair bundles. Proceedings of the National Academy of Sciences109(6), pp.1943-1948.
  2. Salvi, J.D., Maoiléidigh, D.Ó., Fabella, B.A., Tobin, M. and Hudspeth, A.J., 2015. Control of a hair bundle’s mechanosensory function by its mechanical load. Proceedings of the National Academy of Sciences, 112(9), pp.E1000-E1009.

 

Burke Rosen is a first year student in the UCSD Neurosciences Graduate Program. He is currently between rotations and is interested in using clever signals analyses to make somewhat educated guesses about unfathomably complex phenomena. 

 

 

 

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