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Edited by: Pulin Gong, University of Sydney, Australia

Reviewed by: Doekele G. Stavenga, University of Groningen, Netherlands; Daniel Tranchina, New York University, USA

*Correspondence: Zhuoyi Song

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Many diurnal photoreceptors encode vast real-world light changes effectively, but how this performance originates from photon sampling is unclear. A 4-module biophysically-realistic fly photoreceptor model, in which information capture is limited by the number of its sampling units (microvilli) and their photon-hit recovery time (refractoriness), can accurately simulate real recordings and their information content. However, sublinear summation in quantum bump production (

Fly photoreceptors can sample light changes across a truly astronomical input range—from a few photons in nightly shadows to billions in direct sunlight (Van Hateren,

Experiments indicate that each microvillus houses a full set of phototransduction reactants, from the rhodopsin molecules to the light-gated ion channels (Hardie and Postma,

Simulations imply that two mechanisms largely govern a fly photoreceptor's light adaptation: (i) its sample rate (QB rate) saturates, as more microvilli become refractory; and (ii) its sample waveform (QB size) shrinks due to Ca^{2+}-dependent feedback and reduced electromotive force as the cell depolarizes (Juusola and Hardie,

The sublinear bump summation hypothesis states that when more than one photon hits the same microvillus at the same time, multiple rhodopsins can be activated, but the resultant QB will be smaller than the sum of those produced independently. This could reduce the QB/photon gain by several folds (Pumir et al.,

The main aim of this paper is to quantify the probabilities for two or more photons hitting the same microvillus at the same time, and to elucidate what these events would mean to gain control in light adaptation. We do this by using the

In this paper,

A rhabdomere (Figure

Rhabdomeres of different species have different numbers of microvilli, which likely reflects each species' structural adaptations to different lifestyles and habitats. For example, the outer photoreceptors (R1–R6) of a dawn/dusk-active slow-flying fruit fly (

In the next sections, we explore the structural limits of fly photoreceptors' encoding capabilities further. We ask: (i) whether an elementary form of gain control could directly result from the sampling process alone, (ii) and, how much can sublinear bump summation contribute to their light adaptation. But before we answer these questions, we define the underlying assumptions, and derive the relevant probabilities involved in photon sampling.

For deriving the light absorption statistics of a fly photoreceptor, the following assumptions were made explicitly:

Given the assumptions above, if we define a random variable _{u} is the number of microvilli in the rhabdomere (in a _{u} = 30,000), and _{ph} is the number of photons in the light pulse at one time-bin, Δ

Equation (1) can be used to describe the photon absorption probability of a single microvillus. This is because the average probability that one photon hits a given microvillus is _{ph} −

Subject to certain conditions, approximations could be made to _{ph} and _{u} are much larger than _{u}, and 1 ≪ _{u} are satisfied.

In practice, all approximations made in Equation (2) are valid near the region of _{M}, if _{M}, ^{7−9} photons/s to a photoreceptor's receptive field (Warrant and Mcintyre,

For the ease of calculation, a random photon absorption model based on Poisson statistics is sometimes better than that based on binomial statistics, especially at dim light conditions (

In the section Derivation, a single microvillus is viewed as an independent photon-sampling unit. Following assumption (3) and (4), the random number of absorptions for different microvilli (_{m}, _{μ}) are independent, identically distributed Poisson random variables with mean

Here, we propose a practical way to solve this problem by considering the whole rhabdomere as a single unit. From Equation (1), one can calculate

One could denote such a distribution as a compound binomial distribution: _{u}, _{ph}, 1/_{u}). Therefore, the expected number of microvilli that absorbs exactly _{u}_{u}_{n} (_{n} is the maximum number that lets _{u}_{n}) > 1). In theory, if a random variable (x) were drawn from a Poisson distribution, x would not be limited by _{n}. However, in the simulations, this digitization limit exits: the minimum number of microvilli to absorb x photons is one.

An alternative (easier) way to compute the various probabilities of interest is by modeling the photon absorption process as a multinomial process. At each time incident, the distribution of _{ph} photons over _{u} microvilli is multinomial with size parameter equal to _{ph}, and probability vector of length _{u} with each element equal to 1/_{u}.

Assuming that photon-hits to a microvillus are independent across the time-bins, the extension to continuous light stimuli is straightforward. One can simply repeat the same light absorption process at each time-bin. However, the problem that still needs addressing is the sampling rate of the input stimuli; how long should the selected time-bin be?

As light intensity (input) increases, a photoreceptor's voltage response (output) becomes faster, utilizing a broader temporal frequency bandwidth. For slow-flying

When

The question is - how to estimate the total number of absorbed photons as precisely as possible? Considering the attenuating factors down the propagating light path, it is hard to estimate the number of absorbed photons from the photons emitted by the light source. Nevertheless, this number can be extrapolated more conveniently by using intracellular photoreceptor recordings. After prolonged dark-adaptation, QBs can be counted to continuous dim illumination at each second and used for extrapolating the photon-hit rates for brighter light levels (Laughlin and Lillywhite,

Although we use QB counts to determine the light input to the

All these extrinsic and intrinsic changes in the absorption spectra and efficiency may bias the extrapolated photon rates. As this study assesses steady-state adapted photoreceptors, we can ignore nonlinear optical attenuation effects, such as intracellular pupil activation. Nevertheless, while the light intensity values (as extrapolated from the bump counts) may overestimate the photons-hits, these provide the best available and reasonably realistic photon rates (photons/s) for different light inputs, irrespective of the number of QBs they may evoke.

A microvillus can convert a single photon to a QB. But what happens when two or more photons hit it simultaneously? Computational studies imply that a bigger QB might be produced (Figure

We use the gain between the LIC output charge (_{out}) to the light intensity input (_{in}) to quantify a photoreceptor's input/output-relationship. For a discrete light pulse, _{in} is composed of _{ph} photons, and _{out} is the summation of all excited QB charges. If we use _{x} is the number of _{n} is the maximum number that lets _{u}_{n}) > 1. It then follows that:

_{M}. For example, the quantum-gain factor is well below 3% when λ_{M} is less than 0.1 (solid square in ^{6} photons/s). However, it approaches to 35% as λ_{M} increases to 1 (hollow square, corresponding to a species with 1000 microvilli stimulated at 10^{6} photons/s). This means that the _{M} is well below 0.1 for fly photoreceptors, as they have tens of thousands of microvilli to sample the available photons. Even in midday sunshine, quantum-gain factor is low (only ~3% for a ^{6} photons/s; the solid square). Thus,

Substitute Equation (5) to Equation (4), _{out}/_{in} is lower bounded in the case of no-summation:
_{QB} is the total number of QBs, and _{A} is the total number of activated microvilli. As multi-photon-hits induce only one QB, it follows that _{QB} is equal to _{A}, which is less or equal to _{ph}.

From Equation (6), the photoreceptor's input-output gain for a discrete light pulse (_{A}/_{ph} and _{A}/_{ph} is the ratio between the amount of activated microvilli (_{A}) and the number of incoming photons (_{ph}) at a particular time bin (Δ

_{out}/_{in} is upper bounded in the case of linear summation:

To focus on the role of photon sampling on gain control, we define _{A}/_{ph} in no summation. More often, sublinear summation occurs in QB production, and the normalized gain is bounded between _{A}/_{ph} and 1 (Figure

Markedly, the lower bound of the system's normalized gain (_{A}/_{ph}) is not influenced by the QB shape or any subsequent phototransduction processes, but is purely determined by photon sampling (the counted hits). From Section Realization of Random Photon Absorption Model _{A}/_{ph} (random variable x is not limited by _{n}):
_{M} > 0. Owing to multi-photon-hit probabilities, photon sampling causes a nonlinear function of λ_{M}, which varies with the light intensity and the number of photoreceptor microvilli. For a given light intensity, λ_{M} → 0 with more microvilli, and the normalized gain (_{A}/_{ph}) approaches 1, realizing linear photon sampling (Figure _{A}/_{ph} decreases with increasing λ_{M}. For a cell with a fixed number of microvilli, the normalized gain (_{A}/_{ph}) decreases with increasing light intensity, providing an elementary form of gain control in light adaptation (Figure

To quantify multi-photon induced

Following the Poisson distribution, the multi-photon-hit probabilities depend directly upon the photon arrival rate. To illustrate this, we first simulate the photon arrivals to one microvillus. The key parameter is _{M} is less than 1, discrete photon-hits are detected over time (Figures _{M}, the fewer photon-hits there are in a fixed time interval. When λ_{M} is greater than 10, photon-hits fluctuate around the mean (Figures

_{M} = 0.01; _{M} = 0.1; _{M} = 10; _{M} = 100. Notably, when λ_{M} is less than 1, discrete photon-hits are detected over time _{M}, the fewer photon-hits there are in a fixed time interval. But at any particular moment, the probability of multi-photon-hits is less than 1. When λ_{M} is greater than 1, photon-hits fluctuate around the mean _{M}, approximating 100% as λ_{M} approaches 5. As the

The percentage of multi-photon-hits (_{M}) can be calculated by Equation (9):

Clearly, _{M} increases with λ_{M}, which is proportional to incoming photon rate and is in reciprocal relationship to the number of microvilli. So, multi-photon-hit-induced _{ph} and _{u} pairs. For example, a _{ph} = 10 and _{ph} = 100, respectively), the percentage of multi-photon-hits on its microvilli is less than 0.17% (bold values in Table _{ph} = 1000, corresponding to a bright midday: 10^{6} photons/s), _{M} is still less than 2%.

_{ph} photons/ms |
_{u} |
|||||
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10 | 1.66 | 0.33 | 0.08 | 0.03 | 0.01 | |

100 | 15.69 | 3.29 | 0.83 | 0.33 | 0.06 | |

1 k | 87.57 | 29.64 | 8.10 | 3.30 | 1.66 | 0.55 |

10 k | 100 | 99.15 | 61.18 | 29.66 | 15.74 | 5.45 |

100 k | 100 | 100 | 99.99 | 99.15 | 87.67 | 45.47 |

In this section, we quantify how quantum-gain-nonlinearity contributes to gain control (gray area in Figure _{g}), which is the difference between the normalized linear gain and the actual normalized gain (

Using _{g} increases with λ_{M}. It could rise to 35% as λ_{M} approaches 1 (indicated by the hollow square). This means that the _{M} is well below 0.1. Even in midday sunshine, _{g} is low (Figure

^{5} photons/s); ^{5} photons/s). In each sub-figure, the dotted line (blue) is the light input and the solid line (red) is the simulation result. Because these curves overlap perfectly, the blue curves are right-shifted by 1 ms.

Because

In this report, we gave a detailed account of

We used

By reducing

Importantly,

These results clarify that diurnal fly photoreceptors have enough microvilli (30,000–100,000) to maintain high photon-hit rates without

Finally, we speculate that

ZS, MJ designed the study. ZS, YZ constructed the model and performed the analysis. ZS drafted the manuscript with all authors editing it.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

ZS thanks EPSRC-funded 2020 Science fellowship (EP/I017909/1) for funding. MJ thanks these funding sources for supporting this work: the State Key Laboratory of Cognitive Neuroscience and Learning open research, Natural Science Foundation of China Project 30810103906, Jane and Aatos Erkko Foundation Fellowship, Leverhulme Trust Grant RPG-2012-567, and Biotechnology and Biological Sciences Research Council Grants BB/F012071/1, BB/D001900/1, BB/H013849/1, and BB/M009564/1. The authors thank Dr. Mathew Joseph for rechecking the mathematical formulations, Dr. Diana Rien for reading the manuscripts. The authors would particularly thank Dr. Samuel Solomon and the reviewers for critical suggestions.

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We recently developed a full biophysical fly photoreceptor model that can transduce any light intensity time series input into macroscopic output, the light-induced current (LIC) (Song et al.,

These simulations were performed by the procedure described in Figure

We have further shown (Song et al.,

Light input statistics

Number of microvilli (sampling units)

Stochastic refractory period in each microvillus

◦ The success of transducing a photon into a quantum bump depends upon refractoriness of its phototransduction reactions. This means that a microvillus cannot respond to the next photons until its phototransduction reactions have recovered from the previous photon absorption, which takes about 50-300 ms

Stochastic latency distribution (the time delay between photon arrivals to the trigged quantum bump)

Bump waveform adaptation over time

Voltage feedback from the plasma membrane

The details how these mechanisms jointly modulate photoreceptor output dynamics to different light intensity time series input can be found in our previous publications (Song et al.,