Near-field Power Density Mapping of Close-to-body Low-power mmWave Devices Massinissa Ziane, Artem Boriskin, Senior Member, IEEE, and Maxim Zhadobov, Senior Member, IEEE Abstract-This letter introduces a method for power density (PD) measurement of low-power millimeter-wave (mmWave) devices, accounting for antenna/body coupling. This technique employs a thin solid absorbing structure with equivalent scattering properties used to convert the absorbed power into a heat pattern measured by an infrared (IR) camera. The measured IR pattern is then used to reconstruct the PD distribution. The lock-in technique is used to enhance the signal-to-noise ratio (SNR). It consists in filtering the ambient IR noise as well as removing the parasitic heat conduction effect. The reduction of noise enables achieving a measurement sensitivity of the order of 1 mW/cm 2 , substantially overcoming the sensitivity without lock-in (> 10 dB). The proposed approach is experimentally validated for a conical horn antenna at 60 GHz. For the first time, this study demonstrates the sensitivity of IR-based measurements sufficient enough to assess the compliance of medium-and lowpower wireless devices above 6 GHz.

Index Terms-Absorbed power density (APD), dosimetry, millimeter-wave measurements, phantoms.

I. INTRODUCTION

T HE increasing need for high data rates, reliable wireless connection and improved latency is resulting in continuous shifting of operating frequencies towards millimeter waves (mmWaves) [START_REF] Rappaport | Millimeter wave mobile communications for 5G cellular: It will work![END_REF], [START_REF]High five[END_REF]. This band is foreseen for numerous applications such as 5G/xG, telemedicine, remote driving, IoT, smart industry and augmented reality [START_REF] Yin | Multiplexing URLLC traffic within eMBB services in 5G NR: Fair scheduling[END_REF][START_REF] Qi | Integrated sensing, computation and communication in B5G cellular internet of things[END_REF][START_REF] Liu | Resource management for millimeter-wave ultra-reliable and low-latency communications[END_REF]. Near-field measurements in free space or in operating environments (e.g. smartphone close to body) represent an essential step in design as well as in performance and compliance testing of wireless devices [START_REF]Assessment of Power Density of Human Exposure to Radio Frequency Fields from Wireless Devices in Close Proximity to the Head and Body (Frequency Range of 6 GHz to 300 GHz)[END_REF], [START_REF]Measurement Procedure for the Assessment of Specific Absorption Rate of Human Exposure to Radio Frequency Fields from Hand-held and Bodyworn Wireless Communication Devices -Human Models, Instrumentation and Procedures (Frequency Range of 4 MHz to 10 GHz)[END_REF].

Infrared (IR) imaging allows for non-perturbing, isotropic, and broadband 2D measurement of the electromagnetic (EM) power distribution with sub-mm resolution, in the far [START_REF] Levesque | Performance of FPA IR cameras and of their improvement by time, space and frequency data processing: Part II -Application to the thermographic measurement of microwave fields[END_REF] as well as in the near field [START_REF] Ribiere-Tharaud | Wideband validation of a phase retrieval process applied to infrared planar near-field measurements[END_REF]- [START_REF] Prost | Measurement of the amplitude and polarization of the electric field by electromagnetic infrared thermography[END_REF]. To this end, an absorbing dielectric structure is used to convert the EM energy into heat. The temperature rise on its surface generates an IR Color versions of one or more of the figures in this article are available online at http://ieeexplore.ieee.org signal remotely recorded by an IR sensor. The EM power distribution is consequently retrieved by post-processing from the measured IR signal. This method is particularly suitable for power density (PD) measurement at mmWaves due to increasing with frequency losses resulting in a more effective conversion of the EM energy to heat.

Accurate and realistic user exposure assessment requires accounting for the impact of the presence of user on a wireless device under test and PD due to the antenna/body interactions. It was demonstrated that ignoring antenna/body coupling may result in a significant (up to a factor of two) underestimation of the absorbed PD [START_REF] Ziane | Antenna/body coupling in the near-field at 60 GHz: Impact on the absorbed power density[END_REF]. A technique for PD measurement accounting for antenna/body interactions at mmWaves was reported in [START_REF] Ziane | High-resolution technique for near-field power density measurement accounting for antenna/body coupling at millimeter waves[END_REF]. This technique employs a thin solid absorbing structure reproducing the scattering characteristics of human skin. In that study, a high input power (several hundreds of mW, corresponding to an incident PD (IPD) of hundreds of mW/cm 2 ) was required for reliable measurements due to a high level of environmental IR noise. This parasitic noise results in a poor signal-to-noise ratio (SNR), limiting so far application of this technique to characterization of high-power wireless devices.

In this letter, we introduce a novel approach for IR-based measurement of PD with reduced noise resulting in enhanced SNR and sensitivity. Limited sensitivity is the major bottleneck preventing the use of IR imaging for dosimetry and overthe-air (OTA) characterization of medium-and low-power wireless devices (such as smartphones, tablets, etc.). Improved sensitivity achieved in this study enables, for the first time, IR-based characterization of medium-and low-power closeto-body mmWave devices with the total radiated power (TRP) down to mW.

The paper is organized as follows. Section II introduces the measurement setup as well as its key building blocksolid thin dielectric structure (referred hereafter as phantom) -designed to reproduce the scattering characteristics of the human skin and acting as mmWave to IR converter. Section III presents the signal processing technique (lock-in) used to suppress the noise level through spectral filtering. In Section IV, the proposed approach is experimentally validated at 60 GHz. Finally, conclusions are drawn in Section V.

II. MEASUREMENT SYSTEM

A. Experimental Setup

The experimental setup is illustrated in Fig. 1. The antenna under test (AUT) is located at distance d from the surface S1 of the phantom (described in detail in Section II.B). A narrowband mmWave power generator (QuinStar, CA, USA) is used to feed the AUT at 60 GHz through a set of WR15 rectangular waveguides. The waveform and frequency of the input signal are controlled using a programmable power supply (HMP 4040, Rohde and Schwarz, Munich, Germany). The input power level is measured through a -20 dB coupler (V752D, Hewlett Packard, CA, USA) using a power meter (E4418B, Agilent, CA, USA). As a validation example, we used a V-band (50-75 GHz) linearly-polarized conical horn antenna with 21.4 dBi gain.

The heat pattern at the surface S2 of the phantom is recorded by an IR camera (SC5600, FLIR Systems, OR, USA) located 26 cm from the phantom. The camera operates in the 3.5-5 µm range with the spatial resolution of 640×512 pixels (for the selected distance, the pixel size is 0.14 mm).

B. Phantom

The thin solid absorbing structure used for conversion of the mmWave power to heat was designed following the approach detailed in [START_REF] Ziane | High-resolution technique for near-field power density measurement accounting for antenna/body coupling at millimeter waves[END_REF]. The phantom is optimized to reproduce the reflection coefficient of the human skin and therefore to perturb the AUT in the same way as it would be perturbed by the presence of a human body. To this end, the thickness (1.8 mm) and the complex permittivity (ϵ r = 12.5 -3.625j at 60 GHz) are jointly chosen to provide the desired performance. The selected thickness results in a higher temperature rise compared to bulk phantoms thus contributing to enhancement of SNR. Note that, even if for such phantom the complex permittivity of the phantom is different from the one of skin, the PD at the phantom/air interface (S1) is the same as the one at the skin/air interface due to the energy conservation principle.

The spatial-temporal heat pattern remotely recorded by the IR camera at S2 is converted to the PD at S1 following the post-processing procedure detailed in [START_REF] Ziane | Exposure assessments in the near-field accounting for antenna/body interactions at millimeter waves: Absorbed power density reconstruction from E-field vector[END_REF]. The phantom structure enables obtaining similar heat distributions at S1 and S2, thus facilitating the PD retrieval simultaneously decreasing the reconstruction error. Note that the PD retrieval procedure does not require all incident EM power to be absorbed in the phantom (here roughly 95 % of power transmitted through the upper surface is absorbed in the phantom). However, the power dissipation in the phantom should be high enough to obtain a measurable temperature rise.

The phantom structure is fabricated using carbon black powder (40 %) and PDMS (Fig. 2(a)) following the process detailed in [START_REF] Guraliuc | Solid phantom for body-centric propagation measurements at 60 GHz[END_REF]. The phantom is attached to a hollow rectangular frame made of acrylonitrile butadiene styrene (ABS). The power reflection coefficient of the fabricated prototype compared to that of human skin is shown in Fig. 2(b). It is equal to 36.7 % for the normal incidence (0.9 % deviation compared to the Gabriel's dry skin model [START_REF] Gabriel | The dielectric properties of biological tissues: II. Measurements in the frequency range 10 Hz to 20 GHz[END_REF]). The maximum difference of 4.6 % found for TM polarization at the pseudo-Brewster angle of 75 • is attributed to the difference in the imaginary part of carbon-PDMS (ϵ ′′ r = 3.63) compared to skin (ϵ ′′ r = 11.9). Note that for most of practical applications, the angular range of interest is limited to 45 • in respect to the normal incidence. Therefore, the mismatch around the Brewster angle is expected to have a negligible effect.

The thermal conductivity k = 0.48 W•m -1 •K -1 and specific heat capacity c = 1259 J• Kg -1 •K -1 were measured using guarded hot disk and differential scanning calorimetry methods, respectively. The hemispherical spectral normal emissivity (ϵ λ,N ) is equal 0.94 (measured using an IR spectrometer at 20 °C in the 3.5-5 µm range). The convective heat transfer coefficient h of the phantom was set to 5 W•m -2 • • C -1 [START_REF] Hahn | Heat Conduction[END_REF].

III. NOISE REDUCTION

Enhancement of sensitivity and SNR in IR-based PD measurements is crucial for enabling characterization of low-power mmWave signals. One way to achieve this is to reduce the noise level of the measured IR signal by means of spectral filtering using the lock-in method. In this method, the amplitude of the AUT input power is modulated by a periodic waveform in order to: 1) remove the ambient noise, thus increasing the SNR, 2) mitigate the heat conduction effect, thus increasing correlation between the heat pattern and mmWave PD distribution. The steady-state lock-in implemented in this study is detailed hereafter.

A. Periodic Excitation

The signal S(t) is used to modulate the input power (squarewave on/off modulation) at frequency f lock = 1/T lock (Fig. 3(a)):

S(t) = 1 2 A(1 + F sw (T lock , t)), ( 1
)
where A is the peak-to-peak amplitude and F sw (T lock , t) is the square-wave function. S(t) can be analytically expressed as the sum of low-frequency (LF) and high-frequency (HF) signal components (Fig. 3(b) and Fig. 3(c), respectively):

S(t) = S LF + S HF (t), (2)
with S LF = 1 2 A and S HF (t) = 1 2 A • F sw (T lock , t).

B. Steady-state Heat Dynamics

Absorbed by the phantom mmWave power induces temperature rise T (t) (Fig. 4(a)). T (t) is defined as the sum of low-frequency and high-frequency thermal components associated with S LF and S HF (t), respectively. Accounting for the environmental ambient noise, described by Gaussian white noise N (t), the total induced temperature rise T (t) can be expressed as:

T (t) = T LF (t) + T HF (t) + N (t), (3)
where T LF (t) is the low-frequency time-average temperature increase and T HF (t) is the high-frequency oscillating part of T (t). T (t) is measured at steady state, where T HF (t) and T LF (t) are expressed as:

T HF (t) = ∆T 0 • F tw (T lock , t), (4)
and:

T LF (t) = T ss , (5)
where ∆T 0 is the peak-to-peak amplitude of temperature oscillations at steady-state, F tw (T lock , t) is the oscillation function schematically represented by a triangular-wave function in Fig. 4(a), and T ss is the mean steady-state temperature. Under experimental conditions of this study, the steady state is reached after roughly t 0 = 5 min of exposure.

C. Frequency of Amplitude Modulation

∆T 0 represents the temperature rise (peak-to-peak amplitude of T HF (t)) during T lock /2 (Fig. 4(a)). The PD is directly proportional to ∆T 0 . However, this is valid only for sufficiently short T lock . Indeed, increasing T lock would result in the parasitic heat conduction effect. On the other hand, reducing T lock would decrease the useful signal ∆T 0 . The optimal tradeoff between 1) minimizing the heat conduction effect and hence maximizing the correlation between PD and ∆T 0 (> 98%) and 2) enhancing the useful signal amplitude guaranteeing a sufficient IR image contrast (>10 dB) is found for T lock = 5 s (f lock = 0.2 Hz). This tradeoff is derived for the phantom employed in this study and AUT with a peak IPD = 22 mW/cm 2 .

D. Noise Filtering

∆T 0 is de-embedded from T (t) by filtering T LF (t) and N (t) in (3). This is performed using the following steps:

T (f ) = +∞ n=-∞ T (n)e -j2πf n . ( 6
)
Fourier transformation is a linear mathematical operation, hence, using (3) in ( 6) yields:

T (f ) = TLF (f ) + THF (f ) + N , (7)
where TLF , THF , and N are the spectra of T LF , T HF , and N , respectively, schematically represented in Fig. 4(b).

2) Spectral Filtering and Inverse Fourier Transformation: Band-pass filtering at f lock allows to remove TLF and N :

THF (f ) = T (f ) • G(f ), (8)
with

G(f ) = 0 for f ̸ = f lock and G(f ) = 1 for f = f lock .
The inverse DTFT of THF is applied to retrieve T HF and consequently ∆T 0 :

T HF (n) = 2π THF (f )e j2πf n . (9)
A part of N at f lock , which cannot be filtered, is mitigated by maximizing ∆t and f s . To this end, we used ∆t = 180 s, f s = 50 Hz.

IV. EXPERIMENTAL VALIDATION

As a validation example, measurements are performed for a conical horn antenna for A = 36 mW at d = 3.5 mm (Fig. 5). In this configuration, the peak IPD equals to 22 mW/cm 2 . The distance d = 3.5 mm corresponds to a near-field PD pattern with multiple lobes, requiring higher measurement accuracy compared to a far-field configuration where the PD distribution has only one main lobe (e.g. Gaussian-like distribution).

The PD distribution obtained using a non-modulated 36 mW continuous-wave (CW) excitation after 2.5 s = T lock /2 of exposure is shown in Fig. 5(a). Results obtained using lockin demonstrate substantial improvement of the image quality and sensitivity (Fig. 5(b)). The measured PD distributions are compared to PD computed using the finite integration technique (FIT) implemented in CST Microwave Studio (Figs. 5(c,d)).

The PD distributions are compared using a correlation coefficient Υ indicating how closely the two relative distributions fit [START_REF] Holman | Experimental Methods for Engineers[END_REF]:

Υ = m n (X mn -X )(Y mn -Ȳ ) m n (X mn -X ) 2 m n (Y mn -Ȳ ) 2 , ( 10
)
where X and Y are the distributions to compare, X and Ȳ are respectively the mean values of X and Y .

The results demonstrate a very good agreement between the PD distributions measured applying lock-in (and SNR ≈ 11 dB) and the simulated one (Υ = 98.7 %). The proposed technique demonstrates more than 10 dB gain compared to the CW excitation (SNR ≈ 1 dB). Note that a higher SNR can be obtained for a CW signal with a longer duration (e.g. 5.3 dB for 20 s), however the enhanced heat conduction will lead to lower correlation between the measured heat pattern and PD (Υ = 81.9 %).

The sensitivity of the method can be defined as the PD level that can be reliably measured using the proposed technique. Fig. 5(d) (blue curve) demonstrates the sensitivity of about 1 mW/cm 2 , achieved thanks to the noise reduction through lock-in. This represents a significant improvement compared to conventional IR-based measurements [START_REF] Ziane | High-resolution technique for near-field power density measurement accounting for antenna/body coupling at millimeter waves[END_REF], with the sensitivity of the order of 10 mW/cm 2 limited by the level of the ambient noise (Fig. 5

(d) -yellow curve).

The enhanced sensitivity of the proposed technique makes it promising for dosimetry and compliance testing. According to [START_REF]Guidelines for limiting exposure to electromagnetic fields (100 kHz to 300 GHz)[END_REF], the main dosimetry quantity above 6 GHz is the absorbed power density (APD) defined as PD averaged over 4 cm 2 or 1 cm 2 . Thus APD can be calculated by spatial averaging of PD retrieved from high-resolution IR measurements.

V. CONCLUSION

In this study, we introduced a novel approach for highresolution near-field mapping of PD based on IR imaging applied to characterization of low-power mmWave devices accounting for antenna/body interaction. Enhanced sensitivity is achieved by post-processing the measured signal using lockin. The proposed technique is validated at 60 GHz for conical horn antenna. An excellent agreement is found between simulated and measured PD using lock-in (correlation > 98 %). The results show that the IR-based technique can be used for reliable PD measurement at the IPD levels as low as 1 mW/cm 2 (corresponding to sensitivity improvement at least by a factor of 10 compared to conventional IR-based measurements using CW excitation [START_REF] Ziane | High-resolution technique for near-field power density measurement accounting for antenna/body coupling at millimeter waves[END_REF]). For the first time, this study demonstrated the sensitivity of IR-based measurements sufficient enough to assess the compliance of medium-and low-power wireless devices above 6 GHz.

Fig. 1 .

 1 Fig. 1. Conceptual view of the measurement setup.

Fig. 2 .

 2 Fig. 2. Phantom: (a) fabricated prototype, (b) power reflection coefficient for parallel (TM) and perpendicular (TE) polarizations at 60 GHz.

Fig. 3 .

 3 Fig. 3. Modulation signal: (a) S(t), (b) S HF (t), (c) S LF .

1 )

 1 Discrete Time Fourier Transformation: T (t) is a discrete function (T (n) with t = n/f s , where f s is the sampling frequency) recorded at the time interval [t 0 ; t 0 +∆t]. The discrete time Fourier transformation (DTFT) is applied to convert T (n) to the frequency domain ( T ):

  Fig. 5. PD distributions for the conical horn antenna at d = 3.5 mm: (a) measured using non-periodic CW excitation, (b) measured applying lock-in, and (c) simulated. (d) 1D PD profiles along x-axis (y = 0 mm) (left) and y-axis (x = 0 mm) (right) (absolute values are provided for the numerical data and measurement results are normalized to the peak value).

This article has been accepted for publication in IEEE Antennas and Wireless Propagation Letters. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/LAWP.2023.3286944


Aurélien Barbotin
email: aurelien.barbotin@inrae.fr

Cyrille Billaudeau

Erdinc Sezgin

Rut Carballido-López
email: rut.carballido-lopez@inrae.fr

Quantification of membrane fluidity in bacteria using TIR-FCS

Cell membrane fluidity is an important phenotypic feature that regulates the diffusion, function and folding of transmembrane and membrane-associated proteins. It is particularly interesting to study it in bacteria as variations in membrane fluidity are known to affect fundamental cellular processes such as respiration, transport and antibiotic resistance. As such key parameter, membrane fluidity is regulated to adapt to environmental variations and stresses like temperature fluctuations or osmotic shocks. Membrane fluidity has been however scarcely studied quantitatively in bacterial cells, mostly because of the lack of available tools. Here, we developed an assay based on total internal reflection fluorescence correlation spectroscopy (TIR-FCS) to directly measure membrane fluidity in live bacteria via the diffusivity of fluorescent membrane markers. We used this assay to quantify the fluidity of the cytoplasmic membrane of the Gram-positive model bacterium Bacillus subtilis in response to a cold shock, caused by a shift from 37°C to 20°C. In our experimental conditions, steady-state fluidity was recovered within 30 mins, and the steady-state fluidity at 20°C was about half of that at 37°C. Our minimally invasive assay opens up exciting perspectives and could be used to study a wide range of phenomena affecting the bacterial membrane, from disruption by antibiotics, antimicrobial peptides, or osmotic shocks.

Significance

Using fluorescence correlation spectroscopy (FCS) with total internal reflection fluorescence (TIRF) illumination, we measured the diffusion speed of fluorescent membrane markers as a readout for membrane fluidity of growing B. subtilis cells. Quantification of the effect of cold shock provided unique information about the dynamics of the plasma membrane of B. subtilis. The unprecedented capability of TIR-FCS to quantify membrane fluidity in living bacteria opens the door to a whole set of new studies that will shed light on the bacterial plasma membrane and its interactions with the environment.

.

Introduction

The plasma membrane is a component of virtually every living cell, made of a fluid mixture of lipids and proteins, that separates the intracellular and extracellular spaces. The fluidity of the plasma membrane is the physical parameter that defines how fast a given element can diffuse within the membrane at a given temperature. Thus, it is of utmost interest for protein diffusion and biomolecular interactions (1,2). Moreover, membrane fluidity affects protein folding (3)(4)[START_REF] Burgess | β-Barrel Proteins That Reside in the Escherichia coli Outer Membrane in Vivo Demonstrate Varied Folding Behavior in Vitro[END_REF]. In bacteria, membrane fluidity was found to to be critical in both Gram-negative (for respiration in Escherichia coli [START_REF] Budin | Viscous control of cellular respiration by membrane lipid composition[END_REF] and multidrug transport in Methylobacterium extorquens [START_REF] Sáenz | Hopanoids as functional analogues of cholesterol in bacterial membranes[END_REF]) and Grampositive (e.g resistance to antibiotics in Staphylococcus aureus [START_REF] Boudjemaa | Impact of Bacterial Membrane Fatty Acid Composition on the Failure of Daptomycin To Kill Staphylococcus aureus[END_REF] ) bacteria. Membrane fluidity in bacteria was found to vary in response to chemical [START_REF] Vaňousová | Membrane fluidization by alcohols inhibits DesK-DesR signalling in Bacillus subtilis[END_REF] biochemical [START_REF] Popp | The Epipeptide YydF Intrinsically Triggers the Cell Envelope Stress Response of Bacillus subtilis and Causes Severe Membrane Perturbations[END_REF][START_REF] Lee | The role of bacterial lipid diversity and membrane properties in modulating antimicrobial peptide activity and drug resistance[END_REF] or osmotic [START_REF] Loffeld | cis/trans isomerization of unsaturated fatty acids as possible control mechanism of membrane fluidity in Pseudomonas putida P8[END_REF][START_REF] Beney | Influence of the fluidity of the membrane on the response of microorganisms to environmental stresses[END_REF] stresses. Another hint of the importance of membrane fluidity in bacterial cells is the widespread existence of control systems that maintain it by modifying lipid and protein composition [START_REF] Willdigg | Mini Review: Bacterial Membrane Composition and Its Modulation in Response to Stress[END_REF][START_REF] Yoon | Membrane fluidity-related adaptive response mechanisms of foodborne bacterial pathogens under environmental stresses[END_REF]. In particular, the fatty acid composition of phospholipids -the main class of lipids of the plasma membrane -is modified in response to changes in temperature to modulate steric constraints and thereby lipid packing [START_REF] Mendoza | Temperature Sensing by Membranes[END_REF]. Furthermore, proteins such as flotillins [START_REF] Zielińska | Flotillin-mediated membrane fluidity controls peptidoglycan synthesis and MreB movement[END_REF][START_REF] Nickels | Impact of Fatty-Acid Labeling of Bacillus subtilis Membranes on the Cellular Lipidome and Proteome[END_REF] and MreB [START_REF] Strahl | The actin homologue MreB organizes the bacterial cell membrane[END_REF] in B. subtilis are also thought to play a role in membrane fluidity. In the case of extremophiles, the generation of exopolymers [START_REF] Nichols | Effects of Incubation Temperature on Growth and Production of Exopolysaccharides by an Antarctic Sea Ice Bacterium Grown in Batch Culture[END_REF], cryoprotectants and antifreeze proteins [START_REF] Chattopadhyay | Mechanism of bacterial adaptation to low temperature[END_REF] are used by bacteria to protect their membrane against changes in temperature. Existing assays to measure membrane fluidity in live and synthetic membranes include electron spin resonance (ESR) [START_REF] Edgcomb | Electron paramagnetic resonance studies of the membrane fluidity of the foodborne pathogenic psychrotroph Listeria monocytogenes[END_REF][START_REF] Konings | Role of Membrane Lipids and Membrane Fluidity in Thermosensitivity and Thermotolerance of Mammalian Cells[END_REF] and nuclear magnetic resonance (NMR) spectroscopy [START_REF] Zielińska | Flotillin-mediated membrane fluidity controls peptidoglycan synthesis and MreB movement[END_REF], membrane fatty acid analysis [START_REF] Boudjemaa | Impact of Bacterial Membrane Fatty Acid Composition on the Failure of Daptomycin To Kill Staphylococcus aureus[END_REF][START_REF] Loffeld | cis/trans isomerization of unsaturated fatty acids as possible control mechanism of membrane fluidity in Pseudomonas putida P8[END_REF][START_REF] Edgcomb | Electron paramagnetic resonance studies of the membrane fluidity of the foodborne pathogenic psychrotroph Listeria monocytogenes[END_REF], fluorescence assays using environment-sensitive probes like diphenylhexatriene (DPH) or ratiometric probes like Laurdan [START_REF] Vaňousová | Membrane fluidization by alcohols inhibits DesK-DesR signalling in Bacillus subtilis[END_REF] and the measurement of the diffusion speed of a fluorescent tracer using either single particle tracking (SPT) [START_REF] Saxton | SINGLE-PARTICLE TRACKING:Applications to Membrane Dynamics[END_REF], fluorescence recovery after photobleaching (FRAP) [START_REF] Devkota | FRAP: A Powerful Method to Evaluate Membrane Fluidity in Caenorhabditis elegans[END_REF] or fluorescence correlation spectroscopy (FCS) [START_REF] Ponmalar | Modification of bacterial cell membrane dynamics and morphology upon exposure to sub inhibitory concentrations of ciprofloxacin[END_REF]. In microbiology, the most frequently used techniques are fatty acid analysis and DPH anisotropy or ratiometric imaging. Fatty acids analysis informs on the membrane composition but is an indirect readout of fluidity. It gives multidimensional results (relative proportions of different fatty acid with branching and (poly)unsaturation), which can be challenging to directly associate with a change in membrane fluidity, and can in the best case only provide qualitative comparisons of the resulting fluidity. Environment-sensitive fluorescent probes can give useful insights but can only measure relative differences in membrane fluidity. Many probes exist that are sensitive to different parameters of the membrane [START_REF] Ragaller | Dissecting the mechanisms of environment sensitivity of smart probes for quantitative assessment of membrane properties[END_REF][START_REF] Amaro | Laurdan and Di-4-ANEPPDHQ probe different properties of the membrane[END_REF] and their behaviour can be biased by unforeseen interactions [START_REF] Poojari | Behavior of the DPH fluorescence probe in membranes perturbed by drugs[END_REF]. FCS has occasionally been used in a few instances in bacterial membranes, to study protein diffusion [START_REF] Meacci | Mobility of Min-proteins in Escherichia coli measured by fluorescence correlation spectroscopy[END_REF][START_REF] Cluzel | An Ultrasensitive Bacterial Motor Revealed by Monitoring Signaling Proteins in Single Cells[END_REF][START_REF] Dajkovic | Dynamic Organization of SecA and SecY Secretion Complexes in the B. subtilis Membrane[END_REF], RNA concentration [START_REF] Guet | Minimally invasive determination of mRNA concentration in single living bacteria[END_REF], assembly of protein complex [START_REF] Diepold | A dynamic and adaptive network of cytosolic interactions governs protein export by the T3SS injectisome[END_REF] or membrane dynamics in response to antibiotic treatment [START_REF] Ponmalar | Modification of bacterial cell membrane dynamics and morphology upon exposure to sub inhibitory concentrations of ciprofloxacin[END_REF]. These studies were all performed using confocal microscopy which axial resolution is not well suited for measurements in bacteria. The axial resolution of confocal microscopes is comparable to the diameter of most studied bacterial cells (500 nm -1 µm), and this results in problems such as such as having both top and bottom membranes in focus at once or excessive background from out-of-focus membranes. These limitations can be overcome either by using super-resolution [START_REF] Barbotin | z-STED Imaging and Spectroscopy to Investigate Nanoscale Membrane Structure and Dynamics[END_REF] or more simply by using total internal reflection fluorescence (TIRF) microscopy. TIRF significantly improves the axial resolution of a microscope by illumination with an evanescent field that usually significantly decays within a range of 100 nm. Total internal reflection fluorescence correlation spectroscopy (TIR-FCS) [START_REF] Kannan | Electron Multiplying Charge-Coupled Device Camera Based Fluorescence Correlation Spectroscopy[END_REF][START_REF] Kannan | Spatially Resolved Total Internal Reflection Fluorescence Correlation Microscopy Using an Electron Multiplying Charge-Coupled Device Camera[END_REF] was previously used to study molecular dynamics in eukaryotic cells [START_REF] Bag | Imaging FCS delineates subtle heterogeneity in plasma membranes of resting mast cells[END_REF][START_REF] Ng | The Secreted Signaling Protein Wnt3 Is Associated with Membrane Domains In Vivo: A SPIM-FCS Study[END_REF] and to measure the fluidity of flat synthetic membranes [START_REF] Ng | Evidence from ITIR-FCS Diffusion Studies that the Amyloid-Beta (Aβ) Peptide Does Not Perturb Plasma Membrane Fluidity in Neuronal Cells[END_REF][START_REF] Bag | Temperature dependence of diffusion in model and live cell membranes characterized by imaging fluorescence correlation spectroscopy[END_REF]. However, TIR-FCS was not previously applied to bacteria despite TIRF being the standard for membrane investigations in such organisms [START_REF] Yao | Fluorescence Imaging for Bacterial Cell Biology: From Localization to Dynamics, From Ensembles to Single Molecules[END_REF]. TIR-FCS offers several advantages over confocal FCS, besides the unrivalled axial selectivity of TIR illumination: camerabased TIR-FCS also offers massive parallelisation of measurements, since hundreds of FCS curves can be acquired at once instead of a single one on a confocal microscope. TIR-FCS can also easily generate diffusion maps and therefore retrieve spatial information. Finally, TIR-FCS enables, by resampling intensity fluctuations in space after acquisition, the measurement of diffusion speeds at different spatial scales (spot-variation FCS [START_REF] Wawrezinieck | Fluorescence Correlation Spectroscopy Diffusion Laws to Probe the Submicron Cell Membrane Organization[END_REF]). Here, we propose to extend the scope of application of TIR-FCS to measure membrane fluidity in live bacterial cells, exemplified by the Gram-positive leading model organism Bacillus subtilis. To simplify data analysis and fully use the high-throughput capability of imaging FCS, we developed a new FCS quality metric to automatically discard artefactual curves. Using simulations validated by experiments in synthetic samples, we measured the bias induced by the small size and curvature of bacterial membranes on TIR-FCS measurements. We demonstrated the validity of our assay by studying a well-known system: the response of the B. subtilis plasma membrane to a cold shock [START_REF] Konopasek | Cold shock in Bacillus subtilis: different effects of benzyl alcohol and ethanol on the membrane organisation and cell adaptation[END_REF][START_REF] Beranova | Differences in Cold Adaptation of Bacillus subtilis under Anaerobic and Aerobic Conditions[END_REF]. Diffusion measurements of the membrane markers Nile Red and Di4-ANEPPS confirmed the previous knowledge about cold shock recovery and provided unprecedented insights of bacterial membrane dynamics at different temperatures.

Methods

FCS setup

TIR-FCS acquisitions were performed on a Zeiss Elyra PS1 microscope equipped with a 100×/1.46 NA Apochromat oil immersion objective. A typical FCS acquisition consisted of 50000 frames, on a field of view of 128x10 pixels with a pixel size of 160 nm in the object plane and a frame acquisition time of 1.26 ms, the maximum achievable with our camera. Stable focus was ensured using Definite Focus. Detection was performed using an emCCD camera (Andor iXon), using maximum pre-amplification (5×) and electron-multiplying gains (300×) settings. Laser excitation at 561 nm was set, unless specified otherwise, to 5% of the maximum excitation power, corresponding to a power of 460 µW in epifluorescence mode measured in the focal plane of the objective. The excitation area was of approximately 80x80 µm size, leading to an estimated power density of ~70 nW/µm².

Data processing and fitting

Pixels from each image stack were numerically binned 2 by 2, unless specified otherwise. The first 1500 frames of every acquisition were discarded as an occasional loss of focus could lead to artefactual intensity fluctuations in these frames. Intensity timetraces at each binned pixel were corrected for bleaching using a double exponential fit [START_REF] Ries | Accurate Determination of Membrane Dynamics with Line-Scan FCS[END_REF]. Intensity traces at each pixel were correlated using a python implementation of the multipletau algorithm [START_REF] Müller | Python multiple-tau algorithm[END_REF]. FCS curves were fitted using the standard 2D imaging FCS model [START_REF] Ries | Total Internal Reflection Fluorescence Correlation Spectroscopy: Effects of Lateral Diffusion and Surface-Generated Fluorescence[END_REF] except in bacterial cells where the fitting model is described in the 'technical implementation' section:

g xy (τ )= 1 N ( 1 √ π μ (exp(-μ ²)-1)+erf (μ))² μ= a 2 √σ ²+D τ 1
Where N is the average number of molecules in the observation area, a is the effective pixel size (320 nm with 2x2 pixel binning), D the diffusion coefficient, τ the lag time and σ is the standard deviation of the microscope's Point Spread Function (PSF) approximated as a two-dimensional Gaussian function:

PSF (x , y )=exp(- ( x ² + y ²) 2 σ ² ) 2
Calibration of the PSF was done as described in reference [START_REF] Bag | Calibration and Limits of Camera-Based Fluorescence Correlation Spectroscopy: A Supported Lipid Bilayer Study[END_REF]. We measured our PSF σ=0.19 µm, corresponding to a full width at half-maximum (FWHM) of 450 nm, larger than expected using a 1.46 NA oil immersion objective. This enlargement was likely caused by our use of a low magnification tube lens that degraded resolution, as we measured a PSF size σ=0.16 µm when using a higher magnification tube lens. For all acquisitions except in SLBs, we used an intensity threshold set to 80% of the maximum intensity in the field of view (see Supplementary Information for the determination of intensity threshold).

Liposomes preparation

1,2-di-(9Z-octadecenoyl)-sn-glycero-3-phosphocholine (DOPC) and 1-palmitoyl-2-oleoyl-snglycero-3-phosphocholine (POPC) stored in chloroform were purchased from Merck (Darmstadt, Germany) and stored under argon. 50 µL of 10 mg/mL stock were added to a glass tube then dried using Argon under rotation. Lipids were resuspended in 1.6 mL Phosphate-Buffered Saline (PBS), then tip-sonicated for 10 mins in 30 s on/off cycles on ice. Liposomes were labeled before experiments with 1% of 1,2-dioleoyl-sn-glycero-3-phosphoethanolamine-N-(lissamine rhodamine B sulfonyl) (PE-Rhodamine) (Merck) at a concentration of 10µg/mL.

Supported lipid bilayer preparation

Supported lipid bilayers (SLB) were prepared by liposome deposition. We pipetted 20µL of liposome solution to a home-made microfluidic chamber made of a sandwich of a slide and a plasma-cleaned coverslip held together by two strips of molten parafilm. Excess liposomes were abundantly washed using 200µL PBS. The chamber was then sealed using parafilm to prevent evaporation.

Beads-supported lipid bilayer preparation

Beads-supported lipid bilayers were prepared as described elsewhere [START_REF] Beckers | Impact of Nanoscale Hindrances on the Relationship between Lipid Packing and Diffusion in Model Membranes[END_REF]. 10µL of 5 µm uncoated silica beads (BioValley, Nanterre, France) were washed twice in 1 mL PBS, then mixed with 50µL liposomes. The mix was shaken for 20 mins to form BSLBs, then washed twice in 1 mL PBS. 200 µL PBS was left after final wash. 100 µL of BSLBs were then pipetted to a glass-bottom Ibidi chamber for imaging.

Cells preparation and staining

The wild-type laboratory strain 168 trpC2 of B. subtilis was grown and imaged in rich lysogeny broth medium (LB). For measurements at 37°C, 3µL of cells from an overnight culture grown at 30°C were diluted in 2 mL LB and grown at 37°C under agitation for 2h30 until they reached exponential phase (OD 600 ~ 0.3) then labeled with 0.2% (v:v) of either 50 µg/mL Nile Red or 40µg/mL Di4-ANEPPS (Thermofisher) disolved in DMSO. Labeled cultures were left under agitation for 15-20 additional min, then 3 µL were transfered to an agarose pad (1.2% in LB) for immobilisation and covered with a plasma-cleaned coverslip.

For steady-state measurements at 20°C, cells from an overnight culture at 30°C were first diluted (3 µL of overnight culture diluted in 2 mL fresh LB) and grown for about 2 hours at 37°C until reaching exponential phase (OD 600 ~ 0.2), transferred at 20°C under agitation for 4-5h for at least one generation, then labeled as described above.

When cells were transferred from the liquid culture to the agarose-coated slide, we observed temporary membrane readaptation for about 25 mins (Figure S1). This readaptation could be due to osmotic shock [START_REF] Los | Membrane fluidity and its roles in the perception of environmental signals[END_REF], oxidative stress or another cause that remains unknown. We therefore leave cells to settle on the agarose-coated slide for 25 mins on the slide before starting to image. In the case of cold shock, cells were first immobilised on an agarose pad and covered with the coverslip at 37°C for 25 min then transferred at 20°C in the microscope.

Technical implementation

Filtering curves based on goodness of fit

FCS measurements can be subject to artefacts that distort FCS curves, for instance when bright clusters of fluorescent molecules enter the observation area. These curves need to be discarded from analysis to avoid biasing diffusion coefficient estimation. In point FCS, this is often done via manual inspection facilitated by dedicated softwares [START_REF] Waithe | Optimized processing and analysis of conventional confocal microscopy generated scanning FCS data[END_REF]. This approach is however impractical in imaging FCS due to the high parallelisation of FCS curves acquisition resulting in the generation of a high number of FCS curves (typically we acquire 350-500 FCS curves per hour in bacteria, considering only one pixel binning value). The issue of sample-induced artefact in FCS is well acknowledged and solutions to this issue were previously developed. It was notably proposed to compare each curve within a dataset to the averaged curve and to exclude outliers [START_REF] Ries | Automated suppression of sample-related artifacts in Fluorescence Correlation Spectroscopy[END_REF]. This solution is however computationally intensive and might fail if signal levels vary within a dataset, for instance due of cell-to-cell heterogeneity. Another approach consists in rejecting curves with irregular residuals. One method for doing this consists in calculating the χ² goodness of fit ( 30), but it does not handle well noisy curves [START_REF] Ries | Automated suppression of sample-related artifacts in Fluorescence Correlation Spectroscopy[END_REF]. It also requires knowledge of the standard deviation of the FCS curve, which is not always available. The Fourier transform was previously used to detect unevenly distributed residuals, relying however on fine-tuning three empirical parameters and making assumptions on the transit times observed [START_REF] Guet | Minimally invasive determination of mRNA concentration in single living bacteria[END_REF]. As an alternative to these methods, we introduce here a simple error metric based on fitting residuals that quantifies fitting bias.

Considering the mean square error (MSE):

MSE= 1 n ∑ i n (r i ) 2 r i =( y i -ŷ i )/ ŷ 0 3
Where y i is the empirical FCS curve at lag time I, ŷ i is the corresponding fit value and r i the residual.

A high MSE value is indicative of either a high fitting bias in a poorly fitted curve, which needs to be discarded, or of low signal value (54) caused by strong oscillations of the FCS curve around its fit. Expecting strong variations in signal levels within and across acquisitions, due to cell-to-cell variations and inhomogeneous illumination of curved cells in the TIRF field, we designed a metric that measures fit quality with a lower dependency to signal level. This metric, named non-linear mean-square error (MSE nl ) is a weighted sum of residuals, where the weight of each residual is equal to the number of adjacent residuals of same sign n adj,i :

MSE nl = 1 n ∑ i n n adj ,i (r i ) 2 4
Concretely, if residuals r i between i-2 and i+2 are all positive, the fitting bias is strong and the weight n adj,i =4 is high. On the other hand, if the residual r i is positive but the residuals i-1 and i+1 are negative, there is no fitting bias (residuals are oscillating around the mean) and n adj,i =0. To evaluate the capability of MSE nl to evaluate fitting bias, we performed 2 TIR-FCS acquisitions on a flat sample of DOPC SLBs, at either high (740 µW) or low (185 µW) excitation power. The resulting dataset had heterogeneous signal levels representing the expected heterogeneity in biological samples. This dataset being acquired on the same SLB however we expected to find a comparable number of artefactual curves with either excitation intensities. Comparing MSE and MSE nl for every FCS curve in the dataset (Figure 1A), we could observe that excitation intensity was a good predictor of MSE but not of MSE nl . Curves acquired at higher excitation intensity had in average a lower MSE but similar MSE nl , showing a lower dependence of MSE nl with signal levels. We observed a strong correlation between the two metrics for curves acquired at a same excitation intensity, as poorly fitted curves have higher residuals than well-fitted curves at similar signal levels (Figure 1B). However, comparing two FCS curves with same MSE but different MSE nl suggested that MSE nl measures fitting bias irrespective of signal levels, unlike MSE (Figure 1C-D).

Considering that every FCS curve contains imperfections, we then sought to determine the maximum acceptable amount of fitting bias measured by MSE nl . For this, we plotted the measured diffusion coefficient against MSE nl (Figure 1E, bottom). We observed that high fitting bias were correlated to slower diffusion coefficients, themselves caused by artefacts in FCS curves. We confirmed this by plotting the average diffusion coefficient for sets of FCS curves with MSE nl thresholds (Figure 1E, top) and found that FCS curves with a low MSE nl corresponded to the expected diffusion coefficient [START_REF] Tang | Deep learning reduces data requirements and allows real-time measurements in Imaging Fluorescence Correlation Spectroscopy[END_REF]. Using both inspection of individual FCS curves and the scatterplot shown in Figure 1E, we set the MSE nl threshold to the value of 0.015. We kept this threshold throughout this study, in both synthetic and biological samples.

Impact of membrane curvature

Three assumptions made when fitting FCS curves with the model in equation 1 were not verified when doing TIR-FCS in bacteria. First of all, eq. 1 assumes that the diffusion within the observation area is occuring on a 2-dimensional flat surface. Indeed, TIR-FCS measures an average transit time [START_REF] Tang | Deep learning reduces data requirements and allows real-time measurements in Imaging Fluorescence Correlation Spectroscopy[END_REF].

(A) Scatter plot (center) of MSE and MSE nl metrics for each curve acquired in TIR-FCS at low (magenta) and high (green) excitation powers as indicated in the legend, and histograms of the corresponding MSE (bottom) and MSE nl (left) distributions. Colored squares refer to FCS curves shown in panels (B-D). (B-D) Normalised FCS curves (color) and fits (dotted black lines) with (top) fitting residuals, with high MSE and MSE nl (B), medium MSE and high MSE nl (C) and medium MSE and low MSE nl (D). The same scale was used to plot all fitting residuals. (E) Bottom: Scatterplot of measured diffusion coefficient with MSE nl and empirical threshold on fitting quality (dotted vertical line) to discard artefactual FCS curves. Squares represent FCS curves shown in (B-D). Top: Mean +/std of diffusion coefficient within the MSE nl range represented by lateral errorbars (blue) and value reported for the diffusion coefficient of PE-Rhod in a DOPC SLB in ref

in the observation area and then calculates a diffusion coefficient as a ratio between the size of the observation area and the transit time. In eq. 1, it is assumed that the size of the observation area is identical to the size of the area in which molecules diffused (the diffusion area), which is true when imaging a flat surface but is not when imaging a curved surface. In the latter case, the diffusion area is larger than the observation area. Second, it is assumed that intensity fluctuations are only caused by molecules moving across the observation area and Poisson noise. When doing TIR-FCS in a curved membrane, molecules moving laterally also change their axial position which under TIRF excitation determines excitation intensity and therefore induces intensity fluctuations. The third assumption is that the system is open, which means that there is an infinite pool of fluorescent molecules diffusing in an infinite-sized reservoir. This latest assumption is never actually verified but it is a good approximation when the mean square displacement of fluorescent emitters during the time of acquisition is much smaller than the reservoir size. This is not the case for membrane markers in bacteria: the average distance traveled by molecules diffusing at a reasonable 1µm²/s speed over the course of 1 min (our usual acquisition time) is 15 µm, larger than the characteristic dimensions of a B. subtilis cell (typically 5x1µm for exponentially-growing B. subtilis cells in LB at 37°C). To evaluate the potential biases in diffusion coefficient measurements caused by these effects, we simulated diffusion on curved surfaces of finite areas: either on the simplest case of a sphere (that can represent cocci e.g Staphylococcus aureus) or on a cylindrical vessel like the rodshaped B. subtilis. Diffusion on these 3D surfaces were simulated as a Wiener process. We first generated a uniform distribution of initial positions. Position vectors were updated for each step by adding the cross product of the position vector with a random 3D vector of brownian motion, then normalised (see Supporting Materials and Methods for a complete description of the simulation).

From the position of individual emitters determined as trajectories (Figure 2A), we could simulate TIR-FCS experiments having the physical parameters of our setup: framerate (1 ms/frame), PSF size (FWHM of 450 nm), TIRF penetration depth (100 nm). Molecular brightness was set to 20 kHz and simulated diffusion coefficient was set to 1 µm²/s. The number of molecules was set to reach a density of 1.6 molecules/µm², with a minimum of 10 molecules per simulation.

With this simulation framework, we simulated a series of TIR-FCS experiments either on rods (of constant length set to 3 µm) or on spheres of various radii (0.5 to 10 µm) and measured the apparent diffusion coefficient (Figure 2B). We found that in both geometries, the measured diffusion coefficient converged towards the real value for high radius values, which was expected given that a curved membrane of high radius of curvature can be approximated as flat. We found however a significant bias in diffusion coefficient measurement for small radii (below 2 µm). The measurement bias in a rod was well approximated by the square root of the measurement bias in a sphere (Figure 2B, dashdotted line). This can be explained by thinking of curvature as modifying the detection PSF: the detection PSF is modified alongside two dimensions in the case of a sphere and only one dimension in the case of a rod. FCS curves acquired in a rod-shape can therefore be fitted with an updated model based on eq. 1, which is the product of two 1-dimensional fitting models with a fitting bias f accounting for curvature alongside one dimension:

g xy ,rod ( τ)= 1 a ² ( 1 √ π μ 1 (exp(-μ 1 ²)-1)+erf (μ 1 ))( 1 √ π μ 2 (exp(-μ 2 ²)-1)+erf (μ 2 )) μ 1 = a 2 √ σ ²+D τ ;μ 2 = a 2 √σ ²+ D f τ 5
We set the fitting bias f to the value of 2.1 for B. subtilis, corresponding to the bias induced by a curvature of 500 nm radius on a sphere (Figure 2B). We verified using simulations whether rod length influenced the measurement of diffusion coefficient (Figure S4), and found that for a radius of 500 nm only rod lengths below 2.5 µm affected the diffusion measurements, which is below the lengths of B. subtilis cells we measured in this study (Fig. S5). To validate our simulations experimentally, we used POPC Beads-Supported Lipid Bilayers (BSLB) of 5µm diameter, labeled with the fluorescent lipid PE-Rhod, as a system of 2D diffusion of a controlled diameter (Figure 2C). We performed a series of TIR-FCS experiments in these BSLBs and measured the diffusion coefficient with different observation sizes using different pixel binning values (Figure 2D). The apparent curvature increased as the pixel binning increased and this experiment was therefore a good proxy to measure the effect of different curvatures on measured diffusion coefficient. As expected, we observed an increase in diffusion coefficient with increased observation size, corresponding to an increased effect of curvature on the effective PSF. We observed a similar change in diffusion coefficient with observation size on simulated spheres of identical diameter (Figure 2E). This analysis assumed free diffusion occurring in BSLBs, which was previously shown

(A) 3D visualisation of random trajectories generated on a rod (B) Diffusion coefficients measured D measured from simulation of FCS measurements, normalised with simulated value D simulated, in cylindrical (blue) and spherical (orange) coordinates. Dash-dotted red line: square root of normalised diffusion coefficient in a sphere, dotted black line: simulated diffusion coefficient (ground truth). (C) Cartoon of a BSLB (top) and epifluorescence images of BSLBs in an Ibidi chamber (bottom). (D) FCS curves for different observation sizes (obtained by pixel binning) as indicated in the legend, averaged within a 1.28x1.28µm observation area. (E) Diffusion coefficients measured at different observation sizes, normalised with value measured for smallest observation size, from experimental measurements (plain blue line) in 5 µm diameter BSLBs or from simulations in a sphere of same size (dotted red line).

to not be strictly true [START_REF] Beckers | Impact of Nanoscale Hindrances on the Relationship between Lipid Packing and Diffusion in Model Membranes[END_REF]. Nanoscale hindrances were detected in BSLBs using super-resolution spectroscopy [START_REF] Beckers | Impact of Nanoscale Hindrances on the Relationship between Lipid Packing and Diffusion in Model Membranes[END_REF], but these were unlikely to affect diffusion measurements at the larger observation scales used here. Indeed, we observed that the measured diffusion coefficient was constant for pixel sizes comprised between 160 and 480 nm (Figure 2E), consistent with a free diffusion approximation.

Results

Temperature-induced changes in membrane fluidity

The viscosity of biological membranes heavily depends on the ambient temperature. Reducing temperature increases lipid order, thereby decreasing membrane fluidity, up to the point of phase transition from a fluid membrane to a gel-like structure [START_REF] Heimburg | Phase transitions in biological membranes[END_REF]. Poikilothermic ('cold-blooded')

organisms like bacteria which are naturally exposed to ample changes in temperature adapt their membrane composition in order to maintain fluidity to survive to such changes. Bacterial membrane adaptation to temperature has been widely studied, particularly in the Gram-positive model organism B. subtilis, and includes increasing the unsaturated and branched fatty acids. The plasma membrane of steady-state B. subtilis cells contains only low amounts of unsaturated fatty acids [START_REF] Beranova | Differences in Cold Adaptation of Bacillus subtilis under Anaerobic and Aerobic Conditions[END_REF][START_REF] Suutari | Unsaturated and branched chain-fatty acids in temperature adaptation of Bacillus subtilis and Bacillus megaterium[END_REF]. Upon cold shock, the enzyme Des desaturates fatty acids (58) immediately (44)(<30 min).

Long-term membrane readaptation involves branching instead of unsaturation [START_REF] Beranova | Differences in Cold Adaptation of Bacillus subtilis under Anaerobic and Aerobic Conditions[END_REF]. To measure membrane fluidity of exponentially-growing B. subtilis cells in steady-state at 37°C and 20°C, we

subtilis membrane to cold temperature, measured with TIR-FCS. (A-B) representative bright-field (top) with highlighted areas (dotted squares) where TIR-FCS diffusion maps (bottom) of Nile Red in exponentially-growing B. Subtilis were acquired, at 37°C (A) or after 5h at 20°C (B). Scalebars: 5 µm. (C) Diffusion coefficient of Nile Red (left) and Di4 (right) in B. subtilis cells grown and imaged at 37°C (orange) or 5 hours after transfer at 20 °C (blue). Dots: average of individual TIR-FCS acquisitions of one or more cells, 2 biological replicates, n>280 FCS curves per condition. (D) Recovery of diffusion speed of Nile Red upon transfer from 37°C to 20°C, pooling measurements by tranches of 5 mins, n=2 biological replicates. Dotted black line: median diffusion coefficient of Nile Red at 20°C from (C). (E) Normalised FCS curves obtained at 37°C (orange) and 5 mins (green) and 5h (blue) after transfer at 20°C.

first used the membrane marker Nile Red, which has the advantage of being bright, photostable, and long-used in B. subtilis. Individual diffusion maps alone clearly showed that the diffusion speed of Nile Red was slower at 20°C than at 37°C (Figure 3A-B). This observation was verified through multiple acquisitions, showing an about 2-fold reduction in the diffusion coefficient of Nile red at 20°C relative to 37°C (Figure 3C). We verified that cells were healthy and in exponential phase by monitoring their growth before imaging (Figure S6). Rare cells that were not growing in the microscopy field were excluded from the analysis. We also estimated the impact of phototoxicity by monitoring their growth after imaging and found that cells were growing after imaging, although at a slightly slower rate (Figure S6). To verify whether the diffusion speed of Nile Red was controlled by membrane fluidity and not unforeseen interactions, we performed the same experiment with the membrane dye Di4-ANEPPS (Di4). Diffusion speed of Di4 was slower than Nile Red in similar experimental conditions, which could be expected from its larger size (M=318 g/mol for Nile Red and 480 g/mol for Di4). The reduction of diffusion speed measured between 37°C and 20°C was identical for both dyes (Figure 3C and Table 1), suggesting that it was indeed a change in fluidity that led to the observed reductions in diffusion coefficient. Having established a baseline for the diffusivity of Nile Red in B. subtilis at different temperatures, we then sought to observe the remodeling of the membrane in response to a cold shock. For this, we transferred exponentiallygrowing cells labeled with Nile Red at 37 °C to the microscope at 20°C and performed FCS acquisitions on different cells for 1h. As expected, we observed first a reduction in diffusion coefficient caused by the temperature downshift (Figure 3D-E), then a progressive increase likely caused by membrane adaptation. The ~30 mins timescale of membrane adaptation we observed was consistent with the significant membrane fatty acids remodeling within 30 mins of cold shock that was previously observed [START_REF] Konopasek | Cold shock in Bacillus subtilis: different effects of benzyl alcohol and ethanol on the membrane organisation and cell adaptation[END_REF]. The diffusion coefficients measured were fitted using the model of equation 5 accounting for membrane curvature, assuming a constant radius of 0.5 µm and a cell length larger than 2.5 µm (see Figure S4). We verified that these hypotheses were verified by manual measurements of cell width and length (Figure S5).

Discussion

In this paper, we demonstrated how TIR-FCS can be used to quantitatively measure membrane fluidity in bacteria, exemplified by the Gram-positive model organism Bacillus subtilis. For this, we derived a new fit quality metric that greatly simplified data analysis. Using simulations validated by experiments, we estimated the measurement bias caused by the observation of a curved membrane of finite size in a TIRF field, and used this information to perform unbias measurements of diffusion coefficients in the membrane of B. subtilis. We used this assay to measure membrane fluidity, reported by diffusion speed of membrane markers, in the membrane of B. subtilis at different temperatures, during membrane remodeling caused by a cold shock and in a steady-state.

The fitting quality metric we derived in the first section is a fine addition to the collection of readily-available quality metrics for FCS quality assessment. Its reliability across datasets with different signal levels might prove useful in future high-throughput FCS studies, whether camerabased or not [START_REF] Wachsmuth | High-throughput fluorescence correlation spectroscopy enables analysis of proteome dynamics in living cells[END_REF]. Using TIR-FCS instead of conventional confocal FCS offered several advantages, some of them already listed: high-throughput measurements, excellent axial selectivity allowing FCS measurements in single membranes and extraction of spatial information. Besides, using an unpolarised evanescent field with TIRF excitation allows excitation of fluorescent molecules irrespective of their dipole orientation. This is a major advantage when working with membrane markers that intercalate within a membrane and keep a fixed orientation, which can be orthogonal to the polarisation plane of e.g confocal excitation and therefore lead to poor excitation quality and inhomogeneous excitation of observation area.

Simulating TIR-FCS in curved surfaces proved very useful for understanding and measuring the potential biases induced by membrane curvature, inhomogeneous excitation of the curved membrane in the evanescent field, and diffusion in a closed system of small size. The respective impact of these three effects was not disentangled by our simulations and further investigations could quantify the individual impact of each of these three effects on the final measurement. Simple 2-dimensional simulations in finite-sized boxes could readily reveal that the small diffusion areas typical of bacterial membranes can lead to substantial biases in measured diffusion speed (Figure S3). Care must be taken when accounting for membrane curvature with simulations to accurately simulate not only the observed geometry, but also the microscope itself as parameters like the PSF size affect the final bias in diffusion coefficient (Figure S4). Furthermore, we did not account for photobleaching in our simulations. This might affect the measurement bias caused by the closedness of the system: in a closed system, the same molecule is more likely to diffuse repeatdly through an observation area than in an open system. However, if there is significant bleaching, a molecule will likely diffuse only once through the observation area and then bleach. We do not expect this effect to significantly change the predictions of our simulations: bleaching was not simulated but occurred in BSLBs, and both systems exhibited a similar behaviour (Figure 2E).

In this work, we showed that the diffusivity of two different membrane markers decreased similarly in response to a decrease in temperature, suggesting a common determinant to their diffusivity: membrane fluidity. Claiming that this assay measures membrane fluidity however requires a couple of reasonable approximations. Membrane fluidity in the field of microbiology is usually referred to as a single parameter (the physical parameter governing diffusion speed in the plasma membrane). However, it is known that the fluidity in the physical sense affects the diffusivity of objects depending on their hydrophobic radius [START_REF] Petrov | Translational Diffusion in Lipid Membranes beyond the Saffman-Delbrück Approximation[END_REF]. The diffusivity of the membrane markers Nile Red and Di4 is therefore a good proxy for the measurement of membrane fluidity experienced by molecules of comparable size (e.g lipids) but the diffusivity of larger molecules such as transmembrane proteins will be affected by membrane fluidity differently. Relative differences should nevertheless remain the same: the slower diffusion speed of Nile Red at 20°C relative to 37°C indicates that membrane proteins undergoing Brownian motion also diffuse more slowly at 20°C than at 37°C. Furthermore, we did not consider lateral membrane heterogeneities (named functional microdomains in bacteria [START_REF] Bramkamp | Exploring the Existence of Lipid Rafts in Bacteria[END_REF], conceptually equivalent to lipid rafts [START_REF] Sezgin | The mystery of membrane organization: composition, regulation and roles of lipid rafts[END_REF] in eukaryotes) that are thought to recruit certain proteins and lipids in areas of higher molecular order and therefore lower mobility. The diffusivity that we measured in this study likely represents an average of the diffusivity in all domains of the membrane.

Our measurements were performed in the rod-shaped Gram-positive model bacterium B. subtilis. Together with our setup, this offers several advantages: first, the well conserved cell diameter of B.

subtilis across the cell cycle and growth conditions [START_REF] Juillot | A High-Content Microscopy Screening Identifies New Genes Involved in Cell Width Control in Bacillus subtilis[END_REF] facilitates correction for membrane curvature. Second, Gram-positive bacteria lack an outer membrane. More care would need to be taken for labeling Gram-negative bacteria to ensure that the used fluorescent marker labels only the membrane of interest, either the inner or the outer membrane. Third, additional care must be taken when performing FCS experiments in Gram-negative bacteria, as the topography of the outer membrane was previously shown to be affected by various compounds [START_REF] Ponmalar | Modification of bacterial cell membrane dynamics and morphology upon exposure to sub inhibitory concentrations of ciprofloxacin[END_REF][START_REF] Manioglu | Antibiotic polymyxin arranges lipopolysaccharide into crystalline structures to solidify the bacterial membrane[END_REF] which itself affects measurement of diffusion speeds with FCS [START_REF] Gesper | Variations in Plasma Membrane Topography Can Explain Heterogenous Diffusion Coefficients Obtained by Fluorescence Correlation Spectroscopy[END_REF]. Our work confirmed the results of previous experiments on the effect of a cold shock on membrane fluidity. Upon cold shock, membrane fluidity decreases, and quickly increases again thanks to a modification of plasma membrane composition that increases unsaturated and branched-chain fatty acids in the membrane [START_REF] Beranova | Differences in Cold Adaptation of Bacillus subtilis under Anaerobic and Aerobic Conditions[END_REF]. It remained unknown by how much membrane fluidity recovered. Fatty acid profiles only provided qualitative information and conflicting results were obtained with environment-sensitive probes [START_REF] Heřman | Time-resolved polarized fluorescence studies of the temperature adaptation in Bacillus subtilis using DPH and TMA-DPH fluorescent probes[END_REF]. DPH anisotropy suggested an incomplete recovery of B. subtilis membrane fluidity while fluorescence lifetime measurements of the same probe indicated an identical fluidity at 20 and 37°C [START_REF] Heřman | Time-resolved polarized fluorescence studies of the temperature adaptation in Bacillus subtilis using DPH and TMA-DPH fluorescent probes[END_REF]. Our results allow to settle this debate unambiguously: membrane fluidity does not completely recover after a cold shock in B. subtilis: it recovers to the steady-state fluidity at the temperature to which cells were transferred.

Our TIR-FCS assay opens up exciting perspectives in the field of microbiology. The unprecedented ability to quantify membrane fluidity will shed a new light on the biophysics of bacterial membranes and might help to understand severeal cellular processes as well as the mode of action of membrane-targetting antibiotics or antimicrobial peptides. The newly acquired capability to quantify membrane fluidity might also lead to a better understanding of the fundamental role of membrane fluidity in bacterial physiology.

On a rod: the simulation process is very similar. The rod is represented as a cylinder of length L and radius R with two half-spherical parts or radius R at its end (Figure S2B), oriented along the xaxis. The initial distribution of points is done in two steps: a fraction of the total number of the points is distributed on a sphere, while the rest of the points are distributed on a cylinder. The relative fraction of points on the sphere and the cylinder is determined from the relative areas of the spherical and cylindrical parts of the rod. Points drawn on the sphere with a negative x coordinate are moved along the x axis by a distance -L/2, the others by a distance +L/2. The vector position ⃗ r (t ) of each point is then iteratively updated following equation 2 as in the previous section, except that in this case the vector ⃗ r (t ) describes the distance to the medial axis (segment [P 1 P 2 ] in Figure S2B) of the rod and not to the centre of the sphere. Figure S2B illustrates the two different configurations ( ⃗ r 1 (t ) and ⃗ r 2 (t ) ) for the vector ⃗ r (t ) .

Parameters used in the simulations of Figure 2 Particle density was set to the constant value of 1.6 particle/µm², except in smallest simulations for which it was increased to contain at least 10 particles. TIRF penetration depth δz was defined as:

I TIRF ( z)=I 0 exp(-z /δ z)
3 Where I(z) is the depth-dependent TIRF excitation field. To speed up calculations, all particles above 4δz were considered to have a brightness equal to zero and were discarded from the analysis. Analysis was performed with 4x4 binning to an observation area of 320 nm, similar to the one we used in our experiments. An intensity threshold set to 80% of the maximum intensity was also used to analyse simulations as we used in real experiments. Each simulation was performed 9 times. The lateral position of the simulated bacterium was different for each of the 9 simulations to avoid a potential bias.

Influence of size of closedness of the system:

Supp. In order to understand if the closedness of the simulated systems described above and in Figure 2 could lead to a bias in diffusion coefficient estimation with imFCS, we simulated a simple system of 2-dimensional Brownian motion in a homogeneous illumination field. Molecules leaving the system on one edge were reintroduced at the corresponding position on the opposite edge (Figure S3A). When the box became very small, we could observe that FCS curves shifted towards shorter lag times and became distorted (Figure S3B). Fitting curves for different box sizes to extract diffusion coefficients confirmed that smaller box sizes, of areas in the order of magnitude of bacterial membrane areas, indeed induced a bias in diffusion coefficient estimation (Figure S3C). It is therefore very likely that part of the measurement biases in Figure 2 were caused by an effect of the small size of the systems observed.

Influence of rod length and PSF size:

Supp.

Live bacteria

Cell morphology in different conditions

Having found that the width and length of bacterial cells bias diffusion coefficient measurements, we verified whether the morphology of B. subtilis cells changed between the different experimental conditions investigated here. For this, we acquired epifluorescence images for each of these conditions (Figure S5A-B) and measured cell length and width manually using ImageJ. Cell length was determined by drawing a line between the two poles of each cell and measuring the length of this line. We found that the average cell length was identical at 37°C and immediately after cold shock, decreased after 5hours at 20°C (Figure S5C), but not to a point where cell length biased FCS measurements.

Cell width was measured by plotting the intensity profile alongside a line orthogonal to the cell long axis and measuring the peak-to-peak distance. This method led to an underestimation of the real cell width due to off-axis fluorescence emitted by the top and bottom part of the membrane, hence the relative difference with the well-known diameter of B. subtilis of 0.9-1µm (3)(4). It revealed however that as expected cell width did not change significantly between experimental conditions and thus that we could apply the same correction factor accounting for membrane curvature to measurements (Figure S5D). Cell width remained constant during cold shock (Figure S5F) and cell length remained well above the 2.5 µm threshold leading to bias in diffusion coefficient (Figure S4A). Panels C and D of Figure S5 were generated using supplementary ref 1.

Supp.Figure 4: Simulation of the influence of physical parameters on measured diffusion coefficient. (A) Measured

diffusion coefficient with length of rod-shape, for a radius of 0.5 µm. Red dashed line: bias for lengths>2.5 µm. (B) Influence of PSF size (parameter σ in equations 1-2) on measured diffusion coefficient, on a sphere of radius 0.5 µm.

Impact of FCS measurement on doubling time

Using bright-field timelapses, we verified both cell fitness and the impact of phototoxicity on cell growth. For this, we measured the growth rate of cells used in Figure 3C, at 37°C. We acquired for each chain of cells 3 bright-field images (Figure S6A), one at least 3 mins before the beginning of FCS acquisition, one immediately after the FCS acquisition and one at least 3 mins after FCS acquisition. We measured the length of the cell chain in each bright-field image and calculated doubling times between pairs of frames following the equation (under the assumption of constant cell width as is the case in B. subtilis):

T double =Δ t ln(2) / ln (l 2 /l 1 ) 4

Where T double is the doubling time, Δ t is the time between frames 1 and 2, l 1 Supp. and l 2 are the lengths of the cell chain in frames 1 and 2. Cells which doubling was more than twice higher than the nominal doubling time (~20 mins) were considered not exponentially-growing and therefore excluded from the analysis. Comparing pairwise doubling times before and after FCS (Figure S6B), we found that cells kept growing after FCS, yet at a slightly slower rate, suggesting low photoxicity effects.

Detailed implementation of FCS data processing

Determination of intensity threshold

In order to avoid biasing diffusion measurements, we needed to exclude TIR-FCS measurements that were too far from the point of contact between the bacterial cell and the coverslip. An efficient way of doing this consisted in removing pixels with an average intensity below a given threshold, as the excitation of the TIRF field decreases with increased distance to the cell centre. To find an appropriate value for this intensity threshold, we plotted a 2D histogram of intensity (normalised with 98 th percentile) and diffusion coefficient in 6 acquisitions of exponentially-growing B. subtilis labeled with Nile Red at 20°C. We set the threshold to 0.8 so that there was no correlation between diffusion coefficient and intensity (Figure S7).

Bleaching correction and FCS fitting

Bleaching correction was performed using a double exponential fit of the decaying intensity. Intensity timetraces were downsampled 500 times to speed up computations. The resulting traces were fitted with the function: Î (t)=f 0 [ (1-b)exp (-t /τ 1 )+ b exp(-t / τ 2 ) ] +c

5

The original intensity timetrace was then corrected as described in ref (2):

I c (t )= I (t )
√ Î (t)/ Î (0) + Î (0) (1-√ Î (t)/ Î (0))

6

The error function in equation 1 is as :

erf (x)= 2 √ π ∫ 0 x exp(-t ²)dt 7
Supp.

Figure 1 :

 1 Figure 1: Estimating the fit quality of FCS curves acquired on a DOPC SLB labelled with PE-Rhod using MSE and MSE nl . (A) Scatter plot (center) of MSE and MSE nl metrics for each curve acquired in TIR-FCS at low (magenta) and high (green) excitation powers as indicated in the legend, and histograms of the corresponding MSE (bottom) and MSE nl (left) distributions. Colored squares refer to FCS curves shown in panels (B-D). (B-D) Normalised FCS curves (color) and fits (dotted black lines) with (top) fitting residuals, with high MSE and MSE nl (B), medium MSE and high MSE nl (C) and medium MSE and low MSE nl (D). The same scale was used to plot all fitting residuals. (E) Bottom: Scatterplot of measured diffusion coefficient with MSE nl and empirical threshold on fitting quality (dotted vertical line) to discard artefactual FCS curves. Squares represent FCS curves shown in (B-D). Top: Mean +/std of diffusion coefficient within the MSE nl range represented by lateral errorbars (blue) and value reported for the diffusion coefficient of PE-Rhod in a DOPC SLB in ref[START_REF] Tang | Deep learning reduces data requirements and allows real-time measurements in Imaging Fluorescence Correlation Spectroscopy[END_REF].

Figure 2 :

 2 Figure 2: Impact of membrane curvature on TIR-FCS measurements. (A) 3D visualisation of random trajectories generated on a rod (B) Diffusion coefficients measured D measured from simulation of FCS measurements, normalised with simulated value D simulated, in cylindrical (blue) and spherical (orange) coordinates. Dash-dotted red line: square root of normalised diffusion coefficient in a sphere, dotted black line: simulated diffusion coefficient (ground truth). (C) Cartoon of a BSLB (top) and epifluorescence images of BSLBs in an Ibidi chamber (bottom). (D) FCS curves for different observation sizes (obtained by pixel binning) as indicated in the legend, averaged within a 1.28x1.28µm observation area. (E) Diffusion coefficients measured at different observation sizes,normalised with value measured for smallest observation size, from experimental measurements (plain blue line) in 5 µm diameter BSLBs or from simulations in a sphere of same size (dotted red line).

Figure 3 :

 3 Figure 3: Adaptation of B. subtilis membrane to cold temperature, measured with TIR-FCS. (A-B) representative bright-field (top) with highlighted areas (dotted squares) where TIR-FCS diffusion maps (bottom) of Nile Red in exponentially-growing B. Subtilis were acquired, at 37°C (A) or after 5h at 20°C (B). Scalebars: 5 µm. (C) Diffusion coefficient of Nile Red (left) and Di4 (right) in B. subtilis cells grown and imaged at 37°C (orange) or 5 hours after transfer at 20 °C (blue). Dots: average of individual TIR-FCS acquisitions of one or more cells, 2 biological replicates, n>280 FCS curves per condition. (D) Recovery of diffusion speed of Nile Red upon transfer from 37°C to 20°C, pooling measurements by tranches of 5 mins, n=2 biological replicates. Dotted black line: median diffusion coefficient of Nile Red at 20°C from (C). (E) Normalised FCS curves obtained at 37°C (orange) and 5 mins (green) and 5h (blue) after transfer at 20°C.

Figure 2 :

 2 Sketch of the simulation of a Wiener process on curved surfaces. (A): iteration of the Wiener process. B: Sketch of rod-shape simulation.

Figure 3 :

 3 TIR-FCS simulations in a closed box of varying size. (A) sketch of the simulated system with box (black) containing moving particles (blue) and a smaller observation area (centre, orange square) where TIR-FCS is simulated. Top: small box, bottom: large box. A molecule leaving on one side and reentering on the other side is shown with dashed arrows. (B) representative FCS curves obtained when a small (1.6 µm width, orange) and large (160 µm width, blue) simulation box are used. (C) Measured diffusion coefficients in simulated TIR-FCS experiment as a function of box size.

Figure 5 :

 5 Morphology of B. subtilis at different temperatures. (A-B) epifluorescence images of B. subtilis in exponential phase, labeled with Nile Red, at 37°C (A) and 20 °C (B). Scalebars: 5 µm. (C-D) Length (C) and width (D) of B. subtilis measured from epifluorescence images, in exponential phase at 37°C, 20°C, or during cold shock. (E-F) Scatterplots of cell length (E) and width (F) with time at 20°C immediately after cold shock.

Supp.Figure 6 :

 6 Impact of FCS measurements on the growth rate of Nile Red-labeled B. subtilis cells. (A): bright-field images of growing cells acquired before (top), immediately after (middle) and after (bottom) FCS acquisition. Scale bars: 5 µm. (B) Doubling times calculated from cell elongation, before and after FCS acquisition. Black dots: single doubling times measurements, gray lines link doubling times of the same cell chain.

Figure 7 :

 7 Determination of intensity threshold for unbiased diffusion measurement. Correlation between relative pixel intensity (x axis) and measured diffusion coefficient (y axis) visualised as a 2-dimensional histogram in 6 acquisitions of exponentially-growing B. subtilis at 20°C. Vertical dotted red line: selected threshold

Table 1 :

 1 Diffusion coefficients of Nile Red and Di4-ANEPPS measured in B. subtilis at 37°C and 20°C, and ratios of diffusion coefficients at 37°C and 20°C. Presented are the means and standard deviations of the average diffusion coefficients for each acquisition (dots in Figure3C).

	Dye Nile Red Di4-ANEPPS	D 37°C (mean +/-std) 4.4+/-0.3 1.9+/-0.1	D 20°C (mean +/-std) 2.2+/-0.2 0.9+/-0.07	D 37°C /D 20°C 2+/-0.3 2.1+/-0.3

D [µm²/s] # frames Parts. density [parts/µ m²] Length [µm] (rod only) Pixel size [µm] σ psf [µm] δz TIRF [µm] Brightnes s (Hz)

  are listed in the following table:

	Frame rate (kHz)
	1	1	50000	1.6	3	0.08	0.19	0.1	20000

Acknowledgements

This project has received funding from the European Research Council (ERC) under the Horizon 2020 research and innovation program (grant agreement No 772178 to R.C.-L.) and under the Marie Sklodowska-Curie grant agreement No 101030628.

Data availability

Research data will be deposited on Zenodo after publication. Code and manual of the FCS analysis software developed for this project and used throughout this paper can be found at https://github.com/aurelien-barbotin/ pyimfcs . Code for simulations can be found at https://github.com/aurelien-barbotin/geomdsim

Author Contributions

A.B., C.B., E.S. and R.C-L. designed the project and wrote the manuscript. A.B. performed the experiment, analysed data and wrote the code.

Declaration of interests

The authors declare no competing interests.

Supporting material

Simulations

On a sphere: First a set of points representing individual fluorescent emitters were distributed randomly on a sphere, as described in [START_REF] Burgess | β-Barrel Proteins That Reside in the Escherichia coli Outer Membrane in Vivo Demonstrate Varied Folding Behavior in Vitro[END_REF]. The position of each point was described in spherical coordinates. Its azimuthal (θ) and polar (φ) angles were randomly generated using the following equation:

1 where u and v are drawn from uniform random variables with bounds [0,1[. Trajectories were then converted to cartesian coordinates. At a given time t, the vector position ⃗ r (t )=[ x (t ) , y (t ), z (t )] of a point was then updated as follows, as discussed in reference (6):

where R is the radius of the sphere and ⃗ b (t )=[u x (t ),u y (t ),u z (t )] is a three-dimensional random vector drawn from a normal distribution, with each component having a standard deviation of √2 Dt / R , with D the diffusion coefficient. The normalisation factor

is necessary to keep the vector ⃗ r (t +1) on the surface, as the vector ⃗ r (t )∧ ⃗ b (t) is tangential to the curved surface and therefore ⃗ r (t )+⃗ r (t)∧ ⃗ b (t ) is not on the surface (Figure S2A). Under the conditions that the angle between ⃗ r (t ) and ⃗ r (t +1) is small, ‖⃗ r (t +1)-⃗ r (t )‖=⃗ r (t )∧ ⃗ b (t ) and the simulation of brownian motion is accurate.

Supp.Figure 1