H Mishra

M Hehn

S Hage-Ali

S Petit-Watelot

P W Mengue

S Zghoon

H M'jahed

D Lacour

O Elmazria
email: omar.elmazria@univ-lorraine.fr

Micro-structured, multi-layered surface acoustic wave device for multifunctional sensing

A multifunctional sensor based on a surface acoustic wave (SAW) device has been fabricated. It allows independent measurements of temperature and applied magnetic field. Optimization of the multilayered device structure leads to a temperature coefficient frequency of the Love wave resonance reduced to zero. The sensitivity to an applied magnetic field is obtained through magnetostriction of a CoFeB layer. By use of shape anisotropy, the variation of intrinsic magnetic anisotropy with temperature is strongly reduced. On one hand, interrogating the device at the Love wave resonance frequency allows to extract the applied magnetic field independently of the temperature in a [130 -370K] range. On the other hand, the Rayleigh or Leaky waves are less or not sensitive to applied fields but have a high temperature coefficient of frequency. So interrogating the device at the Rayleigh resonance frequency allows to extract the temperature independently of the magnetic field. In addition, the used resonator geometry offers the possibility for future battery less and wireless interrogation.

I. Introduction

Due to their sensitivity to different chemical and physical stimuli, Surface Acoustic Wave (SAW) devices offer excellent capabilities as sensors [Aub10,Tal06,Woj79]. The addition of a magnetostrictive material into the SAW structure allows the magnetic SAW (MSAW) to probe the magnetic field through the magnetostrictive strains and changes to the elastic constants [Kad11,Elh16,Tie08,Pol17,Mis19,Web79,Zho14]. Several studies have been devoted to the magnetic field sensitivy optimization through the device geometry or the choice of material [Zho14,Kitt18]. Progresses towards the use of different elastic waves to improve the magnetic field sensitivity have also been made [Kitt18,Maz20]. However, these studies do not take into account thermal effects. A sensitivity to multiple stimuli, each coupled with the others constitutes a source of complexity when interpreting the output signal of a sensor. A major challenge in magnetic field measurements lies in the large temperature coefficient of frequency (TCF) of SAW devices. From the perspective of applications of MSAW devices, this becomes extremely detrimental, not only in the case of strong temprature variations, but also in an environment where the temperature fluctuates. In addition to the TCF, the magnetic anisotropy controlling the magnetization response under applied field can also be strongly temperature dependent. Generally, an increase in temperature leads to a increase in the internal energy of the ferromagnetic material by virtue of a rise in its entropy and therefore leads to a decrease of the anisotropy constants [START_REF] Getzlaff | Fundamentals of Magnetism[END_REF]. Magnetocrystalline anisotropy, strain induced anisotropy and magnetic field induced anisotropy are all strongly dependent on temperature, either in their intensity or in their direction. As a result, the MSAW response can be strongly altered by the temperature in many ways. Thus, it becomes a prime concern to nullify the effects of temperature on both the magnetic properties and the TCF if we intend to develop an MSAW sensor that is genuinely compensated for temperature in a practically wide temperature range.

There have been several studies aimed at improving the limit of detection (LOD) of SAW devices for magnetic field sensing that are based on extensive phase noise analysis and aim to improve the signal to noise ratio [Kitt18,[START_REF] Durdaut | [END_REF]. These studies are primarily focused on the development of sensor systems including electronic ciruits that optimize the performance of SAW based magnetic field sensors. On the other hand, improving the quality factor (Q) of the SAW device by optimizing its design also leads to enhancement of the LOD. Our focus in this work has been the development of a truly multifunctional SAW sensor that can in future be integrated with such electronic sensor systems to operate wirelessly, and hence we do not discuss the LOD of our devices but rather focus on the structure of the device and the physics behind it.

In our study, we propose a multilayered device structure capable of eliminating the effects of temperature on both the resonance frequency and on the magnetic anisotropy of the sensitive layer. First of all, the positive TCF of a ST-cut quartz crystal has been compensated for by the negative TCF of a ZnO insulating layer. The optimization of its thickness led to a near-zero TCF for the guided Shear wave, i.e. the Love wave. Secondly, we decided to use shape induced anisotropy to influence the anisotropy of a CoFeB magnetostrictive layer. The magnetization of ferromagnetic materials varies only by several % when the temperature is far from its Curie temperature, and thus the anisotropy direction remains unchanged and the intensity of the anisotropy constant varies only by a few % in the [130 -370K] temperature range. In a previous study, we had shown that the intensity of the anisotropy constant can easily be changed by altering the aspect ratio of the CoFeB nanostructures [Pol17]. As a result, the resonator-based device is shown to have three resonances: a Rayleigh wave which is sensitive to temperature and partially sensitive to a magnetic field, a Love wave which is not sensitive to temperature but highly sensitive to magnetic field and a Leaky wave that is highly sensitive to temperature but not to a magnetic field. Based on this, we propose a multifunctional device, enabling the detection of magnetic field as well as temperature simulataneously, yet being independent of each other.

II. Device fabrication

The substrate are single sided polished 42.75 o Y rotated ST-cut quartz crystals. The SAW devices have a resonator geometry with a wavelength (λ) of 10µm and a metallization ratio of 50%. The wave propagation direction is chosen along the (X+90 o ) direction. We fabricated the electrodes and the connecting pads from a 100nm thick Aluminium layer using conventional optical lithography and a lift-off based process. Subsequently, a ZnO thin insulating layer has been deposited onto the device, as a rectangular area covering the device using a lift-off technique. Based on previous optimization studies [Mis20], three ZnO thicknesses of 200nm, 400nm and 600nm were deposited on the devices. The deposition of the ZnO layer was carried out by RF sputtering of a 4-inch ZnO (oxygen rich) target at 150W with 3×10 -3 mbar and 8 cm 3 /min flow rate of both Ar and O2. Finally, using UV lithography, the photoresit layout for deposition of the CoFeB strip structure on top of the ZnO layers was designed as multiple rows of 1.5µm × 500µm bars covering an area of 3120µm × 710µm with a gap of 1µm between each strip. The 100nm blanket or thin film of CoFeB was then deposited by RF sputtering of a 3-inch target at 300 W and 5×10 -3 mbar Ar and the lift-off was performed. Figure 1 (a) shows a schematic of the fabricated device. We also fabricated a multilayered structure of 4mm×4mm dimension, that corresponds exactly to the multilayered structure of the SAW device, but without the Al electrodes, a schematic of which is presented in Figure 1 (b). This sample was fabricated to conduct rigorous magnetometry tests. The topography of the fabricated MSAW device was studied using MFP-3D Infinity Atomic Force Microscope (AFM). Figure 2 summarizes the analysis of the surface of the device with 400nm thick ZnO. As shown in figure 2a, the purple segment shows the location of one IDT of the SAW structure and the green segment shows the location of one CoFeB strip. Of course, the IDT electrodes and the CoFeB strips are separated by 400nm of the ZnO layer to avoid electrical short cuts. The first thing that becomes very apparent with the above figure is the rabbit's ear structure. This is a very peculiar structure that often occurs due to the lift-off procedure leading to the deposition of material on the resist side walls. As can also be seen, the rabbit's ear of the Al electrodes are higher in the region covered by the CoFeB strip meaning that the CoFeB strip is continuous over the Al electrodes. Figure 2(b) gives an analysis of the surface heights along two cross sections. The red area shows the strong imprint of the Al electrodes even at the top surface of the ZnO layer. As mentioned earlier the electrodes are fabricated from 100nm of Al. However, despite depositing 400nm of ZnO over the electrodes, an imprint of 77nm was very clearly measured. This gives a qualitative idea about how undulating the surface of the ZnO is. The blue area gives the thickness of the CoFeB, deposited to form the micro-structured magneto-sensitive layer which is measured to be 97nm as expected. Finally, the width of the Al electrodes was estimated to 2.53µm while the width of the CoFeB strip was measured to be 1.36µm.

III. Magnetometry measurements

One of the key aspects to control the magnetic response of the sensor consists in studying the response of the magnetization under an applied magnetic field. Therefore, the magnetic properties of a 100nm thick CoFeB film deposited on top of 200nm of ZnO have been measured. Figure 3 (a-ii) shows undoubtedly the presence of an easy axis of magnetization (square hysteresis curve) and a hard axis that is perpendicular to the easy axis direction. The origin of this magnetic anisotropy relies either on the strain imposed by the ZnO layer on the CoFeB during growth or on the magnetron stray field during the CoFeB growth. The response along the hard axis is hysteretic around zero applied field. This hysteresis is a priori redhibitory for the realization of a magnetic field sensor. Furthermore, we measured the hard axis response for temperatures ranging from 293 K to 373 K as reported in figure 3 (a-iii). As can be seen, the saturation field decreases by a factor of two when temperature increases from 293 to 373 K. This is a clear evidence that the anisotropy of the full film decreases by a factor of two. Indeed, in the same temperature window, the saturation magnetization, Ms, only varies by 2.1%. This change of saturation field with temperature and the presence of the hysteresis renders this CoFeB layer ineffective for our MSAW sensor.

Considering the small variation of Ms with temperature, we decided to define micro-structures in the magnetostrictive layer with an aspect ratio such that the magnetic anisotropy of CoFeB is provided by the shape anisotropy that varies as Ms. A variation of anisotropy with temperature less than 2.1% is then expected. Furthermore, reducing the width of the CoFeB strip will ensure a more homogenous magnetization rotation and should decrease the magnetic hysteresis.

The width of the strips was measured at 1.36 µm after fabrication. Figure 3 (b-ii) depicts the measurements along the easy and hard magnetization direction. As expected, a clear observation of the easy axis is made along the length of the strips. The hard magnetization axis lies perpendicular to the length of the strips. trade-off for the control of the magnetization is the increased saturation field, its value can be adjusted by the strip aspect ratio. Measurements of the magnetization (refer Figure 3 (b-iii)) at different temperatures show that micro-structuration enables us to obtain a magnetic response that does not change in the 293 K -373 K temperature window. Consequently, by introducing shape anisotropy to control the magnetization in the magnetostrictive layer, we have successfully addressed and found a way to supress the temperature sensitivity of the magnetic anisotropy. Furthermore, the magnetic response is not only stable up to 373 K, but also presents a macro-spin response akin to the ideal magnetization proposed by the Stoner-Wohlfarth model. Combined with the multi-layered device geometry, this crucial factor will now enable us to develop devices which are truly compensated for the effects of temperature. On the other hand however, with respect to the full film situation, the strain induced by the magnetization rotation will certainly lead to a reduction of the MSAW frequency variation with applied field, thereby leading to a comparatively less sensitive device.

IV. RF characterization and TCF measurement

Figure 4a presents the S11 return loss spectrums for the devices with 200nm, 400nm and 600nm of ZnO along with 100nm of micro-structured CoFeB layer over it. As can be noted, the S11 signal from device with 200nm of ZnO is very small compared to the signals from the two other devices. The reason for this was investigated. The low ZnO thickness is causing electrical losses in the device to due proximity effect between the metallic CoFeB and the electrodes. Four resonance frequencies are observed on the devices. In order to validate the nature of the resonance frequencies, we conducted a simulation-based analysis of the deformations associated with each resonance.

The simulations shown in figure 5 have been obtained on the device with 400nm of ZnO. As can be clearly seen, the first resonance peak (329 MHz) corresponds to a Rayleigh wave while the second resonance peak (421 MHz) corresponds to a Love wave. On the other hand, a tiny peak is observed at 431 MHz, that appears to be a mixed mode resonance. Finally, a fourth resonance is observed at 566 MHz. It is interpreted as a Leaky wave. Since the Love wave absorption peak has been identified, we measured the TCF of the Love wave in the various devices and they are reported in figure 4 (b). A clear variation of the TCF can be observed with changing thicknesses of the ZnO layer; wherein, 400nm emerges as the best choice to annihilate the impact of temperature on the amplitude of the frequency response of the Love wave to the strain induced by the CoFeB (TCF of -1.65 ppm/K). On the other hand, this choice of ZnO thickness does not annihilate the response of the Rayleigh and Leaky waves to temperature. Having thus determined the particular device that is best suited for our study (involving Love waves), we also measure the individual TCFs of each remaining resonance peak in this device comparing them the TCF of the Love wave resonance. As shown in figure 6, the first peak that corresponds to a Rayleigh wave has a TCF of -46 ppm/K and while the last peak that corresponds to a Leaky wave has a TCF of -99 ppm/K. If their response does not depend on the magnetic state of the CoFeB, they could give a measure of the temperature of the device. Comparatively the Love wave resonance in the same device should allow the measurement of magnetic field without any temperature effects.

V. Magneto-acoustic measurement

In this section, we study the variations of the Love wave resonance frequency with respect to the intensity of an external magnetic field applied along the hard magnetic axis (figure 7a). In the previous section, we optimized the thickness of the layers to get a zero-TCF structure using 400nm of ZnO and 100nm of CoFeB. Therefore, in this section we consider only this device and study the magneto-acoustic response for the Rayleigh, the Love and the Leaky wave resonance frequencies observed. The MSAW measurements were made using a LakeShore cryogenic probe station (EMPX-HF) connected to a Vector Network Analyzer (VNA: Rohde and Schwartz ZVA67) by a K-cable and a GGB Picoprobes (40 GHz) probe head. During the measurement, the device is placed between the electromagnets in a vacuum sealed chamber, to ensure a thermally stable and isolated environment. The water-cooled electromagnets are capable of providing a bidirectional magnetic field up to 6000 Oe.

The first measurement of the magneto-acoustic response is made on the temperature compensated Love wave observed at 421 MHz (figure 7b). A first important aspect to note here is also the relative absence of a hysteretic behavior. This is achieved because of the exceptional control over the magnetic properties of the CoFeB layer through micro-structuration. Subsequently, in order to validate the magneto-acoustic sensitivity of the Love wave resonance in the device as independent of temperature, we conducted measurements over a wide range of temperatures. The experimental setup for this was the same as described previously; although, we modified the system to a liquid Nitrogen based cooling system for control of the temperature. Figure 7b presents the measurements of the magneto-acoustic response of the device when the field is increased from -5000 Oe to +5000 Oe. As can be clearly seen, the responses appear quite stable and despite a temperature variation of almost 200 K, the resonance frequency without any magnetic field almost does not shift from around -1.65 ppm/K, it was between 293 K and 333 K. There remains a possibility that this value may differ marginally at cryogenic and high temperatures. Additionally, considering that the temperature was maintained manually by optimizing the flow of liquid N2, a variation of around 4 K was observed even during the measurements. Further, the calibration of the setup was only done at room temperature, whereas in order to optimize the responses, it was essential to have a calibration at each temperature. The effects of all of these shortcomings can be seen as noise and minor differences seen in figure 7b. However, despite all the challenges, we can see that in the linear region that is of importance for device applications (approximately between 500 Oe and 1000 Oe), the MSAW responses of all the devices corroborate with negligible differences. This renders the device a highly stable performance from cryogenic to high temperatures thereby greatly expanding the possible areas of applications. In order to complete the magneto-acoustic characterization of the device, we also measured the magneto-acoustic response of the Rayleigh wave and Leaky wave peaks. Figure 7c presents the response of the Rayleigh wave peak. The characteristic 'W-shaped' curve provides a much smaller response as compared to the Love wave in the same device. This low sensitivity to magnetic field, combined with a high sensitivity towards temperature (-46ppm/K) enables the device to be considered as a multifunctional platform to sense magnetic field as well as temperature, both relatively independent of each other. The same conclusion can be held for the Leaky wave (figure 7d). Since we already understand that these secondary resonances are not temperature compensated (non-zero TCF), their responses to magnetic field at different temperatures were not measured.

VI. Conclusions

In this research work, we addressed a major concern to mitigate the effects of temperature on a SAW device used for magnetic field sensing. First of all, the positive TCF of the quartz-based device is compensated by utilizing a multi-layered structure of ZnO and CoFeB. Further, the CoFeB layer was micro-structured in the form of strips to overcome the effects of temperature on the magnetic anisotropy. Additionally, through magnetometry analysis, we have revealed that the device exhibits a hysteresis free behavior. Among the different relevant resonances that were observed in the device at different frequencies, one is related to a Rayleigh or a Leaky wave, sensitive to temperature and negligibly sensitive to a magnetic field; while the second, is a Love wave, with a negligible sensitivity to temperature (-1.65 ppm/K) and a high sensitivity to magnetic field (0.75 MHz/T). Finally, measurements of the magneto-acoustic response at different temperatures validate the capability of our device to operate in temperatures ranging from cryogenic to relatively high temperatures. Thus, in this work, we have demonstrated the capability to develop the SAW device as a multifunctional sensor capable of detecting magnetic field as well as temperature at the same position and each, independent of the effect of the other. As a result, we have established a new state-of-the-art technology for SAW devices to be multifunctional encompassing magnetic field sensitivity and temperature sensitivity in the same device.

Figure 1 :

 1 Figure 1: (a) Schematic of the fabricated MSAW device and (b) the magnetometry sample.

Figure 2 :

 2 Figure 2: (a) Surface topography over a section of device with 400nm of ZnO.(b) Cross sections done on the surface of the device with 400nm of ZnO, using the atomic force microscope. The 'impression' of the Aluminium IDTs (100nm thick) on the 400nm ZnO layer is measured at 77nm while the thickness of the CoFeB layer is measured at 97nm.

Figure 3 :

 3 Figure 3: Normalized magnetization measurements conducted on different samples (a) Full film of CoFeB deposited on 200nm of ZnO; (b) CoFeB strips of 1.36µm width on 200nm of ZnO.

Figure 4 :

 4 Figure 4: (a) Scattering parameter (S11) measured on samples with different ZnO thicknesses after deposition and micro-structuration of 100nm of CoFeB; (b) TCF measured on the Love wave resonances in the three devices.

Figure 5 :

 5 Figure 5: Total displacements observed for the different resonance frequencies in device with 400nm ZnO as obtained through COMSOL simulations.

Figure 6 :

 6 Figure 6: Scattering parameter (S11) measured on samples with 400nm of ZnO after the deposition and micro-structuration of 100nm of CoFeB. For each wave, the values of the TCF have been added.

Figure 7 :

 7 Figure 7: Magneto-acoustic response of the Love wave peak in device with 400nm of ZnO for a magnetic field applied along the hard axis (perpendicular to the CoFeB strips) (a) the geometry of measure; (b) Magneto-acoustic response of the Love wave at different temperatures; (c) Magnetoacoustic response of the Rayleigh wave; (d) Magneto-acoustic response of the Leaky

Acknowledgements

We would like to acknowledge the support from the French PIA project "Lorraine Université d'Excellence" (ANR-15-IDEX-04-LUE) and by the Région Lorraine, ANR JCJC SAWGOOD (ANR-18-CE42-0004-01) the European funds FEDER. Experiments were carried out on IJL Project TUBE-Davm equipment funded by FEDER (EU), Region Grand Est, Metropole Grand Nancy. We also extend our sincere gratitude to the staff from the Competence Centre MINALOR for helpful discussions in fabrication.


T Lagache

D Holcman

Extended Narrow Escape with ManyWindows for Analyzing Viral Entry into the Cell Nucleus

Many viruses must enter the cell nucleus through small nanopores in order to replicate. We model here viral motion as a stochastic process described by the survival Fokker-Planck equation. We estimate the probability and the conditional mean first passage time that a viral trajectory is absorbed at a small nuclear pore before being terminated. The method is based on the explicit Neuman-Green's function. The cell nucleus is model as a three dimensional ball, covered with thousands of small absorbing windows. The minimum distance between them defines the smallest spatial scale that is an unavoidable limit for efficient stochastic simulations. Derived asymptotic formula agree with stochastic simulations and reveal how small and large geometrical parameters define the cytoplasmic stage of viral infection.

Introduction

Understanding the biophysical properties of particles such as molecules, proteins, DNA, RNA, or viruses that are moving inside the crowded cellular environment [START_REF] Medalia | Macromolecular architecture in eukaryotic cells visualized by cryoelectron tomography[END_REF] remains a challenge both experimentally and theoretically. For example, vesicles or RNA granules [START_REF] Dynes | Dynamics of bidirectional transport of arc mrna in neuronal dendrites[END_REF] have to reach small targets in order to deliver their payload or trigger protein synthesis. If small DNA or plasmid can pass the cytoplasmic crowded organization, the motion of larger ones is largerly impaired [START_REF] Dauty | Actin cytoskeleton as the principal determinant of sizedependent dna mobility in cytoplasm: a new barrier for non-viral gene delivery[END_REF]. In some cases, large Brownian particles are transported intermittently along microtubules (MTs) inside the cytoplasm toward the nucleus. Many viruses containing DNA have the ability to hijack the cellular transport machinery to reach a nuclear pore and deliver their genetic material inside the nucleus [START_REF] Sodeik | Mechanisms of viral transport in the cytoplasm[END_REF][START_REF] Greber | A superhighway to virus infection[END_REF]. Viral trajectories can be monitored in vivo using live microscopy and for viruses such as Human Immunodeficiency Virus or the Adeno-Associated Virus [START_REF] Arhel | Quantitative four-dimensional tracking of cytoplasmic and nuclear hiv-1 complexes[END_REF][START_REF] Seisenberger | Real-time single-molecule imaging of the infection pathway of an adeno-associated virus[END_REF], these trajectories consist of alternating epochs of diffusion and directed motion. The precise driving force and physical nature of these trajectories remains unclear. In addition inside the cytoplasm, viruses can be trapped or degraded through several molecular pathways (including the ubiquitin-proteasome).

To quantify the success of the early steps of viral infection, we previously modeled single viral particle as individual stochastic processes [START_REF] Holcman | Modeling dna and virus trafficking in the cell cytoplasm[END_REF][START_REF] Lagache | Quantitative analysis of virus and plasmid trafficking in cells[END_REF]. Our goal was to estimate the success that a trajectory reaches a small window (nuclear pore) located on the nucleus surface (Fig. 1). However, due to small size of the nuclear pore, Brownian simulations are most of the time ineffective to estimate precisely the statistical moments associated with the arrival time distribution. Thus, to study the dependency with respect to geometrical and dynamical parameters, we analyzed the intermittent stochastic dynamics of viruses along MTs and derived asymptotic formula for the conditional mean first passage time (MFPT) τ (n, ) and the probability P (n, ) that a single particle arrives to one over n small targets with radius [START_REF] Holcman | Modeling dna and virus trafficking in the cell cytoplasm[END_REF][START_REF] Lagache | Quantifying intermittent transport in cell cytoplasm[END_REF]. However, these formula are valid when the number of absorbing holes (targets) is not too large and that holes are well separated. We shall extend here this previous analysis to the case of many small holes. This analysis relies on the explicit expansion of the Neumann-Green's function to order three [START_REF] Pillay | An asymptotic analysos of the mean first passage time for narrow escape problems: Part i: Two-dimensional domains[END_REF], and is valid for many interacting small holes [START_REF] Schuss | Diffusion escape through a cluster of small absorbing windows[END_REF][START_REF] Holcman | Diffusion through a cluster of small windows and flux regulation in microdomains[END_REF][START_REF] Cheviakov | An asymptotic analysis of the mean first passage time for narrow escape problems: Part ii: The sphere[END_REF].

In the limit of many absorbing holes distributed on a small spherical target inside a bounded domain (a biological round cell), we compute here a novel mean arrival time formula to an absorbing window, that falls into the Narrow escape time (NET) framework [START_REF] Holcman | Stochastic Narrow Escape in Molecular and Cellular Biology: Analysis and Applications[END_REF] . This computation clarifies the dependency of the NET with respect to the number of holes, their distribution, the radius of the spherical target and the percentage of the target surface covered by holes. This analysis extents the well-known homogenization formula obtained by Berg and Purcell for mixed Neumann-Dirichlet boundary conditions [START_REF] Berg | Physics of chemoreception[END_REF]. The paper is organized as follows. First, we recall our previous model of viral particles and the stochastic description of trajectories. Second, we extend the NET method to many absorbing holes and compute the mean arrival time formula using an interaction matrix between holes. Then, computing the Neumann's function around a small reflecting sphere, we derive asymptotic formula for the probability and the conditional mean time to reach one small window in the limit of a large number of absorbing windows . Finally, we confirm our analysis with Brownian simulations. The new formula improve our previous effort [START_REF] Holcman | Modeling dna and virus trafficking in the cell cytoplasm[END_REF][START_REF] Lagache | Quantifying intermittent transport in cell cytoplasm[END_REF] and can now be used to quantify more precisely the first steps of viral infection in cells. In completely independent line of thoughts, a similar NET formula was recently derived using matched asymptotics for Brownian motion [START_REF] Lindsay | First passage statistics for the capture of a brownian particle by a structured spherical target with multiple surface traps[END_REF]. 2), with a diffusion and a drift terms. They can be absorbed at small windows with radius |Ω| 1/3 located on the surface of the nucleus ∂S a . (Left): simplified geometry: the cell geometry is a ball (radius R) containing a spherical nucleus surface, containing many small absorbing windows. The nucleus ball is of radius a such that a |Ω| 1/3 .

2 Mean time to a small nuclear pore

Intermittent trajectories of a viral particle x(t) are described as realization of the switching stochastic process [START_REF] Lagache | Effective motion of a virus trafficking inside a biological cell[END_REF]

dx =    √ 2Ddw when x (t) is free Vdt x (t) bound , (1)
where w is a standard 3d-Brownian motion, D the diffusion constant and V the velocity of the directed motion along MTs. The switching dynamics depends on the attachment and detachment rates [START_REF] Lagache | Effective motion of a virus trafficking inside a biological cell[END_REF]. In previous work [START_REF] Lagache | Effective motion of a virus trafficking inside a biological cell[END_REF][START_REF] Lagache | Quantifying intermittent transport in cell cytoplasm[END_REF] we have coarse-grained this switching process into by a steady-state stochastic equation

dx = b(x)dt + √ 2Ddw, (2)
where the effective drift b(x) is calibrated using the following criteria: inside the cytoplasm Ω, the mean first passage time of stochastic processes (1) and ( 2) is the same. The steadystate drift b(x) depends on the cell geometry, the number and distribution of MTs and the rates of binding and unbinding to MTs.

To replicate most viruses have to reach a small pore, modeled as an absorbing disk of radius 1 located on the boundary ∂S a of the nucleus. The external cell membrane defines the boundary ∂Ω for the stochastic process equation [START_REF] Berezhkovskii | Boundary homogenization for trapping by patchy surfaces[END_REF]. The cell cytoplasm is 3 represented as the three-dimensional bounded domain Ω, whose boundary is ∂Ω ∂S a . It consists of a reflecting part except on the boundary ∂N a which consists of n small absorbing windows located on the nucleus (fig. 1-left).

We model the degradation activity in the cytoplasm by a steady-state killing rate k(x) and a trajectory described by equation ( 2) can be terminated before reaching the absorbing boundary ∂N a . The survival probability density function (SPDF) is the solution of the forward Fokker-Planck equation (FPE) [START_REF] Holcman | Survival probability of diffusion with trapping in cellular neurobiology[END_REF] ∂p(x, t) ∂t

= ∆p(x, t) -∇ • b(x)p(x, t) -k(x)p(x, t) p (x, 0) = p i (x) (3)
with the boundary conditions:

p(x, t) = 0 on ∂N a and ( 4)

J(x, t).n x = 0 on ∂Ω (∂S a -∂N a ) (5)
where the flux density vector is

J(x, t) = -D∇p(x, t) + b(x)p(x, t). (6)
where n x is the normal vector at point x.

We recall that the mean probability P and the conditional MFPT τ (averaged over a uniform particle distribution) to reach the boundary ∂N a before termination is expressed using the total time spent at a point

p(x) = ∞ 0 p(x, t)dt (7)
and [START_REF] Holcman | Modeling dna and virus trafficking in the cell cytoplasm[END_REF]. The expressions of the probability and arrival time with respect to the solution of the FPE are

q(x) = ∞ 0 tp(x, t)dt (8)
P (n, ) = 1 - Ω k(x)p(x)dx, (9)
and

τ (n, ) = Ω p(x)dx - Ω k(x)q(x)dx 1 - Ω k(x)p(x)dx . ( 10
)
For a drift b (x) = -∇Φ (x), an asymptotic expansion with the small parameter reveals that [START_REF] Holcman | Modeling dna and virus trafficking in the cell cytoplasm[END_REF]

                       P (n, ) = e -Φ 0 D 1 4Dn Ω e -Φ(x) D k(x)dx + e -Φ 0 D τ (n, ) = 1 4Dn Ω e -Φ(x) D dx 1 4Dn Ω e -Φ(x) D k(x)dx + e -Φ 0 D , (11)
where Φ 0 is the constant value of the radial potential Φ(x) on the centered nucleus where the nuclear pores are uniformly distributed. The range of validity of these asymptotic expressions has been explored with Brownian simulations for a single hole [START_REF] Lagache | Quantitative analysis of virus and plasmid trafficking in cells[END_REF]. However, these formulas do not account for the possible interactions between the small absorbing pores, and for a large number of nuclear pores n 1,

lim n→∞,n 2 1 τ (n, ) = 0, (12)
which shows the limitation of the previous formula.

Early results demonstrated that holes' interactions can drastically affect the MFPT, especially when the distance between the holes tends to zero [START_REF] Schuss | Diffusion escape through a cluster of small absorbing windows[END_REF][START_REF] Holcman | Diffusion through a cluster of small windows and flux regulation in microdomains[END_REF]. Refined analysis using analytical expression of the Green's function in the unit disk, sphere or ball based on solving a linear system of equations for the fluxes at each window revealed how the MFPT depends on the position of the windows [START_REF] Coombs | Diffusion on a sphere with localized traps: Mean first passage time, eigenvalue asymptotics, and fekete points[END_REF][START_REF] Cheviakov | An asymptotic analysis of the mean first passage time for narrow escape problems: Part ii: The sphere[END_REF][START_REF] Pillay | An asymptotic analysos of the mean first passage time for narrow escape problems: Part i: Two-dimensional domains[END_REF][START_REF] Chevalier | First-passage quantities of brownian motion in a bounded domain with multiple targets: a unified approach[END_REF]. In particular, the optimal configuration of windows that minimizes the MFPT was related to the classical Fekete problem of minimizing the Coulomb energy on a ball [START_REF] Coombs | Diffusion on a sphere with localized traps: Mean first passage time, eigenvalue asymptotics, and fekete points[END_REF][START_REF] Cheviakov | An asymptotic analysis of the mean first passage time for narrow escape problems: Part ii: The sphere[END_REF][START_REF] Pillay | An asymptotic analysos of the mean first passage time for narrow escape problems: Part i: Two-dimensional domains[END_REF][START_REF] Cheviakov | Narrow-escape problem for the unit sphere: homogenization limit, optimal arrangements of large numbers of traps, and the n(2) conjecture[END_REF]. However, the linear system of equations for the flux at each window does not hold when windows are distributed on the surface of ball with a small radius (of the order of few window size). We shall derive here the correction term that accounts for the nuclear geometry. First, we derive the narrow escape time for a stochastic particle (with a drift) in the presence of a killing field k(x) to reach an absorbing window. Second, by computing analytically the Neumann function for a reflecting ball, we derive a coupled system of equations for the fluxes at each interacting window covering the ball (spherical nucleus). Finally, for a large number of holes, we use a mean-field approximation and obtain formulas for the probability P and the mean time τ , valid for a large range of parameters (window size) and n, generalizing formula [START_REF] Greber | A superhighway to virus infection[END_REF] and extending previous attempts to homogenize mixed Dirichlet-Neumann boundary conditions [START_REF] Berg | Physics of chemoreception[END_REF][START_REF] Zwanzig | Diffusion-controlled ligand binding to spheres partially covered by receptors: an effective medium treatment[END_REF][START_REF] Berezhkovskii | Boundary homogenization for trapping by patchy surfaces[END_REF][START_REF] Reingruber | Narrow escape time to a structured target located on the boundary of a microdomain[END_REF]. We summarize now the main asymptotic formula (86) and (87). We define the function

F (n, a, ) = naD πa + log a + n (1 -2α 0 -α 2 0 log (α 0 )) (13)
with

α 0 =         
a for uniformly randomly distributed windows,

1 √ n for regularly distributed windows. ( 14
)
Using the same notations as in formula [START_REF] Greber | A superhighway to virus infection[END_REF], when the drift is pointing towards the nucleus center and the potential Φ(x) = Φ 0 is constant at nuclear surface, the probability and MFPT formulas are

P = 4πF (n, a, )e -Φ 0 D Ω k(x)e -φ(x) D dx + 4πF (n, a, )e -Φ 0 D , (15)
and

τ = Ω e -φ(x) D dx Ω k(x)e -φ(x) D dx + 4πF (n, a, )e -Φ 0 D . ( 16
)
Finally, we test the robustness of our asymptotic formula against Brownian simulations and apply our formula to describe viral trafficking to a nuclear pore (n ≈ 2, 000 1 [START_REF] Maul | Quantitative determination of nuclear pore complexes in cycling cells with differing dna content[END_REF]) covering the nucleus.

3 Asymptotic derivations of the mean time τ and the probability P

The n-absorbing windows ∂N a = n i=1 ∂Ω i have the same radius , centered at positions (x i ) n i=1 . The steady state SPDF p is solution of equation ( 3) [START_REF] Holcman | Survival probability of diffusion with trapping in cellular neurobiology[END_REF].

The associated Neumann-Green function N (x, x 0 ) is solution of the differential equation [START_REF] Holcman | Modeling dna and virus trafficking in the cell cytoplasm[END_REF] ∆N (x,

x 0 ) = -δ x 0 (x), x ∈ Ω, ∂N ∂n (x, x 0 ) = - 1 |∂Ω| , for x ∈ ∂Ω. (17)
We recall that p

(x) = ∞ 0 p(x, t)dt is solution of equation ∆p (x) -∇ • (b (x) p (x)) -k (x) p (x) = -p i (x) , (18)
where p i (x) is the normalized probability density function for the initial point

Ω p i (x) = 1. ( 19
)
The boundary conditions for equation [START_REF] Lagache | Effective motion of a virus trafficking inside a biological cell[END_REF] are given by

p(x) = 0 on ∂N a = n i=1 ∂Ω i (20)
and

J(x).n x = 0 on ∂Ω (∂S a -∂N a ) (21)
where the steady-state probability flux is given by

J(x) = -D∇p(x) + b(x)p(x). ( 22
)
We recall that the average arrival probability [START_REF] Holcman | Modeling dna and virus trafficking in the cell cytoplasm[END_REF] is given by

P = ∂Na J(x).n x dx = 1 - Ω k(x)p(x)dx. (23)
Thus we shall compute here p. For this, we compute the following integral

I = Ω (∆p(x) -∇ • (b(x)p(x)) -k(x)p(x)) N (x, x 0 )dS x - Ω ∆N (x, x 0 )p(x)dx. ( 24
)
Using the definition of the Neumann-Green function [START_REF] Lagache | Quantitative analysis of virus and plasmid trafficking in cells[END_REF] and that p is solution of equation ( 18), we obtain

I = - Ω p i (x)N (x, x 0 ) + p(x 0 ). ( 25
)
Moreover, using the Green's identity:

Ω (N (x, x 0 )∆p(x) -p(x)∆N (x, x 0 )) dx = ( 26
) ∂Ω N (x, x 0 ) ∂ p(x) ∂n -p(x) ∂N (x, x 0 ) ∂n . ( 27
)
we get another expression for I

I = ∂Ω N (x, x 0 ) ∂ p(x) ∂n -p(x) ∂N (x, x 0 ) ∂n + Ω (-∇ • (b(x)p(x)) -k(x)p(x)) N (x, x 0 )dx. ( 28
)
Using an integration by part, we have

Ω (-∇ • (b(x)p(x))) N (x, x 0 )dx = Ω (b(x)p(x)) ∇N (x, x 0 )dx (29)
-

∂Ω (b(x) • n x ) p(x)N (x, x 0 )dS x . ( 30
)
By definition of the boundary condition of Neumann-Green function N ,

∂Ω p(x) ∂N (x, x 0 ) ∂n dS x = - 1 |∂Ω| ∂Ω p(x)dS x (31)
We conclude that

I = - ∂Na J(x).n x N (x, x 0 )dx + Ω b(x).∇N (x, x 0 )p(x)dx - Ω k(x)p(x)N (x, x 0 )dx + 1 |∂Ω| ∂Ω p(x)dS x . (32)
Finally, merging the two relations (32) and ( 25) for the integral I, we get the identity:

Ω (k(x)p(x) -p i (x)) N (x, x 0 )dx = - ∂Na J(x).n x N (x, x 0 )dx + Ω b(x).∇N (x, x 0 )p(x)dx + 1 |∂Ω| ∂Ω p(x)dx -p(x 0 ). ( 33
)
When the vector field is the gradient of a potential b = -∇Φ, we shall use the following approximation [START_REF] Holcman | Modeling dna and virus trafficking in the cell cytoplasm[END_REF] for the solution

p(x) = C e -Φ(x) D , ( 34
)
where C is a constant that depend on ε, which tends to ∞ when tends to 0. We shall now only retain in relation (33) leading order terms in . For a smooth initial distributions p i , the integral

Ω p i (x)N (x, x 0 )dx = O(1) (35)
is uniformly bounded when → 0 (the proof was given in [START_REF] Holcman | Escape through a small opening: Receptor trafficking in a synaptic membrane[END_REF] by deriving the regular Laplace equation satisfied by the function x 0 → Ω p i (x)N (x, x 0 )dx). All other terms diverge. We shall also consider here the case of a small degradation rate k(x) 1. In that case, we will neglect the term Ω k(x)p(x)N (x, x 0 )dx compared to all other terms in (33). With these approximations, relation (33) can be written at leading order as

p(x 0 ) + O (1) = - ∂Na J(x).n x N (x, x 0 )dS x + Ω b(x).∇N (x, x 0 )p(x)dx + 1 |∂Ω| ∂Ω p(x)dx. ( 36
)
For an initial point x 0 at a distance O (1) away from any absorbing window, the Neumann function N (x, x 0 ) is uniformly bounded for x ∈ ∂Ω a . In addition, integrating relation [START_REF] Lagache | Effective motion of a virus trafficking inside a biological cell[END_REF] over Ω, we obtain

∂Na J(x).n x dS x = 1 - Ω k(x)p(x)dx = P ∈ [0, 1]. ( 37
)
Thus for x ∈ ∂N a the absorbing boundary, the flux J(x).n x = -D ∂ p(x) ∂n x > 0 and thus the integral ∂Na J(x).n x N (x, x 0 )dS x is uniformly bounded in x 0 ∈ Ω. Thus for x 0 at a distance O (1) away from the absorbing windows, relation (36) can be written

1 |∂Ω| ∂Ω p(x)dx + Ω b(x).∇N (x, x 0 )p(x)dx = p(x 0 ), (38)
which is the integral solution p(x 0 ) = C e -Φ(x 0 ) D for the Fokker-Planck equation with no absorbing windows.

1 |∂Ω| ∂Ω p(x)dx + Ω b(x).∇N (x, x 0 )p(x)dx = C e -Φ(x 0 ) D + O (1) . (39)
We shall now take y ∈ ∂N a in relation (36

) 0 = - ∂Na J(x).n x N (x, y)dS x + Ω b(x).∇N (x, y)p(x)dx + 1 |∂Ω| ∂Ω p(x)dx. ( 40
)
Using relation (39), we finally obtain

0 = - ∂Na J(x).n x N (x, x 0 )dx + C e -Φ(x 0 ) D (1 + o(1)), (41)
(because we removed in the two integrals a boundary layer near the absorbing windows).

We shall now compute ∂Na J(x).n x N (x, x 0 )dx = n i=1 ∂Ω i J(x).n x N (x, x 0 )dx, by decomposing the flux

J(x).n x x∈∂Ω i = g i (x) + f i (x), (42)
where the leading order g i (s) with

s = |x -x i | (43) into g i (s) = g i 0 √ 2 -s 2 , ( 44
)
and g i 0 a constant and f i is a regular function such that

0 f i (s)ds = O( g i 0 ). ( 45
)
Choosing y = x i at the center of each absorbing windows, we get that p(x i ) = 0. For i = j, and |x i -x j | and for x ∈ ∂Ω j ,

N (x, x i ) = N (x j , x i ) + O( ). ( 46
)
Consequently, using the flux expansion (42), we get

∂Na J(x).n x N (x, x i )dx = ∂Ω i (g i (x) + f i (x)) N (x, x i )dx + n j=1,j =i (N (x j , x i ) + 0( )) ∂Ω j (g j (x) + f j (x)) dx. ( 47
)
For x i ∈ ∂S a , the Neumann-Green's function N (x, x i ) can be written as [START_REF] Singer | Narrow escape and leakage of brownian particles[END_REF]:

N (x, x i ) = 1 2πD|x -x i | + L(x i ) + N (x i ) 8πD log 1 |x -x i | + ω x i (x), (48)
where L(x i ) and N (x i ) are the principal curvatures of ∂S a at x i and ω x i (x) is the regular part of the Green function, which is bounded for x in Ω. However, when the absorbing small patches are located on the boundary of a small ball of radius a, expansion (48) is not sufficient because the second term -1 4πaD log

1 |x i -x j |
become much larger than the first term

1 2πD|x i -x j | when |x i -x j | ≈ a, and a |Ω| 1 3
. A different expansion is needed that we shall now discuss.

Computing the flux at small absorbing windows located on a small internal ball

The solution of the Neumann's equation

D∆ Ñ (x, x 0 ) = -δ(x -x 0 ), for x ∈ R 3 D ∂ Ñ ∂n (x, x 0 ) = 0, for x ∈ S a . ( 49
) is equal for |x 0 | = |x| = a to (see appendix) Ñ (x, x 0 ) = 1 2πD|x -x 0 | + 1 4πaD log |x -x 0 | 2a + |x -x 0 | . ( 50
)
Thus for x and x 0 in the neighborhood of the sphere S a , we have

N (x, x 0 ) = Ñ (x, x 0 ) + O(1). (51)
Consequently, using the geodesic distance s = d(P, x i ), the flux term in relation ( 47) can now be computed

∂Na J(x).n x N (x, x i )dx = 0 g i 0 √ 2 -s 2 + f i (s) (52) 1 2πDs + 1 4πaD log s 2a + s + O(1) 2πsds + n j=1,j =i (N (x j , x i ) + 0( )) 0 g j 0 √ 2 -s 2 + f j (s) 2πsds.
Using condition (45), we obtain:

∂Na J(x).n x N (x, x 0 )dx = g i 0 D π 2 + 2a log a + O( ) + 2π n j=1,j =i N (x j , x i )g j 0 (1 + O( ))) . ( 53
)
We recall that the constant g i 0 is of order

g i 0 = O 1 n
, and that

N (x j , x i ) = O 1 |x i -x j | = O 1 a , (54)
thus we rewrite the flux condition as

∂Na J(x).n x N (x, x 0 )dx = g i 0 D π 2 + 2a log a (55) + 2π n j=1,j =i N (x j , x i )g j 0 + O a + O 1 n .
Injecting (56) in (41), for y = x i , we obtain the system of n equations for the n+1 variables (g 1 0 , ..g n 0 , C ):

π 2D + 2aD log a g i 0 + 2π n j=1,j =i N (x j , x i )g j 0 = C e -Φ(x i ) D + O (1) . ( 56
)
To close the system of equation, we use the compatibility condition (37) with expression (42) and approximation (34) for the function p:

2π n i=1 g i 0 = 1 -C Ω k(x)e -Φ(x) D dx + O (1) . (57)
Finally, we obtain a linear system of n + 1 equations (56-57) for the flux constant g i 0 , i ≤ i ≤ n and for the constant C summarized here:

               π 2D + 2aD log a g i 0 + 2π n j=1,j =i N (x j , x i )g j 0 = C e -Φ(x i ) D + O (1) , for 1 ≤ i ≤ n 2π n i=1 g i 0 = 1 -C Ω k(x)e -Φ(x) D dx + O (1) (58)
We will now obtain asymptotic estimates for C , P and τ , by solving the linear system of equations (58) in the small limit. We recall the expressions for probability (34) and the arrival time:

P = 1 -C Ω k(x)e -Φ(x) D dx and τ = C Ω e -Φ(x) D dx, (59)
where this last identity is derived in the small killing rate limit |k(x)| 1 [START_REF] Holcman | Modeling dna and virus trafficking in the cell cytoplasm[END_REF]. In the next section, we derive the asymptotic expressions for quantities in (59).

4 Mean field approximation and asymptotics formula for τ and P for n

1
We derive now expressions for P and τ in the limit n 1 and absorbing windows are distributed with respect to a density ρ(x) over the spherical nucleus S a . By summing equation (56) for 1 ≤ i ≤ n, we obtain that

π 2D + 2aD log a n i=1 g i 0 + 2π n i=1 g i 0 n j=1,j =i N (x j , x i ) = C n i=1 e -Φ(x i ) D + O (n) . (60)
When x i is located at the north pole, the distance |x i -x j | with j th located at position x j (θ, φ) is given by |x i -x j | = 2a sin φ 2 and the Neumann function is

Ñ (x j (θ, φ), x i ) = 1 4πaD 1 sin φ 2 + log sin φ 2 1 + sin φ 2 . ( 61
)
We use now that the probability density function ρ i (φ) of the j = i windows (north pole i) is normalized

π 0 2πa 2 ρ i (φ) sin(φ)dφ = 1, ( 62
) thus lim n→∞ 1 n n j=1,j =i N (x j , x i ) = π 0 1 4πaD 1 sin φ 2 + log sin φ 2 1 + sin φ 2 ρ i (φ)2πa 2 sin(φ)dφ = a 2D π 0 1 sin φ 2 + log sin φ 2 1 + sin φ 2 ρ i (φ) sin(φ)dφ. ( 63
)
In addition, we have the limit of the partial sum

lim n→∞ 1 n n i=1 e -Φ(x i ) D = a 2 2π 0 π 0 e -Φ(φ,θ) D ρ(φ)dφdθ, (64)
We now re-write relation (60) using the two integrals

I i 1 = π 0 1 sin φ 2 + log sin φ 2 1 + sin φ 2 ρ i (φ) sin(φ)dφ, (65)
I 2 = 2π 0 π 0 e -Φ(φ,θ) D ρ(φ, θ) sin(φ)dφdθ, ( 66
) so that π 2D + 2aD log a n i=1 g i 0 + naπ D n i=1 g i 0 I i 1 = C na 2 I 2 + O (n) . (67)
For identically distributed windows 1 ≤ i ≤ n, I i 1 = I 1 . Using the compatibility condition (57) in equation (67), we obtain

1 4D + 1 4πaD log a + na 2D I 1 1 -C Ω k(x)e -Φ(x) D dx = C na 2 I 2 + O (n) . ( 68
)
Thus we obtain to leading order

C = πa + log a + 2nπa 2 I 1 πa + log a + 2nπa 2 I 1 Ω k(x)e -Φ(x) D dx + 4πna 3 D I 2 . ( 69
)
To further compute for the probability P and the MFPT τ using expression (69), we shall now consider two distributions of windows:

1. Random distribution, 2. Uniform distribution.

Effect of randomly distributed narrow windows located on a sphere on the arrival time

When there n 1 non-overlapping windows randomly distributed on the sphere, the probability distribution of windows is given by

ρ(φ, θ) = ρ i (φ) = 1 4πa 2 1 {φ>2 arcsin( a )} , (70)
for all 1 ≤ i ≤ n. The condition φ > 2 arcsin a ensures non-overlapping. Using the change of variable y = sin φ 2 , we re-write integral (65)

I 1 = 1 πa 2 1 a 1 y + log y 1 + y ydy = 1 2πa 2 x + log (1 + x) + x 2 log x 1 + x 1 a , (71)
that is

I 1 = 1 2πa 2 1 -2 a - 2 a 2 log a . (72)
In addition, we have for the second integral (66)

I 2 = 1 4πa 2 2π 0 π 2 arcsin( a ) e -Φ(φ,θ) D sin(φ)dφdθ. ( 73
)
Replacing in equation (69), I 1 and I 2 by expressions (72) and (73), we obtain to leading order for randomly distributed windows,

C rand = 1 Ω k(x)e -Φ(x) D dx + C(n, ε) 2π 0 π 2 a e -Φ(φ,θ) D sin(φ)dφdθ , (74)
where

C(n, ε) = naD πa + 1 -n 2 a 2 log a + n 1 -2 a . ( 75
)
4.2 Effect of regularly distributed windows located on the surface S a on the mean time

For small windows regularly distributed on a sphere, the density was computed in [START_REF] Cheviakov | Narrow-escape problem for the unit sphere: homogenization limit, optimal arrangements of large numbers of traps, and the n(2) conjecture[END_REF]:

ρ(φ) = 1 {φ>arccos(1-2 n )} 1 4πa 2 , ( 76
)
leading to with the first integral (65) to

I 1 = 1 2πa 2 x + log (1 + x) + x 2 log x 1 + x 1 1 2 arccos(1-2 n ) , (77)
and for n 1

I 1 = 1 2πa 2 1 - 2 √ n + log(n) 2n + o( log(n) 2n ). ( 78
)
In addition, for second integral (66) we have

I 2 = 1 4πa 2 2π 0 π 2 arccos(1-2 n ) e -Φ(φ,θ) D sin(φ)dφdθ. ( 79
)
Replacing in equation ( 69), I 1 and I 2 by expressions (78) and ( 79) respectively, we obtain

C hom = 1 Ω k(x)e -Φ(x) D dx + C ,n 2π 0 π 4 √ n e -Φ(φ,θ) D sin(φ)dφdθ . (80)
where

C ,n = naD πa + log √ n a + n 1 -2 √ n (81)
Using formula (74) and (80) in expression (59), we obtain the asymptotic expressions for the probability and the condition MFPT that a stochastic particle reaches a small windows

P = F (n, a, ) 2π 0 π α 0 e -Φ(φ,θ) D sin(φ)dφdθ Ω k(x)e -Φ(x) D dx + F (n, a, ) 2π 0 π α 0 e -Φ(φ,θ) D sin(φ)dφdθ , (82)
and

τ = Ω e -φ(x) D dx Ω k(x)e -Φ(x) D dx + F (n, a, ) 2π 0 π α 0 e -Φ(φ,θ) D sin(φ)dφdθ (83)
where

F (n, a, ) = naD πa + log a + n (1 -2α 0 -α 2 0 log (α 0 )) (84)
and

α 0 =         
a for randomly distributed windows, 1 √ n for regularly distributed windows.

(85)

When the drift is pointing towards the nucleus center and the potential Φ(x) = Φ 0 is constant at the nuclear surface, then the probability and MFPT formulas reduce to

P = 4πF (n, a, )e -Φ 0 D Ω k(x)e -φ(x) D dx + 4πF (n, a, )e -Φ 0 D , (86)
and

τ = Ω e -φ(x) D dx Ω k(x)e -φ(x) D dx + 4πF (n, a, )e -Φ 0 D . ( 87
)
When the drift Φ Sa restricted to S a has a single global minima Φ m at position x 0 (φ 0 , θ 0 ) ∈ S a , we approximate integral I 2 using Laplace's method. In the small diffusion limit D Φ(x) and large n, we get

I 2 = 1 4πa 2 2π 0 π 2 arccos(1-2 n ) e -Φ(φ,θ) D sin(φ)dφdθ ≈ D 4a 2 det -H Φ Sa (x 0 ) e -Φm D (88)
where det H Φ Sa (x 0 ) is the determinant of the Hessian matrix of potential Φ Sa at x 0 . The probability and MFPT to a nuclear pore are then given by

P = πDF (n, a, ) det -1 H Φ Sa (x 0 ) e -Φm D Ω k(x)e -Φ(x) D dx + πDF (n, a, ) det -1 H Φ Sa (x 0 ) e -Φm D , (89)
and

τ = Ω e -φ(x) D dx Ω k(x)e -Φ(x) D dx + πDF (n, a, ) det -1 H Φ Sa (x 0 ) e -Φm D . (90)
A second Laplace's method can be used to estimate the volume integral. If the global minimum Φ Ω is attained at a point

x g ∈ Ω, Ω e -φ(x) D dx ≈ (πD) 3/2 det [H Φ (x g )] e -Φ Ω D . ( 91
)
We conclude this section by indicating that the formulas presented above can be used to estimate the probability and the mean time for a viral particle to reach a nuclear pore inside the nucleus.

Effect of changing the window coverage on the escape time

For a large number of windows n 1, distributed over a small target S a of a domain Ω and covering a proportion σ = nπ 2 |∂Sa| of the surface ∂S a of S a , the leading order term in the NET for a Brownian particle to one of the small window was computed asymptotically using electrostatic consideration [START_REF] Reingruber | Narrow escape time to a structured target located on the boundary of a microdomain[END_REF]

τ ES = |Ω| D 1 C Sa + f (σ) 4n , ( 92
)
where |Ω| is the volume, C Sa the capacity of the surface ∂S a containing the absorbing windows, and f (σ) an unknown function that whose leading order is given by f (σ) = 1 as σ tends to 0 [START_REF] Berg | Physics of chemoreception[END_REF]. When the surface S a is a sphere of radius a, the capacitance C Sa = 4πa, and the MFPT is given by

τ ES = |Ω| D 1 4πa + f (σ) 4n . (93)
Here, for a Brownian particle (no drift and no killing measure), the MFPT (equation ( 83)) reduces to

τ Φ=0,k=0 ≈ |Ω| D 1 4πa + 1 4n 1 - n πa 2α 0 -α 2 0 log(α 0 ) + 1 n log a . ( 94
)
Thus, we identify the function

f (σ) = 1 -8 σ π + aπ (1 -4σ) log a + o a , (95)
when non-overlapping absorbing holes are randomly distributed and

f (σ) = 1 -4 √ σ π + aπ log √ σ + o a , (96)
when absorbing holes are regularly distributed (equally spaced). We end this section with two remarks. First, for a coverage σ << 1, 8 σ π < 4

√ σ π and the MFPT of a Brownian particle to an absorbing hole is larger for randomly compared to regularly distributed (equally spaced) holes. This regular distribution shall be close to the arrangement of Fekete points on a sphere that minimizes the MFPT [START_REF] Cheviakov | Narrow-escape problem for the unit sphere: homogenization limit, optimal arrangements of large numbers of traps, and the n(2) conjecture[END_REF]. Second, formulas (95) and (96) derived here by accounting for two window coverage predict that the MFPT formula is different than previously reported based on an effective approximation where f (σ) = 1 -σ [START_REF] Zwanzig | Diffusion-controlled ligand binding to spheres partially covered by receptors: an effective medium treatment[END_REF] or on an interpolation procedure using Brownian simulations for which f (σ) = 1 -σ 1 + 3.8σ 1.25 [2]. This latter difference may arise from the differences in the windows' arrangement.

Comparison of asymptotic formula with respect to Brownian simulations

We compare our asymptotic formula for the probability (82) and conditional MFPT (83) with Monte-Carlo simulations when the number of holes n, their coverage of the nuclear surface σ = nπ 2 4πa 2 , or the distance r 0 between drift and cell centers vary. We model the cell as a ball of radius R and the nucleus as a centered sphere S a (radius a) uniformly covered by n small absorbing pores (radius ) (Figure 1 right). Stochastic particles are reflected on the external membrane r = R and on r = a except on all windows ∂N a = n i=1 ∂Ω i , centered at random locations (x i ) n i=1 . To explore how the number of holes and their coverage impact the probability and conditional MFPT, we first use a constant radial drift B directed toward the nucleus (with a potential Φ(r) = -Br). We consider a constant killing rate k(x) = k 0 and consequently, using function

G(D, B, a) = 4πe -Ba D |Ω| D B a 2 + 2 D B 2 a + 2 D B 3 (97)
expressions (86-87) simplify to

P = e -Ba D τ Φ=0,k=0 (G(D, B, a) -G(D, B, R)) k + e -Ba D , (98)
and

τ = τ Φ=0,k=0 (G(D, B, a) -G(D, B, R)) τ Φ=0,k=0 (G(D, B, a) -G(D, B, R)) k + e -Ba D . (99)
In Figure 2a, we compare these expressions to stochastic simulations for an increasing number of holes while maintaining constant the ratio σ = nπ 2 4πa 2 = 2% of the nucleus surface covered by the absorbing windows. This ratio was calibrated using a surface covered by 2, 000 pores of 25nm diameter on the nucleus of a chinese hamster ovary cell [START_REF] Maul | Quantitative determination of nuclear pore complexes in cycling cells with differing dna content[END_REF]) (parameters are summarized in table 1).

We evaluate the convergence of the Monte-Carlo simulations by computing the difference between two independent runs and stop the iteration when it is less than 5%. The converge is attained for N > 10, 000 runs with a time step dt < 0.001/ √ n. These values show that Monte-Carlo simulations become rapidly intractable as the number of absorbing holes increases, confirming the need of asymptotic formula to analyse the passage time to many small holes. We limited our simulations to the range 1 ≤ n ≤ 30. In this range, we observe that the asymptotic formula we derived match the simulation results. The remaining difference between the simulation results and the formula is probably due to higher order terms in asymptotic formula, that were recently discussed in [START_REF] Lindsay | First passage statistics for the capture of a brownian particle by a structured spherical target with multiple surface traps[END_REF]. Our analysis show that the conditional MFPT (respectively the probability) rapidly decreases (respectively increases) with the number of holes, the MFPT decreasing by 40% for n = 30 (MFPT = 160s) compared to n = 1 (MFPT = 270s) and the probability being more than doubled (55% for n = 30 compared to 26% for n = 1). We also compare Monte-Carlo simulations to previous formula [START_REF] Greber | A superhighway to virus infection[END_REF] that does not account for hole interactions (f (σ) = 0 in expression (95)), and observe a significant mismatch even for few (n > 5) absorbing holes.

In a second ensemble of simulations, we kept constant the number of absorbing holes (n = 10) and we vary the coverage by absorbing windows σ = n 2 a 2 from 2% to 50% (fig. 2b). We observe that formula (99) and (98) with the coverage function f (σ) given by expression (95) match simulations even for high coverage (σ ≤ 30%) before a small difference appears. We also plotted MFPT and the probability with no correction for holes' coverage (f (σ) = 1), and observe that they deviate significantly from simulations for σ ≥ 5%. We also studied the conditional MFPT and probability for off-center drift. This question is particularly relevant in modeling viral trafficking as the vector field for the drift is centered at the centrosome, which close to the nucleus. We kept n = 10 and σ = 2% and vary the distance r 0 between the domain and and the drift center from r 0 = 0 to r 0 = R = 20µm (Fig. 2c). The asymptotic formula (82) and (83) for the MFPT and the probability that a stochastic virus reaches an absorbing pore can be made more explicit by using a potential with an off-center constant drift B equals to

Φ(r 0 , r, φ) = B r 2 + r 2 0 -2rr 0 cos(φ). ( 100
)
Using this expression, the surface integral π 0 e -Φ(r 0 ,r,φ) D sin(φ)dφ and the volume integral R a π 0 e -Φ(r 0 ,r,φ) D r 2 sin(φ)dφdr in expressions ( 82) and ( 83) can be computed numerically.

These formula match the results of stochastic simulations in the entire interval of r 0 ∈ [0, 20µm]. The variance increases drastically with the distance r 0 . This high variability is due to the random configuration of the n = 10 holes that vary between simulations runs: if an absorbing window is located next to the drift center, the MFPT will be lower than the results obtained from an homogeneous distributed windows.

Finally, it has been hypothesized that large viruses characterized by small diffusion constant could be trapped to the centrosome and need to be propel actively to get back-transported toward the nucleus [START_REF] Burckhardt | Drifting motions of the adenovirus receptor car and immobile integrins initiate virus uncoating and membrane lytic protein exposure[END_REF]. We thus plotted here the MFPT and the probability that a large virus with decreased diffusion D = D/10 = 0.13µm 2 s -1 reaches a nuclear pore with respect to the distance r 0 between the centrosome and the nucleus. We found that the probability decreases drastically for r 0 ≥ 5, and falls to approximately 0 for r 0 ≥ 8µm suggesting that large viruses should use a different routing strategy, involving a combination of passive diffusion and retrograde transport to move from the centrosome toward the nucleus surface. We conclude that Monte-Carlo simulations have shown that the asymptotic formula derived here are valid in large range of parameters. Moreover, the geometry of the cell, the nuclear pores configuration and the viral motion affect the ability of a single virus to reach a nuclear pore to deliver its genome.

Conclusion

We extended here the asymptotic formula for the NET and the probability that a stochastic viral particle reaches a nuclear pores originally developed in [START_REF] Holcman | Modeling dna and virus trafficking in the cell cytoplasm[END_REF]. We also obtained a refinement of the classical Berg-Purcell formula for a Brownian particle to get absorbed on a partially reflected sphere.

The present approach applies to an intermittent dynamics with alternative periods of free diffusion and directed motion along MTs. This dynamics characterizes a large class of cellular transport. When the intermittent particle can be degraded (through the ubiquitin-proteasome machinery or trapped in the crowded cytoplasm), the asymptotical formula for the probability P and the mean time τ to reach a small absorbing target among n accounts for the geometrical interactions between the windows. When the targets co-localize on a small domain S a , the asymptotics of P and τ are derived in the limit |Sa| |Ω| 1. We confirmed the validity of asymptotics formula for the probability P and the mean time τ respectively using Brownian simulations. These formulas provide an estimate for the arrival time of DNA viruses to a small nuclear pore. Contrary to the classical narrow escape asymptotic where the leader order term contains most of the geometry, here the O(1)-term accounts for the interactions between windows, discussed in [START_REF] Pillay | An asymptotic analysos of the mean first passage time for narrow escape problems: Part i: Two-dimensional domains[END_REF][START_REF] Cheviakov | An asymptotic analysis of the mean first passage time for narrow escape problems: Part ii: The sphere[END_REF]. For viruses, where the number of nuclear pores is very high (n ≈ 2, 000, σ = 2% [START_REF] Maul | Quantitative determination of nuclear pore complexes in cycling cells with differing dna content[END_REF]) making Brownian simulations intractable, our formula predict that τ = 90 seconds and P = 74% which is close to the MFPT τ = 78 seconds and the probability P = 78% to reach a completely absorbing nucleus. Moreover, these estimates are far from the ones derived in [START_REF] Holcman | Modeling dna and virus trafficking in the cell cytoplasm[END_REF] with formula [START_REF] Greber | A superhighway to virus infection[END_REF] for a single big pore covering σ = 2% of the nucleus ( τ = 268 seconds and P = 26%). This shows that the fragmentation of absorbing targets in many smaller targets renders the overall absorption process much more efficient. Finally, MTs along which viruses are transported point towards the centrosome which is close to the nucleus leading to an off-center effective potential. Our formula shows that the distance of drift to cell center decreases the prob- 99)) and probability (equation (98)) formula (blue) is compared to stochastic simulations (black, bars = ±1 standard error) for an increasing number of absorbing windows, distributed randomly. The coverage ratio σ = nπ 2 4πa 2 of absorbing windows to nuclear surface is fixed to σ = 2% [START_REF] Maul | Quantitative determination of nuclear pore complexes in cycling cells with differing dna content[END_REF]. Plot of the asymptotics formula (9) (corresponding to f (σ) = 0 in equation ( 95)) that neglects windows interactions (red). b-Centered drift, number of pores n = 10 fixed. Same as in a, but now the number of random, absorbing pores is fixed to n = 10. We plot the formula without coverage correction (f (σ) = 1 in equation ( 95)). c-Non-centered drift, coverage σ = 2% and number of pores n = 10 fixed. We shifted the drift-center at a distance 0 ≤ r 0 ≤ R from the center and compare MFPT (83) and probability (82) as in a. The coverage ratio σ = 2% and the number of pores n = 10 are kept constant. The asymptotic formula for a smaller diffusion D = 0.13µm 2 s -1 = D/10 and a ratio B/D = 1.5µm -1 are shown (dashed blue). We used 10000 stochastic trajectories with a time step dt = 0.001/ √ n. Parameters are summarized in table 1.

ability that a virus reaches a nuclear pore before being degraded, and that large viruses with small diffusion coefficient are even more affected. Indeed, for a distance r 0 = 6µm just above the nucleus radius a = 5µm, our formula predicts that the viral probability P is decreased to 56%, and falls to 4.8% and 0.0004% when the diffusion coefficient is divided by 10 (D = 0.13µm 2 s -1 ) or 100 (D = 0.013µm 2 s -1 ). The latter small diffusion coefficient has been reported for many large viruses (table 1 in [START_REF] Sodeik | Mechanisms of viral transport in the cytoplasm[END_REF]), and other mechanisms such as retrograde transport of viruses from the centrosome to the nucleus might be involved in viral trafficking to optimize the delivery of their genome to the nucleus. It would certainly be valuable to extend the present work by considering the exact mechanism, where large molecules are being imported through nuclear pores. Such model should account for cargo molecules that transport large molecules though the pore, which is an energy costly process (ATPdependent). Such model should be further coarse-grained into a steady-state Robin boundary condition, instead of using the pure absorbing condition at the windows, as we did here. The constant in the Robin term should then be carefully derived from the model equations, see also [START_REF] Lindsay | First passage statistics for the capture of a brownian particle by a structured spherical target with multiple surface traps[END_REF].

Parameter Description Value D

Diffusion constant of the virus D = 1.3µm

Appendix

We derive in this appendix the asymptotic of the Neumann's function N (x i , x j ) for two absorbing patches of size ε located on the surface of a small ball of radius a. Expansion (48) is insufficient and we shall now derive a two parameter expansion in a and ε. We start by observing that the log-term -1 4πaD log

1 |x i -x j |
can be much larger than the leading order term 1 2πD|x i -x j | when |x i -x j | ≈ a, for a |Ω|

To compute the log-term, we first decompose Ñ (x, x 0 ) = 1 4π|x-x 0 | + Φ(x, x 0 ) where Φ is solution of the system: ∆Φ(x, x 0 ) = 0, for x ∈ R 3 ∂Φ ∂n (x, x 0 ) = -∂ ∂n

1 4π|x -x 0 | , for x ∈ S a . (103)
To solve equation ( 103), we choose a coordinate system for which the point source x = x 0 is on the positive z-axis. Since aΦ = 0 and Φ is axisymmetric, it has the series expansion Φ(x, x 0 ) =

that is

I (ρ) = 1 ρa 1 + |x 0 | 2 ρ 2 a 4
-2 |x 0 |ρ a 2 cos(θ)

1 2 . ( 113
)
Because lim ρ→∞ l(ρ) = 0, we have:

l(ρ) = - ∞ ρ I (

Figure 1 :

 1 Figure 1: Schematic representation of the cell cytoplasm as a 3-dimensional domain Ω. (Right): Stochastic trajectories, modeled by equation (2), with a diffusion and a drift terms. They can be absorbed at small windows with radius |Ω| 1/3 located on the surface of the nucleus ∂S a . (Left): simplified geometry: the cell geometry is a ball (radius R) containing a spherical nucleus surface, containing many small absorbing windows. The nucleus ball is of radius a such that a |Ω| 1/3 .

Figure 2 :

 2 Figure 2: Conditional MFPT τ and probability P for a virus to reach an absorbing nuclear pore. a-Centered drift, absorbing surface coverage ratio σ = 2%. For a centered drift, the MFPT (equation (99)) and probability (equation (98)) formula (blue) is compared to stochastic simulations (black, bars = ±1 standard error) for an increasing number of absorbing windows, distributed randomly. The coverage ratio σ = nπ 2 4πa 2 of absorbing windows to nuclear surface is fixed to σ = 2%[START_REF] Maul | Quantitative determination of nuclear pore complexes in cycling cells with differing dna content[END_REF]. Plot of the asymptotics formula (9) (corresponding to f (σ) = 0 in equation (95)) that neglects windows interactions (red). b-Centered drift, number of pores n = 10 fixed. Same as in a, but now the number of random, absorbing pores is fixed to n = 10. We plot the formula without coverage correction (f (σ) = 1 in equation (95)). c-Non-centered drift, coverage σ = 2% and number of pores n = 10 fixed. We shifted the drift-center at a distance 0 ≤ r 0 ≤ R from the center and compare MFPT (83) and probability (82) as in a. The coverage ratio σ = 2% and the number of pores n = 10 are kept constant. The asymptotic formula for a smaller diffusion D = 0.13µm 2 s -1 = D/10 and a ratio B/D = 1.5µm -1 are shown (dashed blue). We used 10000 stochastic trajectories with a time step dt = 0.001/ √ n. Parameters are summarized in table 1.

1 3 .

 3 Consequently, we shall reexamine the log-term expansion. For x and x 0 in the neighborhood of the sphere S a , we expand the Neumann function N (x, x 0 )N (x, x 0 ) = Ñ (x, x 0 ) + O(1),(101)where Ñ (x, x 0 ) is solution with D = 1 of the Laplace equation∆ Ñ (x, x 0 ) = -δ(x -x 0 ) for x ∈ R 3 ∂ Ñ ∂n (x, x 0 ) = 0 for x ∈ S a .

∞b

  n=0 n (|x 0 |) P n (cos(θ)) |x| n+1 , (104) where P n are the Legendre polynomials of integer n, θ is the angle between x and the north pole and b n (|x 0 |) are coefficients, determined from boundary condition (103). For x ∈ S a and ρ = |x|,∂Φ ∂n (x, x 0 ) = ∂Φ ∂ρ (ρ = a) = -∞ n=0 (n + 1) b n (|x 0 |) a n+2 P n (cos(θ). (105) For |x| < |x 0 |, we have the 0 | n+1 P n (cos (θ)) ,(106)which leads to the boundary condition:-∂ ∂ρ 1 4π|x -x 0 | (ρ = a) = -1 4π ∞ n=0 na n-1 |x 0 | n+1 P n (cos (θ)) .(107)Injecting relations (105-107) into the boundary condition (103), we obtain that for all n ≥ 0:b n (|x 0 |) = 1 4π na 2n+1 (n + 1)|x 0 | n+1 . (108)The Neumann function Ñ (x, x 0 ) is then given by: 1)|x| n+1 |x 0 | n+1 P n (cos(θ)), |x| n+1 |x 0 | n+1 -a 2n+1 (n + 1)|x| n+1 |x 0 | n+1 P n (cos(θ)). (110)Using expansion (106), we have for the first term of (110),1 4π ∞ n=0 a 2n+1 |x| n+1 |x 0 | n+1 P n (cos(θ)) = a 4π|x 0 ||x -a 2 x 0 |x 0 | 2 | .(111)To compute the second termI(ρ) = -∞ n=0 a 2n+1(n+1)ρ n+1 |x 0 | n+1 P n (cos(θ)), we differentiate I (ρ) = ∞ n=0 a 2n+1 ρ n+2 |x 0 | n+1 P n (cos(θ)) = a ρ|x 0 ||x -a 2 x 0 |x 0 | 2 | ,

s)ds = - ∞ ρ ds sa 1 + |x 0 | 2 s 2 a 4 -2 |x 0 |s a 2 |x 0 |ρ a 2 ( 1 - 2 2- 2 a 2 ( 1 - 1 -|x 0 ||x| a 2 cos(θ) + 1 + |x 0 ||x| a 2 2 -2 |x 0 ||x| a 2 cos

 14221222112122 cos(θ)) 1 -|x 0 |ρ a 2 cos(θ) + 1 + |x 0 |ρ a |x 0 |ρ a 2 cos(θ) obtain the expression of the Neumann function Ñ (x, x 0 ) and the exact dependency with respect to the inner ball of radius a:Ñ (x, x 0 ) = 1 4π|x -x 0 | + a 4πD|x 0 ||x -a 2 x 0 |x 0 | 2 | cos(θ))x and x 0 are on the sphere S a , |x 0 | = |x| = a, we haveÑ (x, x 0 ) = 1 2π|x -x 0 | + 1 4πa log |x -x 0 | 2a + |x -x 0 | .(117)

Table 1 :

 1 Numerical parameters used for Brownian simulations

	2 s -1
	for the Associated-Adeno-Virus [27]

Acknowledgments: T.L. is supported by a FRM post-doctoral fellowship and a grant from Philippe Foundation. D. H. research is supported by a Marie-Curie fellowship.