Spyridon Patmanidis
email: spatmanid@gmail.com

Alexandros C Charalampidis
email: alexandros.charalampidis@centralesupelec.fr

Ioannis Kordonis
email: jkordonis1920@yahoo.com

Georgios D Mitsis
email: georgios.mitsis@mcgill.ca

George P Papavassilopoulos

Tumor Growth Modeling: Parameter estimation with Maximum Likelihood methods

Keywords: Tumor Growth Modeling, Nonlinear Systems, Parameter Estimation, Maximum Likelihood, Extended Kalman Filter

Background & Objective: In this work, we focus on estimating the parameters of the widely used Gompertz tumor growth model, based on measurements of the tumor's volume. Being able to accurately describe the dynamics of tumor growth on an individual basis is very important both for growth prediction and designing personalized, optimal therapy schemes (e.g. when using model predictive control).

Methods: Our analysis aims to compute both the growth rate and the carrying capacity of the Gompertz function, along with the characteristics of the additive Gaussian process and measurement noise of the system. Three methods based on Maximum Likelihood estimation are proposed. The first utilizes an assumption regarding the measurement noise that simplifies the problem, the second combines the Extended Kalman Filter and Maximum Likelihood estimation, and the third is a nonstandard exact form of Maximum Likelihood estimation,

Introduction

Cancer is a disease that can affect most of the tissues in the human body. It is defined by uncontrolled cell growth and the potential to invade surrounding tissues. As an additional feature, many cancers, but not all, can also experience migration of tumor cells from the primary site to a distant one where they settle, a phenomenon called metastasis. Cancer also includes benign tumors that show uncontrolled cell growth but no invasion, and in most cases do not threaten the patient's life [START_REF] Barillot | Computational Systems Biology of Cancer[END_REF]. According to [START_REF] Stewart | World Cancer Report[END_REF], cancer is a major cause of morbidity and mortality, affecting populations in all countries and regions. The predicted global cancer burden is expected to exceed 20 million new cases annually by 2025, compared with the estimated 14.1 million cases worldwide in 2012. Treatment depends on the type of cancer and commonly consists of one or a combination of therapies, such as surgery, chemotherapy, radiotherapy, immunotherapy and hormone therapy.

Mathematical modeling combined with optimal control techniques have been proposed to improve the outcome of chemotherapy or radiation therapy [START_REF] Hadjiandreou | Mathematical modeling of tumor growth, drug-resistance, toxicity, and optimal therapy design[END_REF][START_REF] Dua | Optimal delivery of chemotherapeutic agents in cancer[END_REF][START_REF] Barbolosi | Optimizing drug regimens in cancer chemotherapy: a simulation study using a pk-pd model[END_REF][START_REF] Rockne | Advances in Radiation Oncology[END_REF] predict the success or failure of immunotherapy [START_REF] Babbs | Predicting success or failure of immunotherapy for cancer: insights from a clinically applicable mathematical model[END_REF], and illustrate the potential synergism between different treatment methods [START_REF] Serre | Mathematical modeling of cancer immunotherapy and its synergy with radiotherapy[END_REF]. Recent findings also suggest that mathematical modeling can provide useful clues about the impact of surgery on metastasis and may help to predict the risk of cancer spread [START_REF] Benzekry | Modeling spontaneous metastasis following surgery: An in vivo-in silico approach[END_REF]. An extensive literature review on the topic of how mathematical modeling can be applied to deliver better drug administration regimes is presented in [START_REF] Michor | Improving cancer treatment via mathematical modeling: Surmounting the challenges is worth the effort[END_REF].

Tumor growth modeling is a topic that has been studied extensively. The authors of [START_REF] Araujo | A history of the study of solid tumour growth: The contribution of mathematical modelling[END_REF] present a concise history of the study of solid tumor growth by reviewing some of the important mathematical contributions since the early decades of the twentieth century. More recent studies about mathematical models that are used to describe tumor growth can be found in [START_REF] Enderling | Mathematical modeling of tumor growth and treatment[END_REF][START_REF] Byrne | Modelling aspects of cancer dynamics: a review[END_REF][START_REF] Gerlee | The model muddle: In search of tumor growth laws[END_REF][START_REF] Benzekry | Classical mathematical models for description and prediction of experimental tumor growth[END_REF]. In the present study, the Gompertz model is utilized to describe tumor growth. The Gompertzian equation originates from the actuarial model developed by Benjamin Gompertz in 1825 [START_REF] Gompertz | On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies[END_REF], and has been also applied to the study of growth in biological and economic contexts [START_REF] Winsor | The gompertz curve as a growth curve[END_REF]. It has been shown that the Gompertz model can describe the growth of several tumor types [START_REF] Laird | Dynamics of tumour growth[END_REF]. This is also indicated by several recent studies where the Gompertz model is compared with other classical mathematical models (e.g. exponential, Von Bertalanffy, power law, etc) [START_REF] Benzekry | Classical mathematical models for description and prediction of experimental tumor growth[END_REF][START_REF] Hartung | Mathematical modeling of tumor growth and metastatic spreading: Validation in tumor-bearing mice[END_REF][START_REF] Sarapata | A comparison and catalog of intrinsic tumor growth models[END_REF]. Therefore, it is a frequent choice for modeling tumor growth in recent experimental studies [START_REF] Hadjiandreou | Mathematical modeling of tumor growth, drug-resistance, toxicity, and optimal therapy design[END_REF][START_REF] Dua | Optimal delivery of chemotherapeutic agents in cancer[END_REF][START_REF] Barbolosi | Optimizing drug regimens in cancer chemotherapy: a simulation study using a pk-pd model[END_REF], [START_REF] Achilleos | Multiprocess dynamic modeling of tumor evolution with bayesian tumorspecific predictions[END_REF][START_REF] Loizides | Model-based tumor growth dynamics and therapy response in a mouse model of de novo carcinogenesis[END_REF][START_REF] Chignola | Estimating the growth kinetics of experimental tumors from as few as two determinations of tumor size: implications for clinical oncology[END_REF].

According to the Gompertz model, the growth of a solid tumor is characterized by two main features, the growth rate and the carrying capacity. The growth rate refers to the time needed for the tumor cells to produce new cells, while the carrying capacity to the fact that a tumor cannot grow indefinitely.

When fitting experimental tumor data to the Gompertz equation, several studies [START_REF] Loizides | Model-based tumor growth dynamics and therapy response in a mouse model of de novo carcinogenesis[END_REF][START_REF] Chignola | Estimating the growth kinetics of experimental tumors from as few as two determinations of tumor size: implications for clinical oncology[END_REF][START_REF] Talkington | Estimating tumor growth rates in vivo[END_REF] have used the findings of Brunton and Wheldon [START_REF] Brunton | Prediction of the complete growth pattern of human multiple myeloma from restricted initial measurements[END_REF][START_REF] Brunton | Characteristic species dependent growth patterns of mammalian neoplasms[END_REF][START_REF] Brunton | The gompertz equation and the construction of tumour growth curves[END_REF], that the carrying capacity of a tumor is a species-specific parameter. However, the theoretical assumption that any given tumor will approach the same asymptotic value has not been proven experimentally.

Tumor growth modeling is inherently difficult as cancer is a complex bio-logical phenomenon. The fact that its presentation, development and outcome varies from one patient to the other necessitates the need of tumor growth modeling to be done on an individualized basis, in order to better understand the underlying biology and design better therapies. Towards this direction, nonstandard, problem specific techniques can provide very accurate models and improve the modeling procedure. An accurate patient-specific growth model combined with mathematical models that simulate the pharmacokinetic and pharmacodynamic processes of drugs and optimal control techniques, can help to predict and optimize the patient's response to various dose schedules and treatment combinations. Moreover, since clinical trials are limited due to their high cost, combining mathematical modeling with numerical simulations is a low cost and safe (since there is no immediate threat for the patient) way to evaluate different strategies.

In this study, it is assumed that both the growth rate and the carrying capacity are unknown (the proposed methods, could also be used if the carrying capacity is considered a species-specific parameter). Using the Gompertz equation, a stochastic state-space model with additive Gaussian process and measurement noise is created. Based on this stochastic model, we try to estimate the growth rate, the carrying capacity and the noise characteristics of tumors. Synthetic data are used because they provide the major advantage that the methods' ability to estimate the unknown parameters can be evaluated objectively. The methods we use to estimate the unknown parameters are based on Maximum Likelihood estimation [START_REF] Kay | Fundamentals Of Statistical Signal Processing[END_REF][START_REF] Ljung | System Identification: Theory for the User[END_REF][START_REF] Schön | System identification of nonlinear statespace models[END_REF]. According to Kay [START_REF] Kay | Fundamentals Of Statistical Signal Processing[END_REF], this technique is overwhelmingly the most popular approach to obtain practical estimators.

Additionally, the successful application of Maximum Likelihood estimation on problems within a biological or economic context where the estimation of the unknown parameters of stochastic Gompertz-type systems is required [START_REF] Garg | Maximum-likelihood estimation of the parameters of the gompertz survival function[END_REF][START_REF] Lambert | Modeling of nonlinear growth curve on series of correlated count data measured at unequally spaced times: A full likelihood based approach[END_REF][START_REF] Gutiérrez | Inference in gompertz-type nonhomogeneous stochastic systems by means of discrete sampling[END_REF],

suggests that it can also be successfully applied for the study of tumor growth.

The main focus of this work is to examine the cases where the characteristics of one or both the process and measurement noise are unknown. A prior work for the case where the characteristics of both process and measurement noise are known has already been published (see [START_REF] Patmanidis | Comparing methods for parameter estimation of the gompertz tumor growth model[END_REF]). For the sake of completeness, we also include this case here. Three methods are described in this manuscript.

The first will be referred to as Naive Maximum Likelihood (NaML). In this method, a simplification considering the measurement noise has been made in order to obtain the system's states and avoid solving time consuming recursive equations. The second method uses the Extended Kalman Filter (EKF) [START_REF] Charalampidis | Development and numerical investigation of new non-linear kalman filter variants[END_REF][START_REF] Charalampidis | Computationally efficient kalman filtering for a class of nonlinear systems[END_REF][START_REF] Maybeck | Stochastic Models, Estimation, and Control, Mathematics in science and engineering[END_REF],

to compute estimates of the system's states and Maximum Likelihood estimation to compute the parameters that maximize the joint density of the observations, and will be referred to as EKF-ML. The third one, which will be referred to as Numerical Maximum Likelihood (NuML), is the method that overall yielded the best results. This method uses numerical integration to compute the integrals that compose the joint density function. We also propose an effective way to reduce the interval of integration and thus the method's execution time, without any significant approximation error. Overall, we were able to achieve good model prediction and accurate parameter estimation.

In the following sections, we present the model used to describe the tumor growth (Section 2), analyze the proposed methods (Section 3.1) and the test cases (Section 3.2), describe the simulations and present their results (Section 4), as well as explain our findings (Section 5) and discuss future work (Section 6).

Tumor Growth Model

The Gompertz model has been widely used because of its simplicity and its ability to describe experimental data reasonably well. The discrete time state-space representation of the model is given by:

x k+1 = f (x k , θ 1 , θ 2 ) = θ 2 exp log x k θ 2 exp - 1 θ 1 T , (1)
where x k (mm 3 ) is the tumor size, θ 1 (days) is a constant related to the proliferative ability of the cells, θ 2 (mm 3 ) is the carrying capacity ( lim

k→∞ x k = θ 2 ),
T (days) is the time interval between k and k + 1, and k ∈ N [START_REF] Dennis | Density-dependent state-space model for population-abundance data with unequal time intervals[END_REF]. For simplicity reasons, throughout this work, a constant sampling period T is assumed.

Nonetheless, the results of the following sections can be applied to the case of nonconstant sampling intervals.

Assuming random additive process and measurement noise the model can be formulated as follows:

x k+1 = f (x k , θ 1 , θ 2 ) + w k , (2a)
y k = x k + υ k , (2b)
where y k is a measurement taken at time step k. The random variables w k and υ k , k ∈ N are mutually independent and normally distributed with zero mean and unknown standard deviation: w k ∼ N (0, θ 3 x θ4 k ) and υ k ∼ N (0, θ 5 x θ6 k ). The notation X ∼ N (µ, σ) is used in this manuscript to denote a normally distributed random variable X with mean µ and standard deviation σ. Measurements are available from time k=1 onwards and known x 0 , y 0 are assumed.

Because of the random components w k and υ k , the model of Eq. ( 2) also corresponds to the description:

x k+1 ∼ p θa (x k+1 |x k ), (3a)
y k ∼ p θb (y k |x k ), (3b)
where p θa (x k+1 |x k ) is the probability density function describing the dynamics for given values of x k and p θ b (y k |x k ) is the probability density function describing the measurements [START_REF] Schön | System identification of nonlinear statespace models[END_REF], [START_REF] Charalampidis | Development and numerical investigation of new non-linear kalman filter variants[END_REF]. In Eq. (3a), θ a = [θ 1 ; θ 2 ; θ 3 ; θ 4 ] T , where

θ a ∈ Θ a with Θ a ⊆ R 4 >0
denoting a compact set of permissible values of the unknown vector θ a and in Eq. (3b),

θ b = [θ 5 ; θ 6 ] T , where θ b ∈ Θ b with Θ b ⊆ R 2 >0
denoting a compact set of permissible values of the unknown vector θ b . We also define the vector θ = θ a × θ b , where θ ∈ Θ with Θ ⊆ R 6 >0

Methods

The problem addressed in this section is to obtain an estimate θ based on N measurements Y N = [y 1 , . . . , y n ]. However, as the value of the state variable x k is not directly measurable, Maximum Likelihood estimation cannot be applied directly to Eq. (2b) and Eq. (2b). To overcome this difficulty, in Section (3.1), we propose three methods. The first is a naive approach where, for the sake of simplicity, it is assumed that there is no measurement noise and thus the states x k equal the measurements y k . In the second approach, an Extended Kalman Filter is used to estimate the unknown states x k and then Maximum Likelihood 120 to estimate the system's parameters. Lastly, the third approach, decomposes the likelihood of the observations and approximates it by using one-dimensional recursive numerical integration.

Parameter Estimation Techniques

Naive Maximum Likelihood 125

In this method, because of the assumption that there is no measurement noise, Maximum Likelihood estimation can be applied directly, as if the measurements were taken from a system of the form:

x k+1 = f (x k , θ 1 , θ 2 ) + d k , (4a)
y k = x k , (4b)
where d k ∼ N (0, σ k ) and σ k = θ 3 x θ4 k + θ 5 x θ6 k . θ is the estimate of the vector θ that maximizes the joint density (likelihood) p θ (Y N ) of the observations:

θ = arg max θ∈Θ p θ (Y N ). (5)
Using Bayes' rule, the likelihood p θ (Y N ) can be decomposed according to

p θ (Y N ) = N k=1 p θ (y k |Y k-1 ) = N k=1 p θ (x k |x k-1 ), (6)
where x 0 is considered known and p θ (x k |x k-1 ) is coming from Eq. (4a).

Extended Kalman Filter -Maximum Likelihood

In this method, the EKF is used to provide an approximation xk of the state x k . In order for the filter to approximate the non-linearities of the system dynamics, a linearized version of the nonlinear system model around the last state estimate is created. Once the state estimates XN = [x 1 , . . . , xn ] are available, the maximum likelihood approach can be used to find an estimate θa of the vector θ a , that maximizes the joint density p θa ( XN ) of the state estimates: θa = arg max θa∈Θa p θa ( XN ), [START_REF] Babbs | Predicting success or failure of immunotherapy for cancer: insights from a clinically applicable mathematical model[END_REF] and an estimate θb of the vector θ b , that maximizes the joint density

p θ b (Y N )
of the measurements:

θb = arg max θ b ∈Θ b p θ b (Y N ). ( 8
)
If Bayes' rule is applied, the likelihoods p θa ( XN ) and p θ b (Y N ) can be decomposed according to

p θa ( XN ) = N k=1 p θa (x k |x k-1 ) (9)
and

p θ b (Y N ) = N k=2 p θ b (y k |x k ), (10)
where p θa (x k |x k-1 ) comes from Eq. (2a), p θ b (y k |x k ) from Eq. (2b) and x0 is considered known.

The EKF used in this work is an implementation of the classical EKF [START_REF] Charalampidis | Computationally efficient kalman filtering for a class of nonlinear systems[END_REF].

Moreover, it has been observed that when this procedure is used repeatedly, θ will converge to a better estimate than the initial one. The reason behind this behavior is that the vector θ is considered known for the EKF and the state estimation is based on this θ. However, when the state estimates are used to maximize Eq. ( 9) and ( 10) the updated θ is a better estimate of θ and when the EKF is used again with this new θ, the new states xk will be improved estimates of the real states x k , and so on. In the first iteration, the estimate θ that results from NaML is used for the EKF, and in the following iterations the latest estimate θ of Eq. ( 7) and ( 8) is used. This procedure stops when two consecutive estimates do not diverge over a specific threshold or after a preset number of iterations has been reached.

Numerical Maximum Likelihood

This method is applied to the model described in Eq. ( 2) and is used to find an estimate θ of the vector θ that maximizes the joint density p θ (Y N ) of the observation:

θ = arg max θ∈Θ p θ (Y N ). (11)
Bayes' rule can be used in order to decompose the joint density according to

p θ (Y N ) = p θ (y 1 ) N k=2 p θ (y k |Y k-1 ), (12)
where

p θ (y k+1 |Y k ) = p θ (y k+1 |x k+1 )p θ (x k+1 |Y k )dx k+1 , (13)
p θ (x k+1 |Y k ) = p θ (x k+1 |x k )p θ (x k |Y k )dx k , (14)
and

p θ (x k+1 |Y k+1 ) = p θ (y k+1 |x k+1 )p θ (x k+1 |Y k ) p θ (y k+1 |Y k ) . (15)
The distributions in Eq. ( 13) to ( 15) can be computed iteratively. Numerical integration can be used to compute the integrals of Eq. ( 13) and ( 14). The interval of integration is [0, X max ], where X max is the maximum volume a tumor can attain, and the definite integrals that come of, can be computed by using the trapezoidal rule:

b a f (x)dx ≈ (b -a) f (a) + f (b) 2 . ( 16
)
In Eq. ( 16), the smaller the interval [a, b] is, the smaller the approximation error will be. The integral defined in [0, X max ] can be divided into subintegrals, so that the total integral comes as the summation of the individual subintegrals.

145 However, due to the size of [0, X max ] and the number of subintegrals needed for an accurate approximation, such an approach is completely ineffective, because making a huge number of calculations at every iteration is prohibitive. An effective way to limit the interval of integration is of great importance, because it reduces the number of subintegrals needed for the approximation and thus 150 the execution time.

Using the Gaussian property of the noise, the interval of integration can be reduced in order to make the numerical integration by using the trapezoidal rule applicable. At every iteration step, if x k is considered a normally distributed random variable with mean µ = y k and standard deviation σ = σ 0 y e0 k

( x k ∼ N (µ, σ)), then [µ -5σ, µ + 5σ] [0,
X max ] can be used as the interval of integration, which is significantly smaller than [0, X max ]. In this manuscript, the maximum obtainable values of θ 5 and θ 6 are used as σ 0 and e 0 respectively.

The integral defined in the new interval can be divided in subintegrals, that can be approximated by applying the trapezoidal rule. As a result, the distributions of Eq. ( 13) to ( 15) can be approximated very accurately.

Test Cases

In this work, we present four test cases. In all cases, the system parameters are unknown. The test cases differ in terms of the available information about the process and measurement noise characteristics. Case 1 has been studied in [START_REF] Patmanidis | Comparing methods for parameter estimation of the gompertz tumor growth model[END_REF] and is included in order to be compared with the three new cases.

Case 1 -Known Noise Characteristics

In this case, the unknown parameters are the growth rate and the carrying capacity (θ 1 , θ 2 ), while the characteristics of the process (θ 3 , θ 4 ) and measurement noise (θ 5 , θ 6 ) are considered known. This is the simplest case, because only two parameters need to be estimated. Parameters θ 3 , θ 4 , θ 5 and θ 6 are set to their real values and the argument that maximizes the likelihood

p θ (Y n ) is θ = [ θ1 ; θ2 ] T .

Case 2 -Known Process Noise Characteristics

In this case, the unknown parameters are θ 1 , θ 2 , θ 5 and θ 6 , while θ 3 and θ 4 are considered known. Parameters θ 3 and θ 4 are set to their real values and the argument that maximizes the likelihood p θ (Y n ) is θ = [ θ1 ; θ2 ; θ5 ; θ6 ] T .

Case 3 -Known Measurement Noise Characteristics

In this case, the unknown parameters are θ 1 , θ 2 , θ 3 and θ 4 , while θ 5 and θ 6 are considered known. Parameters θ 5 and θ 6 are set to their real values and the argument that maximizes the likelihood p θ (Y n ) is θ = [ θ1 ; θ2 ; θ3 ; θ4 ] T .

Case 4 -Unknown Noise Characteristics

In this case, all parameters (θ 1 , θ 2 , θ 3 , θ 4 , θ 5 and θ 6 ) are unknown and the techniques are used as described previously in this section.

Implementation

The computational methods presented in this section, were implemented in MATLAB (ver. R2015a, The MathWorks Inc., Natick, IL) by writing the suitable functions. In order to maximize the likelihoods of Eq. ( 6), ( 9), [START_REF] Michor | Improving cancer treatment via mathematical modeling: Surmounting the challenges is worth the effort[END_REF] and ( 12) the function fmincon [START_REF]Matlab Optimization Toolbox User's Guide[END_REF] was used to find the argument that minimizes the negative logarithm of the likelihood p θ (Data):

θ = arg min θ∈Θ -log p θ (Data) , (17)
where Data stands in for Y N or XN depending on the applied method and the parameters that need to be estimated. Maximizing the log-likelihood is equivalent since the logarithm is a monotonic function. It has also been proved numerically using the definition of convexity (e.g. [START_REF] Bertsekas | Nonlinear Programming[END_REF]), that the likelihood functions of this manuscript are generally not convex. In order to reduce the possibility of getting stuck to a local minimum, we ran fmincon for many different random initial conditions. The simulations were performed using an Intel Core i7-6700K @ 4.00GHz and 16GB of DDR4 @ 3200 MHz.

Simulations & Results

In this section, we present the results from the simulations in MATLAB.

The growth of 100 cancer tumors was simulated using Eq. (2a) and (2b). Each tumor's growth rate θ j 1 and its carrying capacity θ j 2 , as well as the process and measurement noise characteristics θ j 3 , θ j 4 , θ j 5 , θ j 6 (j = 1, ..., 100 is the identification 11 number of the test subject) were chosen randomly from uniform distributions.

The minimum and maximum values for the parameters (see Table 1) were chosen so that the obtained dataset appears as realistic tumor measurements. The sampling time between two consecutive measurements is T = 2 (days) and the number of available measurements is N = 30.

MAPE = 1 j j i=1 APE i (%), (18)
and the standard deviation by:

STD = 1 j -1 j i=1 |APE i -MAPE| 2 1/2 . ( 19
)
In Eq. ( 18) and ( 19)

APE i = A i -E i A i , (20)
where A i is the actual value of the parameter, E i is the estimated value and j(= 100) is the number of the tumors.

As mentioned in Section 3.3, regarding the minimization of Eq. ( 17), fmincon has been executed for many different random initial conditions. For all the random initial condition, the difference between the parameter estimates is very small, and especially for parameters θ 1 and θ 2 is practically negligible. Fig. 4 shows the tumor growth progression for two randomly chosen test subjects, and the ability of Gompertz curves to fit the data. These curves are created at different time-steps, meaning that the number of available measurements differs. The unknown parameters were estimated with the NuML, which is the method that yielded the best results. However, both NaML and EKF-ML yield similar results. The red curve is created by the parameters estimated from all the measurement. The parameters used for the green curve are reestimated for every new measurement (practically, the one-step ahead prediction of the growth). For the blue curve, the estimate is obtained by the three initial measurements. Likewise, for the magenta, the cyan and the black curve, the measurements until the 50%, 75% and 100% of the carrying capacity respectively are used in order to obtain an estimate for the unknown parameters and predict future growth.

Discussion

Despite the large number of studies in tumor growth, the widely used theoretical assumption that the carrying capacity is a species-specific parameter has not been proved experimentally. On the other hand, even though there is strong evidence that tumor growth is subject-specific since trying to estimate the carrying capacity may lead to unrealistic results, it is preferred to keep a fixed value for this parameter [START_REF] Loizides | Model-based tumor growth dynamics and therapy response in a mouse model of de novo carcinogenesis[END_REF]. In this manuscript, our simulation results

show that the methods proposed in Section 3.1 can estimate accurately both the growth rate and the carrying capacity. Realistic estimates for the carrying capacity are obtained even when the number of available measurements is very small.

Four different cases have been studied depending on the available information considering the noise. The metrics we used to evaluate the methods' performance regarding the estimation of the growth dynamics on these cases are presented in Fig. 1. The most important test case is the fourth, where both the process and measurement noise characteristics are unknown. This case corresponds to the problem needed to be solved when dealing with experimental data. Even when the noise type is known, it is still difficult to know the characteristics of the noise. The other three cases were studied in order to test if knowing the exact values for the noise characteristics could lead in a more accurate estimation of the growth dynamics. However, for all three methods, the results show that most times, knowing the exact values of the noise characteristics does not considerably improve the estimation of the growth rate and carrying capacity. On the contrary, considering the noise characteristics unknown seems to give flexibility in finding estimates for the noise characteristics that describe the noise realization better than the real values, which leads to more accurate estimates for the growth parameters. Eq. ( 17) seems to be more sensitive to parameters θ 1 and θ 2 and can be further minimized by choosing parameter values for θ 3 , θ 4 , θ 5 and θ 6 , that differ from the real parameters. As a result, the fact that the noise characteristics cannot be estimated accurately does not affect the methods' ability to provide accurate estimates for the growth rate and the carrying capacity.

In this work, three methods have been used to estimate the unknown parameters of the model described by Eq. (2a) and (2b). The simulation results are shown in Fig. 2. Overall, NuML performs better than the other two methods regarding the estimation of growth rate and carrying capacity. As regards the noise characteristics, there was no method that could estimate all the noise pa-rameters consistently better than the others. In Fig. 2c, we can see that NuML performs better in estimating parameters θ 3 and θ 6 , while EKF-ML is better at estimating θ 4 and θ 5 .

Another important finding is that as new measurements become available, the estimates of growth rate and carrying capacity converge to a value that is better than the previous values, but this does not happen for the estimates of the noise characteristics. Of course, after the carrying capacity has been reached, the new measurements do not provide much information about the growth rate, so the growth rate estimates do not improve significantly after that point on. In the same manner, the initial measurements do not provide a lot of information about the carrying capacity. As we can see, the estimation of carrying capacity improves a lot after the 75% of the tumor's maximum size has been reached, and when the tumor has fully grown it can be estimated very accurately. However, even though the initial estimates of the carrying capacity are not very accurate, they still provide realistic values for this parameter.

NaML is not as accurate as NuML and EKF-ML in estimating parameters θ 1 and θ 2 . This is due to the simplification concerning the way noise is added to the system. However, this method is very useful because the estimates are reasonably good, and most importantly, the execution time is extremely low (less than 0.15 secs when 30 measurements are available). The NaML estimates can be used as initial conditions for the other methods, as it has been observed that having a good initial guess instead of random one reduces the execution time of NuML and the number of iterations needed for the EKF-ML to converge. Especially for NuML, there are cases where the execution time is reduced by half, when the NaML estimates are used. The EKF-ML method seems to perform better than NaML in estimating the growth rate and carrying capacity. The execution time of this methods is low (about 1.75 secs when 30 measurements are available), but not as low as the execution time of NaML. The NuML method gives the best estimates for the parameters θ 1 and θ 2 compared to NaML and EKF-ML. Even though all three methods can estimate the growth of a tumor very well, the estimates from NuML create curves that fit better to the mea-sured data. Moreover, the one-step ahead prediction performance has promising potential. However, the execution time of NuML is very high (about 125 secs when 30 measurements are available) compared to the ones from NaML and EKF-ML due to the recursion and the computationally demanding approximations of the integrals, but yet far away from being prohibitive. Fig. 5 shows the mean execution time of EKF-ML and NuML at every iteration step. In our experiments, NuML improved the estimation accuracy of θ 1 and θ 2 by 0.3 to 2.3% and 0.3 to 1.75% respectively compared to EKF-ML. Even though the previous numbers imply that the improvement is small, waiting for a couple of minutes in order to obtain a better estimate should not discourage the use of NuML.

Cancer is a complex biological phenomenon that is affected by various factors and occurs on many levels [START_REF] Hanahan | Hallmarks of cancer: The next generation[END_REF]. The lack of appropriate experimental data makes infeasible to fully validate dynamic cancer models, which would be the most principled approach for capturing tumor growth [START_REF] Loizides | Model-based tumor growth dynamics and therapy response in a mouse model of de novo carcinogenesis[END_REF]. Any attempt to model cancer progression has to rely on incomplete information. Due to the fact that no such thing as a universal model for all cancers exists and since all models are bound by limitations, the model selection is usually based on the type of data that are used. In this study, the Gompertz model is utilized to describe the tumor growth. The main criticism of this model is that the relative tumor growth rate becomes arbitrary large for small tumor volumes [START_REF] Benzekry | Classical mathematical models for description and prediction of experimental tumor growth[END_REF]. Additionally, it does not model any biological mechanism that underly cancer, which means that it does not provide any insight about the biological procedures. It has been shown that it provides best fit for breast and lung cancer [START_REF] Benzekry | Classical mathematical models for description and prediction of experimental tumor growth[END_REF] and is very popular for large animal tumor datasets, but it has not been shown that it can model accurately all types of cancer. However, it is a simple model that is suitable for macroscopic tumor growth modeling. The fact that it can be used in optimal control algorithms during treatment optimization makes this model very popular. It has been shown to be appropriate for the prediction of the average growth behavior of a tumor and that it fits tumor growth datasets with accuracy [START_REF] Benzekry | Classical mathematical models for description and prediction of experimental tumor growth[END_REF][START_REF] Laird | Dynamics of tumour growth[END_REF][START_REF] Wheldon | Mathematical models in cancer research[END_REF][START_REF] Norton | Predicting the course of gompertzian growth[END_REF]. Moreover, the experimental data that we plan to work with in our future work and on which the synthetic data used here are based are of macroscopic tumor size. In the majority of studies on this type of data the Gompertz model is used. Thus, we consider that our choice to use this model is well justified. More information about the advantages and limitations of some of the most popular tumor growth models and the possible reasons why the Gompertz model ended up being the most widely applied one, can be found in the work of P. Gerlee [START_REF] Gerlee | The model muddle: In search of tumor growth laws[END_REF]. Nevertheless, the model's selection does not restrict us from utilizing other models in the future if needed, as the methods we propose are not limited by the model's choice.

The evidence that cancer is a subject-specific disease [START_REF] Loizides | Model-based tumor growth dynamics and therapy response in a mouse model of de novo carcinogenesis[END_REF] along with observations that the clinical response of individual patients to therapy still remains uncertain [START_REF] Volm | Prediction of cancer drug resistance and implications for personalized medicine[END_REF]

Conclusion

In this work, we developed three methods in order to estimate the unknown parameters of the Gompertz function and the characteristics of the Gaussian process and measurement noise. Synthetic data were used to test the methods' effectiveness and accuracy. It has been found that tumor growth can be modeled adequately, even when the noise characteristics cannot be estimated accurately. Simulation results show that, overall, NuML performs better compared to NaML and EKF-ML. NuML can also provide realistic estimates for the growth dynamics, even when a few measurements are available.

During our work with the synthetic data, we observed that applying nonstandard techniques provided improved estimates and better fitting curves compared to classical estimation techniques (e.g. Least Squares). This work constitutes an important step towards our future research, the application and testing of the methods we propose on experimental cancer data. Even though these techniques are computationally demanding, nowadays the available computational power is more than enough. So, since even the slightest improvement in tumor growth prediction could be really important, we believe that trading speed off for accuracy is completely justified.

Fig. 1 presents

 1 Fig. 1 presents MAPE and the standard deviation of the APE for the growth rate and the carrying capacity at each iteration step (each time a new measure-

4 (Fig. 1 .

 41 Fig.1. MAPE and STD of parameters θ 1 and θ 2 for the NuML method. 1a and 1b show the MAPE for θ 1 and θ 2 respectively, while 1c shows the STD of θ 1 and θ 2 .

Fig. 2 . 1 and θ 2

 212 Fig.2. MAPE of the unknown parameters for test case 4. 2a and 2b show the MAPE of θ 1 and θ 2 respectively. 2c shows the MAPE of θ 3 , θ 4 , θ 5 and θ 6 . In 2c, the line representation refers to the NuML method, the line-asterisk to the NaML method and the line-square to the EKF-ML method.

Fig. 3 .

 3 Fig.3. STD of the unknown parameters for test case 4. 3a and 3b show the STD of θ 1 and θ 2 respectively. 3c shows the STD of θ 3 , θ 4 , θ 5 and θ 6 . In 3c, the line representation refers to the NuML method, the line-asterisk to the NaML method and the line-square to the EKF-ML method.

Tumor 2 Fig. 4 .

 24 Fig. 4. Growth curves for random test subjects. In each subfigure, the x marks represents y k while the colored curves describe the tumor growth estimation based on the available measurements. The unknown parameters have been estimated with the NuML.

27

 27

Fig. 5 .

 5 Fig. 5. Execution time for EKF-ML and NuML.

Table 1

 1 Minimum and maximum values for each parameter

	Parameters θ 1	θ 2	θ 3 θ 4 θ 5 θ 6
	Minimum	6 320 1 0.3 2 0.2
	Maximum 16 800 4 0.5 6 0.5

In order to evaluate the performance of the methods described in Section 3.1 and applied to the cases of Section 3.2, the Mean Absolute Percentage Error (MAPE) and the standard deviation of the Absolute Percentage Errors (APE) are used. The MAPE is given by:

Acknowledgment Spyridon Patmanidis wishes to thank the Special Account for Research of the National Technical University of Athens for supporting his doctoral research through a scholarship of excellence. The work of Alexandros Charalampidis has been supported by the European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 705982.

Conflict of interest

The authors of this manuscript declare that they have no financial and personal relationships with other people or organizations that could inappropriately influence their work.


Thibault Courtheoux

Alghassimou Diallo

Arun Prasath Damodaran

David Reboutier

Erwan Watrin

Claude Prigent
email: claude.prigent@univ-rennes1.fr

Aurora A kinase activity is required to maintain an active spindle assembly checkpoint during prometaphase

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Aurora kinases are key regulators of mitosis that fulfil complementary functions, as suggested by their localisation on mitotic structures; Aurora A localises at the centrosome and spindle poles, whereas Aurora-B and Aurora-C localise on chromosomes and at the midbody [START_REF] Giet | Aurora kinases, aneuploidy and cancer, a coincidence or a real link?[END_REF]. Aurora-B and Aurora-C belong to the chromosome passenger complex (CPC) (Carmena et al., 2012b); their kinase activities are required for chromosome condensation by phosphorylating histones [START_REF] Hsu | Mitotic phosphorylation of histone H3 is governed by Ipl1/aurora kinase and Glc7/PP1 phosphatase in budding yeast and nematodes[END_REF][START_REF] Wilkins | A cascade of histone modifications induces chromatin condensation in mitosis[END_REF], for the spindle assembly checkpoint (SAC) by phosphorylating and regulating Zwint-1 [START_REF] Andrews | Aurora B regulates MCAK at the mitotic centromere[END_REF][START_REF] Kasuboski | Zwint-1 is a novel Aurora B substrate required for the assembly of a dynein-binding platform on kinetochores[END_REF], for cytokinesis by phosphorylating the central spindlin component [START_REF] Guse | Phosphorylation of ZEN-4/MKLP1 by aurora B regulates completion of cytokinesis[END_REF], and for the abscission checkpoint [START_REF] Mathieu | Aurora B and cyclin B have opposite effects on the timing of cytokinesis abscission in Drosophila germ cells and in vertebrate somatic cells[END_REF][START_REF] Norden | The NoCut pathway links completion of cytokinesis to spindle midzone function to prevent chromosome breakage[END_REF]. Aurora A is involved in G2/M transition [START_REF] Dutertre | Phosphorylation of CDC25B by Aurora-A at the centrosome contributes to the G2-M transition[END_REF][START_REF] Seki | Bora and the kinase Aurora a cooperatively activate the kinase Plk1 and control mitotic entry[END_REF], centrosome maturation [START_REF] Hannak | Aurora-A kinase is required for centrosome maturation in Caenorhabditis elegans[END_REF], and spindle assembly during prometaphase [START_REF] Roghi | The Xenopus protein kinase pEg2 associates with the centrosome in a cell cycle-dependent manner, binds to the spindle microtubules and is involved in bipolar mitotic spindle assembly[END_REF]. The SAC is a surveillance mechanism that monitors the attachment of kinetochores to microtubules during the process of bipolar spindle assembly [START_REF] Rieder | Stuck in division or passing through: what happens when cells cannot satisfy the spindle assembly checkpoint[END_REF]. Every mitotic chromosome made of two sister chromatids possesses one kinetochore per chromatid. Once every chromosome kinetochores are attached to microtubules, the SAC is switched off, chromosome segregation occurs and the cell enters anaphase [START_REF] Foley | Microtubule attachment and spindle assembly checkpoint signalling at the kinetochore[END_REF]. The SAC components are localised at the kinetochores, a macromolecular structure organised in different layers of protein complexes. Starting at the level of the centromeric chromatin, the histone H3 variant centromere protein A (CENP-A) defines the localisation of the constitutive centromere-associated network (CCAN), which is composed of 16 CENPs. This complex provides a platform on which to build an interface between the kinetochore, the microtubules and the KMN network, which is made up of Knl1, Mis12, and Ndc80.

At the molecular level, the SAC maintains the ubiquitin ligase Anaphase Promoting Complex/Cyclosome (APC/C) in an inactive state through sequestration of its activator Cdc20 by the protein Mad2 [START_REF] Nilsson | The APC/C maintains the spindle assembly checkpoint by targeting Cdc20 for destruction[END_REF]. The MCC (Mitotic Checkpoint Complex) responsible for the SAC signal in human cells is composed of the four proteins Mad2, BubR1, Bub3, Cdc20. Mad2 localises to unattached kinetochores and is released upon attachment by microtubules [START_REF] Buffin | Recruitment of Mad2 to the kinetochore requires the Rod/Zw10 complex[END_REF][START_REF] Chen | Association of spindle assembly checkpoint component XMAD2 with unattached kinetochores[END_REF]. When all kinetochores are attached to microtubules, the inhibition of Cdc20 by Mad2 is relieved, allowing Cdc20 to bind to and activate the APC/C, which in turn ubiquitinates cylin B and securin for their degradation by the proteasome. Without securin, the separase becomes active and cleaves the cohesin subunit Scc1/Rad21/Mcd1, triggering sister chromatid separation [START_REF] Uhlmann | Sister-chromatid separation at anaphase onset is promoted by cleavage of the cohesin subunit Scc1[END_REF]. APC/C also targets cyclin B1 for degradation, which induces the inhibition of CDK1 activity and allows the cell to exit mitosis [START_REF] Thornton | Securin and B-cyclin/CDK are the only essential targets of the APC[END_REF]. BubR1 and Mad2 are present on unattached kinetochores and participates in SAC activity [START_REF] Fang | Checkpoint protein BubR1 acts synergistically with Mad2 to inhibit anaphase-promoting complex[END_REF]. Although the regulation of the SAC has been attributed to Aurora-B rather than Aurora A, data suggest that Aurora A might also participate in SAC regulation. For example, phosphorylation of the centromere histone H3 variant CENP-A on serine 7 by Aurora A is required to localise Aurora-B at the kinetochore [START_REF] Kunitoku | CENP-A phosphorylation by Aurora-A in prophase is required for enrichment of Aurora-B at inner centromeres and for kinetochore function[END_REF]. CENP-A is considered as a platform on which to build the kinetochore where the mitotic checkpoint complex (MCC) is assembled. Unfortunately, inhibition of Aurora A kinase leads to defects in spindle assembly, which hinders any study of the role of the kinase in regulating the SAC [START_REF] Hoar | MLN8054, a small-molecule inhibitor of Aurora A, causes spindle pole and chromosome congression defects leading to aneuploidy[END_REF]. To overcome this problem, we used two independent approaches that allow Aurora A inhibition in a timely and precise manner (analogsensitive inhibition and specific inhibitor). Using analog-sensitive inhibition method, we previously showed that the kinase activity of Aurora A is required for central spindle assembly through the phosphorylation of dynactin subunit p150Glued [START_REF] Reboutier | Aurora A is involved in central spindle assembly through phosphorylation of Ser 19 in P150Glued[END_REF]. Here, we report that, in the absence of Aurora A activity, cells exhibit defective chromosome Journal of Cell Science • Accepted manuscript congression, premature entry into anaphase and delocalisation of Mad2 from kinetochores to centrosomes, demonstrating for the first time that Aurora A activity is required to maintain an active SAC during prometaphase.

Method

Cell culture

For analog-sensitive inhibition, three U20S cell lines were used: a normal U2OS cell line as control and two stable cell lines expressing a wild-type version of Aurora A tagged with GFP (wt-Aurora A) and an allele-sensitive version of Aurora A tagged with GFP (as-Aurora A).

Both versions of Aurora A were expressed under the control of the Aurora A endogenous promoter [START_REF] Reboutier | Aurora A is involved in central spindle assembly through phosphorylation of Ser 19 in P150Glued[END_REF]. These cell lines were maintained in McCoy's 5A medium containing GluthaMAX (Gibco, Invitrogen) supplemented with 10% FBS (PAA) and 1% penicillin-streptomycin (Gibco, Invitrogen). In addition, the wt-Aurora A and as-Aurora A cells were grown in the presence of 1.25 mg/ml geneticin (G418 sulphate, PAA).

Hela tubulin-GFP and U2OS Aurora A-GFP under endogenous promotor were grown in a humidified incubator at 37°C and 5% CO 2 in Dulbecco's modified Eagle medium (Gibco, Invitrogen, Carlsbad, USA) supplemented with 10% fetal calf serum (PAA Laboratories) and 1% penicillin/streptomycin (Gibco).

Small interfering RNA and transient transfections

The oligonucleotide sequence of the siRNA targeting Aurora A (Qiagen) was 5'-AAATGCCCTGTCTTACTGTCA -3' [START_REF] Reboutier | Aurora A is involved in central spindle assembly through phosphorylation of Ser 19 in P150Glued[END_REF]. Transfection was performed using JetPRIME (Polyplus transfection). Depletion of endogenous Aurora A kinase was controlled by Western blot in every experiment as described by [START_REF] Reboutier | Aurora A is involved in central spindle assembly through phosphorylation of Ser 19 in P150Glued[END_REF].

Western blot

Cells were harvested by treatment with trypsin-EDTA (Gibco, Invitrogen) and resuspended in McCoy's 5A medium containing GluthaMAX, centrifuged at 1200 rpm at 4°C for 3 min, and washed three times in PBS. The cell pellet was resuspended in Laemmli buffer, sonicated, and incubated for 5 min at 95°C. Proteins in the cell extracts were separated on 12.5% SDS-PAGE and transferred to a nitrocellulose membrane. After transfer, the membrane was blocked with 3% non-fat milk in TBST for 1 h at room temperature followed by an overnight incubation in TBST containing 3% non-fat milk and primary antibodies. The primary antibodies were mouse anti-Aurora A 5C3 (1/100) (Cremet et al. 2003); rabbit anti-Mad2 (1/1000, Covance); rabbit anti-HCAP-D2 (1/10000) (Collas et al. 1999); mouse anti-Cyclin B1 (1/1000, Santa Cruz); mouse anti-HA (1/1000, Covance); mouse anti-β-tubulin (1/2000, Sigma-Aldrich); rabbit anti Phospho-Aurora A/B/C (1/1000, Cell Signaling); rabbit antiphospho-histone H3 (H3S10, 1/1000, Euromedex). The secondary antibodies were anti-mouse or anti-rabbit IgG coupled to peroxidase (1/5000 and 1/110,000, Jackson Laboratories). Finally, the membranes were processed for chemiluminescent enhancement with Dura or Pico (Thermo Fisher Scientific) before film exposure.

Cell synchronisation

MLN8237 approach: The cells were grown on coverslips in 12-well plates. Cells were then synchronized in late G2 (G2/M transition) by treatment with 100 nM nocodazole for 12 hours at 37°C. Analog sensitive approach: The cells were grown on coverslips in 12-well plates. They were depleted of endogenous Aurora A using siRNA as described by [START_REF] Reboutier | Nucleophosmin/B23 activates Aurora A at the centrosome through phosphorylation of serine 89[END_REF]. Cells were then synchronised in late G2 (G2/M transition) by treatment with 2 M CDK1 inhibitor RO3306 (Calbiochem) at 37°C for 5 h. Next, they were released in late prometaphase by Journal of Cell Science • Accepted manuscript washing out RO3306 three times for 4 min. Finally, the cells were treated with 10 M 1-Na-PP1 to inhibit as-Aurora A, fixed for 10 min at either -20°C in 100% cold methanol or at room temperature in 4% paraformaldehyde, and processed for immuno-fluorescence microscopy.

Immunofluorescence

MLN8237 approach: Cells fixed in -20°C methanol for 5 min and treated as described in [START_REF] Courthéoux | Cortical dynamics during cell motility are regulated by CRL3(KLHL21) E3 ubiquitin ligase[END_REF]. For GFP stable cell line, GFP nano-booster (chromotek) was used at the secondary antibody incubation step. Images were taken with the new LSM800 Airyscan (Zeiss Inc. and processed with FIJI (imageJ). For figures 3, 4 and 5, the Fast-Airyscan mode was used with optimum parameter settings. Lasers power and acquisition settings are the same for all conditions within an experiment. Enlarged pictures in Figures 3C and5A are z-projection of 5 z stacks.

Analog sensitive approach: Cells fixed in methanol or paraformaldehyde as described above were washed with PBST buffer (PBS containing 0.1% Triton X100) and permeabilised by incubation in MBS buffer (100 mM PIPES pH 6.8, 1 mM MgCl2, 0.1 mM CaCl2, 0.1% Triton X100). The cells were then incubated in PBST containing 1% BSA for 1 h at room temperature, followed by 1 h incubation at room temperature in PBST containing 1% BSA and primary antibodies. After several washes in PBST containing 1% BSA, the cells were incubated with secondary antibodies. Images were taken using a Coolsnap ES (Photometrics) equipped Leica DMRXA2 microscope and image acquisition software MetaVue (Molecular Devices, Inc.). The images were deconvolved by the Metamorph software (Molecular Devices, Inc.) and edited using Adobe Photoshop CS3.

Journal of Cell Science • Accepted manuscript

Antibodies

We used antibodies directed against: mouse anti--tubulin B-5-1-2 (1/2000, Sigma); rat anti--tubulin (YL1/2 1/500, Millipore), rabbit anti-Mad2 (1/500, Covance), human anti-CREST centromere protein (1/3000, Antibodies Incorporated); mouse anti-Aurora B (1/1000, BD).

Secondary antibodies: rabbit and mouse CF TM (1/1000, Biotium). Alexa Fluor 488 donkey antirat IgG (1/1000), Alexa Fluor 555 donkey anti-mouse IgG (1/1000), or Alexa Fluor 555 goat anti-rabbit IgG (1/1000) (Invitrogen). Finally, samples were mounted with Vectashield or Vectashield-DAPI (1/1000, Vector Laboratories) for DNA staining.

Quantification of fluorescence signals

MLN8237 approach: For quantitative analysis in Fig. 3C and Fig. 4C of Aurora A and Mad2 intensities at kinetochore, kinetochore volume was isolated using "3D object counter" from FIJI on thresholded CREST z-stack images. Kinetochore volumes were imported in "3D ROI manager". Average intensity per kinetochore were collected and normalized to the maximum mean from NOCO (for Mad2) or MLN condition (respectively showing the highest Mad2 or Aurora A mean intensity per kinetochore). Aurora A and Mad2 kinetochore values were pooled per condition and showed as box plots.

Analog sensitive approach: The integrated densities of Mad2 (equatorial plates and bipolar spindle poles) were determined from deconvolved images taken with the Leica DMRXA2 (63°objective) using the FIJI (ImageJ) 1.46i software (NIH).

Journal of Cell Science • Accepted manuscript

Live cell imaging

MLN8237 approach: HeLa cell line expressing Tubulin-GFP [START_REF] Reboutier | Aurora A is involved in central spindle assembly through phosphorylation of Ser 19 in P150Glued[END_REF] was grown on Zeiss High resolution cover sleeps for 24h then synchronized as describe in Fig. 1A.

Video microscopy was performed on Spinning disk CSU-X1 on Nikon Ti-E. Seven z-stack (3 m step) were collected every 3 min using 63X magnification objective. Z-stack were zprojected and movie were analysed using FIJI. Anaphase was detected manually (as 00:30 in control condition Fig. 1B). Timing was collected for experiments, pooled and presented as box plot (Fig. 1C).

Analog sensitive approach: Endogenous Aurora A was depleted from both stable U2OS cell lines expressing wt-Aurora A and as-Aurora A using siRNA as described by [START_REF] Reboutier | Aurora A is involved in central spindle assembly through phosphorylation of Ser 19 in P150Glued[END_REF]. The cells were then incubated in 100 ng/ml (0.2 M) nocodazole or 5 M taxol until the end of the time lapse (16 h). After 13 h incubation in nocodazole, the cells were treated with 10 M 1-Na-PP1 and filmed for 3 h. The addition of 1-Na-PP1 marks T0 for each movie. Video microscopy was performed on a Leica DM IRB microscope equipped with a 63x/1.4 oilobjective and a Coolsnap HQ camera (Photometrics) driven by Metamorph software (Molecular Devices, Inc.). Images were acquired every 5 min and analysed using the same software.

Statistical analysis

All experiments have been performed at least three independent times and T-test on the mean was performed using Igor (Wavemetrics).

Journal of Cell Science • Accepted manuscript

Other statistical analyses were performed using Tukey's Honestly Significant Difference (HSD) test in R 2.13.0 software (R Core Group). Values were represented as histograms or box plots.

Results

Aurora A inhibition causes premature exit from mitosis of cells arrested in pro-metaphase due to SAC activation.

We previously developed an allele-sensitive Aurora A (as-Aurora A), of which the activity can be inhibited by an ATP analogue [START_REF] Reboutier | Aurora A is involved in central spindle assembly through phosphorylation of Ser 19 in P150Glued[END_REF], allowing specific inhibition of Aurora A in a very narrow window of time during cell cycle progression. We took advantage of this system to determine whether Aurora A activity plays a role in the SAC. We used a cell line expressing a GFP-tagged wild-type version of Aurora A (WT-U2OS) and another cell line expressing a GFP-tagged allele-sensitive Aurora A (AS-U2OS). Only as-Aurora A is sensitive to the ATP analogue 1-Na-PP1 [START_REF] Reboutier | Aurora A is involved in central spindle assembly through phosphorylation of Ser 19 in P150Glued[END_REF]. Importantly the ectopic kinase was expressed under the control of its own minimum promoter [START_REF] Reboutier | Aurora A is involved in central spindle assembly through phosphorylation of Ser 19 in P150Glued[END_REF][START_REF] Tanaka | Cell-cycle-dependent regulation of human aurora A transcription is mediated by periodic repression of E4TF1[END_REF]. We depleted the endogenous Aurora A via RNA interference, controlled the efficiency of the depletion (Fig. 1A andB) and left the cells with ectopic Aurora A. The cells were then treated with the microtubule poisons nocodazole at 100 nM for 13-hours. This compound affects microtubule dynamics and arrests the cells in prometaphase of mitosis for several hours by maintaining the SAC active [START_REF] Rieder | Stuck in division or passing through: what happens when cells cannot satisfy the spindle assembly checkpoint[END_REF]. To directly test whether Aurora A kinase activity was required in this cell cycle arrested cells, each cell line AS-U2OS and WT-U2OS was treated with 10 M 1-Na-PP1 for 30 min in the presence of nocodazole and filmed during 3 hrs. In the presence of the inhibitor only the WT-U2OS cells containing an active Aurora A kinase remain arrested in prometaphase, while the cells containing an inactive Aurora A kinase exit from mitosis as soon as we added the as-Aurora A Journal of Cell Science • Accepted manuscript inhibitor (Fig. 1A and1C). To control that Aurora A inhibition did not stabilize kinetochoremicrotubule attachments we repeated the experiment with 3.3 M nocodazole that not only maintains the SAC active but also depolymerizes all mitotic microtubules [START_REF] De Brabander | Taxol induces the assembly of free microtubules in living cells and blocks the organizing capacity of the centrosomes and kinetochores[END_REF][START_REF] Jordan | Effects of vinblastine, podophyllotoxin and nocodazole on mitotic spindles. Implications for the role of microtubule dynamics in mitosis[END_REF]. We obtained the same result. When treated with 3.3 M nocodazole or with 3.3 M nocodazole plus 10 M 1NaPP1 the two cell lines AS-U2OS and WT-U2OS behave the same way, they arrest in prometaphase (Fig. 1D). Only when cells treated with nocodazole and 1-Na-PP1 were depleted of endogenous Aurora A, cells expressing wt-Aurora A (insensitive to 1-Na-PP1) remained in mitosis, whereas cells expressing as-Aurora A (inhibited by 1-Na-PP1) exited mitosis in less than 1 h Fig. 1C andD).

We also counted the number of mitotic cells upon depletion of endogenous Aurora A and treated with nocodazole and 1-Na-PP1. The number of cells in mitosis was much higher in cells expressing wt-Aurora A (78.29%, +/-4.65, n=46) than in cells expressing as-Aurora A and treated with 1-Na-PP1 (9.41%, +/-1.99, n=62): 7-times higher in the presence of nocodazole (Fig. 1E). 21.71% of the cells expressing active wt-Aurora A escape the nocodazole block and exit mitosis within 1 h, whereas the percentage of cells escaping the block reached 91.59 % upon inhibition of Aurora A (Fig. 1E). Similar observations were made when cells were treated with taxol; 40% of cells escaped mitosis in the presence of active Aurora A, whereas 85% escaped mitosis in the absence of active Aurora A (data not shown).

These finding demonstrated that inhibition of Aurora A induces cells to exit mitosis in condition

where the SAC should prevent it.

Journal of Cell Science • Accepted manuscript

Premature exit and abortive cytokinesis

To confirm and further characterized observations above, we simplified the approach and used the Aurora A inhibitor MLN8237 at a concentration (50 nM) that affects only Aurora A and not B and C [START_REF] Asteriti | The Aurora-A inhibitor MLN8237 affects multiple mitotic processes and induces dose-dependent mitotic abnormalities and aneuploidy[END_REF]. We also decided to used low concentration of nocodazole to avoid a complete depolymerisation of microtubules, but a concentration that remains sufficient to keep the SAC active [START_REF] Wang | Checkpoint genes required to delay cell division in response to nocodazole respond to impaired kinetochore function in the yeast Saccharomyces cerevisiae[END_REF]. HeLa cells expressing tubulin-GFP were treated with 100 nM nocodazole for ten hours, released in prometaphase for 15 min, then incubated with 50 nM MLN8237, 20 nM nocodazole or both and filmed (Fig. 2A).

As already reported, Aurora A inhibition affects spindle morphology in particular spindle length [START_REF] Bird | Building a spindle of the correct length in human cells requires the interaction between TPX2 and Aurora A[END_REF], Fig. 2B compare green lines 1 & 2) but does not affect mitotic timing (Fig. 2C andD). Low dose of nocodazole (20 nM), however, affects drastically spindle formation (Fig. 2B yellow line) and as it maintains the SAC active, induces a mitotic arrest (Fig. 2C andD). By adding MLN on these mitotic arrested cells (20 nM nocodazole and 50 nM MLN8237), we confirm our first observation (Fig. 1), almost 75 percent of mitotic cells exit from mitosis (Fig. 2D) without having been able to assemble a proper bipolar spindle structure. Nonetheless, these cells that prematurely exit mitosis do not divide (Fig2B red line).

Instead, we observed a deformation of the cell cortex that might reveal cell attempt to complete mitosis and to divide. Cytokinesis eventually aborted and the tetraploid cell re-adhered to the cover slip (Fig. 2B red line).

Because these phenotypes were reminiscent to that Aurora-B inhibition, we decided to control whether Aurora-B could be affected during Aurora A inhibition. Cells were synchronized in prophase and release in presence on MG132 (50 minutes, a proteasome inhibitor) to inhibit anaphase execution. Then cells were treated with DMSO, MLN, Nocodazole, MLN and nocodazole (Fig. 3A andB). In control cells Aurora B signal was detected in between CREST (a marker of centromere) signals indicating that the kinase is localised in between kinetochore Journal of Cell Science • Accepted manuscript pairs as previously reported (Fig. 2B and Fig. 3C right graphics, line scans). We did not detect any overlapping area between Aurora A and Aurora-B. Nor did we detect any modification of Aurora-B localisation under MLN8237 treatment (line scans, Fig. 3C) or using the as-Aurora A/1NaPP1 approach (not shown).

To control the specificity of MLN8732 towards Aurora A kinase activity, we compared the auto-phosphorylation state of each Aurora kinases (A, B and C) in the absence or in the presence of 50 nM MLN8237 using anti-phosphoserine antibodies in immunoblots (Fig. 3D). Aurora A auto-phosphorylation was almost completely lost under MLN8237 treatment, demonstrating that Aurora A was inactivated. In the same condition, we did not observe any change in Aurora B or C auto-phosphorylation. Additionally, the phosphorylation level of serine10 on histone H3, an in vivo substrate of Aurora B and C was not affected by MLN8732 (Fig. 3D).

These data indicate that the condition we used to inhibit Aurora A kinase does not affect Aurora B localisation or activity.

Inactivation of Aurora A disrupts the localisation of Mad2.

To obtain molecular insight into the role of Aurora A in mitotic exit in the presence of abnormal spindle we investigated the behaviour of Mad2, one major MCC component that localises on unattached kinetochores during prometaphase and that signals the presence of an active SAC.

Using the as-Aurora A approach, we observed that without any treatment, 90% of the control cells had Mad2 at the kinetochores in prometaphase. In the presence of the as-Aurora A inhibitor 1-Na-PP1, 87% of cells expressing wt-Aurora A had Mad2 at the kinetochores, whereas only 10% of cells expressing as-Aurora A did (Fig. Sup 1A andB). Western blot analysis revealed that Mad2 protein levels were not affected by Aurora A inhibition, indicating Journal of Cell Science • Accepted manuscript that the absence of the protein at the kinetochores was not due to a reduced amount of protein (Fig. 1B). Interestingly, in these last cells Mad2 was accumulating at the centrosomes (Fig. Sup 1A) as revealed by a quantification of Mad2 signal at the kinetochore and centrosome (Fig. Sup 1C).

We then used MLN8237 treatment to confirm these data by following the previously described experimental approaches (Fig. 3A). Then in the presence of MG132 the cells were treated with either 50 nM MLN8237 or 20 nM nocodazole or both and images were acquired using super resolution microscopy (Zeiss Airyscan) to evaluate the localisation of Mad2 in late prometaphase (Fig. 4A). Like previously reported by others [START_REF] Bird | Building a spindle of the correct length in human cells requires the interaction between TPX2 and Aurora A[END_REF], inhibition of Aurora A leads to shorter spindles indicative of a good efficiency of MLN8237. By using super resolution images and quantitative analysis, mean Mad2 intensity per kinetochore (stained with CREST) was calculated, pooled and compared. As expected, we observed a recruitment of Mad2 at kinetochores under nocodazole treatment (Fig. 4B). Interestingly, MLN8237 treatment reduces Mad2 signal levels at kinetochore in nocodazole treated as well as in untreated cells (Fig. 4B). Like observed in the as-Aurora A approach, this decrease of Mad2 at kinetochores is concomitant with a massive recruitment of Mad2 at centrosomes (Fig. 4A red arrows, Fig. 4B andC).

This data indicates that Aurora A activity is required to maintain Mad2 on unattached kinetochores thereby possibly contributing to maintain the SAC active.

Journal of Cell Science • Accepted manuscript

Kinetochore localisation of Aurora A

Because we observed Mad2 localisation at kinetochores is disrupted upon Aurora A inhibition, we investigated whether the Aurora A itself could be detected at or near the kinetochores.

Although it is thought that Aurora A localises at the centrosome while Aurora-B at the kinetochores during prometaphase, a presence of the Aurora A at kinetochores has been previously reported [START_REF] Katayama | Aurora kinase-A regulates kinetochore/chromatin associated microtubule assembly in human cells[END_REF]. Using super resolution, we investigated Aurora A localisation around kinetochores during kinetochores-microtubules attachment process. Like above, we used cells synchronized with nocodazole and blocked in metaphase by 50 nM of the proteasome inhibitor MG132 (Fig. 3A). We investigated the localisation of Mad2 and Aurora A along the axis of microtubules attached to kinetochores (red discontinued lines in c panels of Fig. 5A). Microtubules are visualised by tubulin staining and kinetochores by CREST staining.

In control cells, we observed Aurora A signal following that of microtubules with an intensity (reflecting protein levels) regularly decreasing until reaching the kinetochore. Two kinetochores are shown in figure 5A in the first row. Mad2 is absent on the first kinetochore on the left but present in the one on the right signalling the presence of an active SAC (Fig. 5A panel 1b). The drawing scheme representing the area occupied by the CREST signal (kinetochore) in purple and by Aurora A signal in red reveals an area occupied by both signals at the external surface of the kinetochore where the microtubules attach (Fig. 5A panel 1e).

After MLN8237 treatment, as observed above, Mad2 was not found at kinetochores anymore (Fig. 5A panel 2b). Interestingly, MLN8237 treatment also affects Aurora A localisation, the kinase signal accumulated at kinetochores (Fig. 5A In the presence of both nocodazole and MLN8237, only kinetochores still linked to microtubules showed Aurora A accumulation (Fig. 5A panel 4c and4d, Fig. 5B fourth row), suggesting Aurora A accumulation at kinetochores relies on the presence of microtubules.

We tested this hypothesis by investigating the level of Aurora A signal at kinetochores using the same methodology we used to quantify Mad2 (Fig. 3C). We observed an increase of Aurora A signal at kinetochores when cells were treated with either MLN8237 or nocodazole. This indicate that inhibiting Aurora A or maintaining the checkpoint active triggers Aurora A localisation at kinetochores. In cells treated with MLN8237 and nocodazole together, the level of Aurora A to kinetochore further increased again (Fig. 5A panel 4d and4e, Fig. 5B fourth row, Fig. 5C). This data strongly suggests that Aurora A uses microtubules to localize at kinetochores.

Discussion

Until very recently, all of the studies describing the function of Aurora A have used loss of function by RNA interference, mutations in animal models, expression of a dominant negative version of the kinase, or gain of function by over-expressing wild-type Aurora A or a mutated version that become hyperactive or non-degradable [START_REF] Glover | Mutations in aurora prevent centrosome separation leading to the formation of monopolar spindles[END_REF][START_REF] Littlepage | Identification of phosphorylated residues that affect the activity of the mitotic kinase Aurora-A[END_REF][START_REF] Sasai | Targeted disruption of Aurora A causes abnormal mitotic spindle assembly, chromosome misalignment and embryonic lethality[END_REF][START_REF] Schumacher | A highly conserved centrosomal kinase, AIR-1, is required for accurate cell cycle progression and segregation of developmental factors in Caenorhabditis elegans embryos[END_REF][START_REF] Zhou | Tumour amplified kinase STK15/BTAK induces centrosome amplification, aneuploidy and transformation[END_REF]. Because Aurora A is essential for centrosome maturation, the phenotype observed in all of these cases corresponds to a defect in spindle assembly and the cell never reaches metaphase [START_REF] Berdnik | Drosophila Aurora-A is required for centrosome maturation and actin-dependent asymmetric protein localization during mitosis[END_REF][START_REF] Hannak | Aurora-A kinase is required for centrosome maturation in Caenorhabditis elegans[END_REF]. We investigated whether Aurora A is involved in the SAC by specifically inhibiting Aurora A activity after the centrosomes have matured and the cell reaches metaphase.

We succeeded by taking advantage of the as-Aurora A isoform already described (Reboutier et Journal of Cell Science • Accepted manuscript al., 2013). And we confirmed our results using the Aurora A specific inhibitor MLN8237.

Although Aurora A has never been directly involved in kinetochore functions, it phosphorylates CENP-A in prophase to allow Aurora-B localisation at kinetochores [START_REF] Kunitoku | CENP-A phosphorylation by Aurora-A in prophase is required for enrichment of Aurora-B at inner centromeres and for kinetochore function[END_REF].

Aurora A also phosphorylates Haspin in late G2, participating indirectly to the phosphorylation of Threonine 3 of Histone H3 and to the recruitment of Aurora-B to kinetochores and more largely to the localisation of CPC and SAC proteins at kinetochores [START_REF] Yu | Aurora-A promotes the establishment of spindle assembly checkpoint by priming the Haspin-Aurora-B feedback loop in late G2 phase[END_REF]. Aurora A was also recently reported to phosphorylate Hec1 and to associate with the inner centromere Aurora-B partner INCENP, when overexpressed, to localise at the mitotic chromosome kinetochore [START_REF] Deluca | Aurora A kinase phosphorylates Hec1 to regulate metaphase kinetochore-microtubule dynamics[END_REF].

Eventually, Aurora A and Aurora-B share substrates with phosphorylation events occurring where the kinases are located. Aurora A phosphorylates MCAK and PLK1 at the centrosome [START_REF] Macurek | Polo-like kinase-1 is activated by aurora A to promote checkpoint recovery[END_REF][START_REF] Seki | Bora and the kinase Aurora a cooperatively activate the kinase Plk1 and control mitotic entry[END_REF][START_REF] Zhang | Aurora A phosphorylates MCAK to control ran-dependent spindle bipolarity[END_REF], whereas Aurora-B phosphorylates the same two proteins on the same residues but at the kinetochore [START_REF] Andrews | Aurora B regulates MCAK at the mitotic centromere[END_REF]Carmena et al., 2012a;[START_REF] Lan | Aurora B phosphorylates centromeric MCAK and regulates its localization and microtubule depolymerization activity[END_REF]. Therefore, both Aurora A and Aurora-B act in concert on the same substrates but at different locations to coordinate mitotic progression. This coordination was demonstrated further in DT40 cells in which Aurora A was depleted (KO) and Aurora-B inhibited. The authors found cooperation between both proteins in the coordination of chromosome segregation in metaphase and microtubule depolymerisation in anaphase [START_REF] Hégarat | Aurora A and Aurora B jointly coordinate chromosome segregation and anaphase microtubule dynamics[END_REF].

In the present study and in contrast to Aurora A KO, Aurora A was present and active until very late prometaphase. Its inhibition in that precise window of time led to a premature exit from mitosis with reduced Mad2 at kinetochores. In the presence of nocodazole or paclitaxel, which maintains the SAC active due to Mad2 recruitment at the kinetochores [START_REF] Rieder | Stuck in division or passing through: what happens when cells cannot satisfy the spindle assembly checkpoint[END_REF], inhibition of Aurora A led to the removal of Mad2 from the kinetochores, illustrating the extinction of the checkpoint, and a relocalisation of Mad2 to the centrosomes. This indicates that Aurora A activity is required to maintain Mad2 at non-attached kinetochores during prometaphase, ensuring that the SAC remains active. Whether this is a direct or indirect effect of Aurora A remains unknown.

Consistent with such role of Aurora A, we also observed the kinase inhibition (MLN8237) leads to its retention at kinetochores. Consistently with a recent report by DeLuca and collaborators [START_REF] Deluca | Aurora A kinase phosphorylates Hec1 to regulate metaphase kinetochore-microtubule dynamics[END_REF] who identified HEC1 (Ndc80 component) as Aurora A kinase substrate, we detected Aurora A at the outer kinetochores. This localisation of Aurora A seems to be very dynamic and the kinase might in fact shuttle at this localisation to phosphorylate substrates. Indeed, we observed that an inhibition of Aurora A increases its localization to kinetochores as if the inactive kinase remained blocked on its a substrate [START_REF] Widmann | The kinase activity of human Rio1 is required for final steps of cytoplasmic maturation of 40S subunits[END_REF]. This localisation of Aurora A under MLN8237 shed light on a possible cooperation between both Aurora A and B in regulating kinetochores bi-orientation leading to SAC silencing. Aurora A localises at the outer kinetochore while Aurora-B at the inner kinetochore (Fig. 6) Aurora A has been observed at kinetochores where it participates in kinetochore/chromatin microtubule nucleation [START_REF] Katayama | Interaction and feedback regulation between STK15/BTAK/Aurora-A kinase and protein phosphatase 1 through mitotic cell division cycle[END_REF]. Consistent with a potential function in the SAC, overexpression of Aurora A has been shown to be sufficient to override the SAC in the presence of taxol [START_REF] Anand | AURORA-A amplification overrides the mitotic spindle assembly checkpoint, inducing resistance to Taxol[END_REF][START_REF] Dutertre | Phosphorylation of CDC25B by Aurora-A at the centrosome contributes to the G2-M transition[END_REF]. Increase taxol sensitivity was also observed in pancreatic cancer cell lines upon Aurora A depletion [START_REF] Hata | RNA interference targeting aurora kinase a suppresses tumor growth and enhances the taxane chemosensitivity in human pancreatic cancer cells[END_REF]. More recently, it has been reported that the phosphorylation of p73 by Aurora A contributes to the breakdown of the Mad2-Cdc20 complex releasing Cdc20 to degrade cyclin B and to securin, providing a molecular explanation for the SAC override [START_REF] Katayama | Aurora kinase-A inactivates DNA damage-induced apoptosis and spindle assembly checkpoint response functions of p73[END_REF]. Although resistance to taxol treatment has been observed in breast cancer cells over-expressing Aurora A, this is only true in estrogen receptor (ER)-positive tumours and not in ER-negative tumours [START_REF] Noguchi | Predictive factors for response to docetaxel in human breast cancers[END_REF]. The reason for this difference remains to be elucidated, but the Journal of Cell Science • Accepted manuscript phosphorylation by Aurora A of Ser167 and Ser305 in ER might be involved [START_REF] Zheng | Aurora-A is a determinant of tamoxifen sensitivity through phosphorylation of ERα in breast cancer[END_REF].

Reports have also indicated a bypass of the SAC upon Aurora inhibition, but the inhibitors used were not specific to one Aurora kinase and the bypass was eventually attributed to Aurora-B inhibition. For example, the use of MLN8054, a more specific inhibitor of Aurora A than Aurora-B, revealed that inhibition of Aurora A accelerates mitosis exit in the presence of nocodazole or taxol [START_REF] Wysong | The inhibition of Aurora A abrogates the mitotic delay induced by microtubule perturbing agents[END_REF]. The concentration of MLN8054 used in the study was optimised for Aurora A inhibition, but the observed phenotype mimicked Aurora-B inhibition [START_REF] Tyler | VX-680 inhibits Aurora A and Aurora B kinase activity in human cells[END_REF].

Here, we used two strategies to specifically inhibit Aurora A to clearly demonstrate for the first time that inhibition of Aurora A is sufficient to inhibit SAC activity and exit from mitosis. We demonstrated that the mechanism involved is not SAC override, but SAC inactivation. The molecular mechanism underlying the involvement of Aurora A in maintaining an active SAC during pro-metaphase remains to be elucidated, and a search for Aurora A substrates has begun.

This report is the first to demonstrate a role of Aurora A kinase activity at the kinetochore for the SAC maintenance. Aurora A is a major target in cancer therapy with several inhibitors currently in clinical trials [START_REF] Kollareddy | Aurora kinase inhibitors: progress towards the clinic[END_REF]. A precise understanding of the multiple functions of Aurora A during mitosis will undoubtedly help in designing drug associations and increase the efficiency of chemotherapeutic strategies to eliminate cancer cells [START_REF] Bush | AMG 900, a small-molecule inhibitor of aurora kinases, potentiates the activity of microtubuletargeting agents in human metastatic breast cancer models[END_REF][START_REF] Schmidt | Aurora B kinase controls the targeting of the Astrin-SKAP complex to bioriented kinetochores[END_REF].

Journal of Cell Science • Accepted manuscript

  panel 2d and 2e, Fig. 5B second row). Nocodazole treatment at low dose on the other hand, induced a massive recruitment of Mad2 at kinetochores (Fig. 4A panel 3b and 3e) but also a recruitment of Aurora A (Fig. 5A panel 3d and 3e, Fig. 5B third row).

FiguresFigure 1 :

 1 Figures

Figure 2 :

 2 Figure 2: Aurora A inhibition during mitosis causes SAC override. (A) Methods used for

ACKNOWLEDGMENTS

Thibault Courtheoux was a fellow of the Region Bretagne and FHU CAMIn. Alghassimou Diallo was a fellow of the Guinée government, some data in the paper forms part of his PhD thesis prepared in the Institute of Genetics and Development of Rennes and defended at the University of Rennes 1 in December 18th, 2013. We thank the MRic facility (Microscopy Rennes Imaging Center, Biosit, IBISA), Stéphanie Dutertre (Biosit), and Jacques Pécreaux for helpful advice on integrated density. This work was supported by the Centre National de la Recherche Scientifique, Université de Rennes 1, Institut National du Cancer, Ligue Nationale Contre le Cancer (Equipe Labelisée 2014), and Agence Nationale de la Recherche (Aurora). Journal of Cell Science • Accepted manuscript showing representative case of mitotic progression under CTL, MLN, Nocodazole or Nocodazole and MLN treatment. Scale bar: 5m. (C) and (D) Quantification of mitotic duration (C) and mitotic behaviour (D) after release (CTL n=49, MLN8237 n=49, NOC=49), green: mitosis to cytokinesis, yellow: mitotic delay and red: abnormal cytokinesis. Journal of Cell Science • Accepted manuscript

Journal of Cell Science • Accepted manuscript

J . Cell Sci. 131: doi:10.1242/jcs.191353: Supplementary information Journal of Cell Science • Supplementary information