Jérôme Dejeu
email: jerome.dejeu@univ-fcomte.fr

Dr Mikhael Bechelany
email: mikhael.bechelany@univ-montp2.fr

Elise Berodier
email: elise.berodier@gmail.com

Patrick Rougeot
email: patrick.rougeot@femto-st.fr

Johann Michler
email: johann.michler@empa.ch

Michaël Gauthier
email: michael.gauthier@femto-st.fr

Nanostructured Non-adhesive Surfaces for Micro-and Nanomanipulation

Adhesion between microgripper end-effector and a nano/micro-object is a main topic for manipulation in micro-and nanoscale. Tuning this force is a great challenge. Adhesion force is directly linked to the chemical composition and the surface roughness of both, the object and the gripper.

Recently, we proposed a multispheres Van der Waals force model able to predict this force. The surface used was structured by an array of polystyrene spheres with radii from 35 nm to 2 µm. The experimental pull-off forces have confirmed our model. In this present work, we analyzed other innovating structure such as non-closed packed polystyrene (PS) spheres and organized Si Nanostructures, formed by chemical etching. The adhesion values of the pull-off force measured on these structures were very low (in the range of 2 to 10 nN), and suggest that these new structures have non-adhesive properties. A new model taking in account the roughness and the organization of the PS spheres and Si Nanostructures has been developed to predict these new properties.

INTRODUCTION

The assembly and micromanipulation of nano/micro-components is a major issue. Manufactured products are getting smaller while they combine more and more utilities. The major issue lies in the drastic reliance of the micro/nano-objects behavior and the surface forces. 1 The manipulation of these objects involves the handling, positioning, and releasing while avoiding any disturbances of the surface forces such as capillary, van der Waals, or electrostatic forces. [START_REF] Gauthier | Submitted to[END_REF] Several application fields are concerned such as telecommunications, bioengineering, or usually all Micro-Electro-Mechanical Systems (MEMS). In the microworld, surface forces are non-negligible compared to volume forces (weight, buoyancy, etc.). Microhandling methods are mostly based on two general approaches which consist in exploiting the surface forces 3-5 (e.g. capillary grippers) or reducing the surface forces (e.g. reducing adhesion on tweezers). This paper deals with the second challenge and especially on the reduction of adhesion between tweezers and the grasped object.. One approach to reduce the adhesion force has been developed in liquid and dry medium by surface structuring 1,[6][7][8] or chemical functionalization. [9][10][11][12][13] Surface nanostructuring is able to reduce the contact area between the gripper and the objects, and therefore decreases the contact area and van der Waals forces. Two approaches allow the surface structuring: topdown 14,15 and bottom-up approaches 16,17 . There are also methods based on the combination of both approaches. 18,[START_REF] Zhong-Ze | Artificial silver ragwort surface[END_REF] Among these methods, the nanospheres lithography [START_REF] Deckman | Natural lithography[END_REF] has received great consideration as a result of its simplicity compare to conventional lithography techniques. Using this methods, patterning of a wide variety of solid substrates has been achieved including metals, [21][22][23][24] semiconductors, [25][26][27] and ceramics. 28 Recently, we reported the influence of the structuring surface by closed-packed polystyrene (PS) spheres on the adhesion forces. We proposed a multispheres van der Waals force model which may suggest the existence of an optimal value of the sphere radius that would minimize the adhesion. 8 This model has been extended to demonstrate the existence of a minimum independent of the diameter and the nature of the spheres. 1 In this paper, and in order to understand the influence of organization, nature of nanostructures and roughness on adhesion properties, measurement has been performed on nonorganized Nanowires (NWs), non-closed packed array of PS spheres and organized Si NWs produced by metal induced chemical etching. The measurement shows that these new structures have non-adhesive properties. A new model has been developed to predict the adhesion force between a rough non-closed packed and a microsphere. The pull-off forces predict by the model are compared to experimental data and are concordance.

EXPERIMENTAL SECTION

Materials

Commercially available PS microspheres suspension (d ≈ 900 nm, 2.53 wt% aqueous dispersion, and d ≈ 200 nm, 2.53 wt% aqueous dispersion) was used (Polysciences, Inc., Eppelheim, Germany) as received. Acetone, toluene, hydrofluoric acid (HF, 48 %), Ammonium hydroxide (NH 4 OH,25 %), Sulfuric acid (H 2 SO 4 , 98 %), sodium dodecyl sulfate (≥ 99.0 %), Nitric acid (HNO 3 , 70 %) , Hydrochloric acid (HCl, 37 %)and Hydrogen peroxide (H 2 O 2 , 30 %), were purchased from Aldrich. ptype Si wafers (> 10 .cm), (100) crystal orientation from Silicon Materials, USA.

Synthesis of non-organized and non-structured NWs

P-type Si wafers were used as substrates. Prior to patterning, the Si specimens were pre-cleaned in acetone to remove the organic contaminants, and were then heated in air at 600 °C for 10 min to increase the thickness of the oxide layer. The substrate was then treated with a 1:1:5 solution of NH 4 OH (25 %), H 2 O 2 (30 %), and water at 80 °C for 15 min to obtain a hydrophilic Si surface. 23 Hydrophilic surfaces were formed by the terminal silanol (SiOH) groups. After this pre-treatment, a monodisperse suspension of PS microspheres was spin coated onto the substrate. The suspension was dried in air at Room Temperature (RT), and the spheres arranged themselves into a closed-packed structure of twodimensional ordered lattices due to attractive capillary forces. In order to fabricate an Au thin film pattern, the dense packed PS spheres are used as a shadow mask. Thin Au films were deposited onto the Si substrate through the PS honeycomb shadow mask using a Balzers SCD 040 sputter coater at a discharge current of 25 mA in a vacuum with pressure below 1 Pa for the desired time. After sputtering, the PS mask was removed by immersing the specimens in 97 % toluene in an ultrasonic bath. Finally, the specimens were annealed at 1000 °C for 1 h (heating rate 10 °C min -1 ) under vacuum and then cooled down to RT.

Structured surfaces: Organized non-closed pack of PS spheres

The two dimensional (2D) PS sphere ordered lattices were achieved by self-assembly using commercially available PS spheres suspension of 900 nm and 200 nm of diameter onto silicon substrate.

In a standard procedure, Si wafers were cut into 1.5 cm² pieces and were pre-cleaned in acetone for 5 min and in 1 wt% HF for 5 min to remove both organic contaminants and the native oxide. Additionally, the specimens were immersed in H 2 O 2 /H 2 SO 4 solution for 10 min and afterwards treated by O 2 plasma to produce a hydrophilic surface. After this pre-treatment, a monodisperse suspension of 10 µL of PS spheres was released onto the substrate. Then the sample was immersed into deionized water. Two droplets of sodium dodecyl sulfate solution (10 %) were deposited onto the water surface and the sample was removed out of the water, holding tilted in order to remove PS spheres in excess to the PS monolayer. After the complete drying of the substrate, the spheres were reduced by Reactive Ion Etching process (RIE) in order to obtain a non-closed packed array of spheres. The spheres were finally stuck on Si substrate by a thermal treatment at 100 °C for 30 min.

Structured surfaces: Organized non-closed pack of Si NSs

A non-closed packed array of spheres was prepared as mentioned before. A thin Au film was deposited then by sputtering. The sputtering was carried out at a discharge of 25 mA in a vacuum with the pressure below 0.1 mbar. By this Physical Vapor Deposition (PVD) step a mask of Au was obtained and the thickness of this layer was measured around 400 nm between spheres. 27 The specimens covered by Au were etched in a solution of HF/H 2 O 2 (4 M/0.88 M) in water at room temperature. After removing the metal with aqua regia (HNO 3 /HCl in a volume ratio of 1:3) solution and immersing the sample into toluene at 50 °C to dissolve the PS spheres, Si samples were dried and fully characterized.

Characterization

Samples were characterized by Scanning Electron Microscopy (SEM, Hitachi S-4800), and energy dispersive X-ray spectroscopy (EDX, Genesis 4000 EDAX).

Force Distance Measurement by Atomic Force Microscopy

Characterization of the pull-off force was performed with a commercial atomic force microscope (AFM; stand-alone SMENA scanning probe microscope NT-MDT, Russian). The experiments were done under a controlled environment with a laminar flow (humidity 30 % and 25 °C) on the Nanorol platform (http://nanorol.cnrs.fr/events.php). The rectangular silicon AFM cantilever, whose stiffness is 0.3 N/m, was fixed, and the substrate moved vertically. The same kind of cantilever was used for all experiments.

As the objective of this work is to improve the reliability of micro-object manipulation, interactions have been studied between a micrometric sphere and a structured surface. Measurements were in fact performed with a cantilever where a sphere (radius r 1 ) was glued in place of the standard AFM tip. In order to measure interaction forces in an experimental comparable to micromanipulation framework, the radius r 1 of the sphere glued on the AFM is 5µm. The force calibration was performed for each cantilever with its resonance frequency, and 10 measurements were done at different locations on the same sample with a driving speed of 200 nm/s.

RESULTS

Synthesis methods and Morphologies

Non-organized NWs. In order to understand the influence of organization of Si NWs on their adhesion properties, non-organized NWs has been synthesized as reported elsewhere. 23 SEM characterizations show the formation of a forest of NWs (Figure 1). EDX characterization shows that these NWs are composed of oxidized silicon.

Non-closed pack of PS spheres.

From one closed-packed PS sphere arrays on substrate, several nonclosed-packed arrays were produced by reducing the initial sphere diameter with RIE process (Figure 2). 900 nm diameters were shrunk to 800 nm, 710 nm, 560 nm, 495 nm for respectively 1 min, 2 min, 5 min and 7 min of RIE process (Figure 2a-e).

Non-closed packed of organized Si nanostructures. Non-organized Si nanostructures has been synthesized as reported elsewhere. 24 Figure 3 shows the regular pattern of Si NWs synthesized by selective metal inducing chemical etching process. The advantage of this method as compared to other lithographic methods is that it is simple and fast. In addition, both diameter and density of PS sphere can be controlled to a certain extent, so that pattern variations can be easily achieved. By modifying the size of the PS spheres, the dimension of the Si NWs and their spatial density can be changed. 24 Figure 3 shows Si NWs arrays resulting from the use of PS spheres with diameters of 200 nm, and 900 nm respectively. The diameter of the resulting Si NWs is 70 nm (Figure 3a) and 400 nm (Figure 3b) and the spatial density is ~2.89 x 10 9 cm -2 (Figure 3a) and ~1.43 x 10 8 cm -2 (Figure 3b-d), respectively. By decreasing the size of the PS spheres used in the beginning, we decrease the dimension of the Si NWs and increase their density. We note here that it is also possible to tune the diameter of the NWs by conserving the same spatial density using different conditions of RIE as well as the length using different immersing times during metal assisted chemical etching (Figure 3c,d). EDX measurement shows the presence only of oxygen and silicon in the specimens.

Adhesion measurement

Organized and non-organized Si NWs. The pull-off force was measured by AFM for different silica types: wafer, non-organized and organized NWs. The adhesion curves were presented in Figure 4. The pull-off force was influenced by the texture of the silica (wafer or nanowire). The adhesion force is lower on surface with nanowires than on silica wafer. The reason is the decrease of the contact surface.

On the non-organized nanowires curve (Figure 4a), the release between the probe and the surface is performed step by step and the curve presents discontinuous jumps. This phenomenon is already observed in the case of adhesion force measurement on carbon and boron nitride nanotubes. [29][30][31] The previous measurements are summarized and compared with silicon surface in Table 1.

The adhesion force with silica layer as homogeneous or NWs decreasing the pull-off compares to silicon surface which is near 250 nN. The surface with organized nanowires has a pull-off force inferior to the other silica materials. The Si NWs values of the pull-off force are very weak close to the detection limit of the AFM. Consequently the comparison between the different sizes of Si NWs cannot be done due to the lack of accuracy for this range of values. However the low pull-off forces from NWs surface confirm that nanostructure decreases greatly the adhesion forces. In addition, the diameter and the length of the Si NWs seem to have no effect on the pull-off force value.

Non-closed pack of PS spheres. It was found in a previous study on closed pack of PS spheres arrays

that the sphere size influences the pull-off force. The decrease of PS spheres size reduces the adhesion force near 100 times as compare to uncoated substrate. We can expect the same behavior with the nonclosed pack of PS spheres. The force measurements on the non-closed packed samples are presented in Figure 5 and are summarized in Table 2. The pull-off force, after shrinkage, is lower than that of the initial sphere. Indeed, for 900 nm diameter, the adhesion is 149 nN and decreases up to 10 nN with spheres of 495 nm of diameter.

DISCUSSION

Non-organized NWs

Compared to organized Si NWs, non-organized NWs present a different behavior in term of adhesion force. The discontinuous jumps observed in Figure 4 can be explain by two phenomena depending on the contact number between the probe and the NWs. If this number is one, i.e. just one nanowire is attaching to the probe; the discontinuous jump is explained by Ishikawa et al. 29 . The number of discontinuous jumps increases with the length of the NWs. The first jumps are a mesoscale peeling. The small stick-slip behaviors are also observed at the slope, which exhibit a nanoscale intermittent peeling without sliding. After, the jump corresponds to a conformational transition of the nanowires. The following slope, with no stick-slip behaviors, is the sliding of the contact point between the nanowire and the silicon substrate or the stretching of the NW. In the work reported by Ishikawa et al. 29 , just one slope without stick-slip has been observed. In our experiment the graph of Figure 4 shows three slopes without nanoscale intermittent peeling. This finding could be related to the fact that the probe is in contact with several NWs and the adhesion force is performed between the AFM cantilever and several NWs. In this case, the retract curve (Figure 6) is explained as previously, mesoscale intermittent peeling (B-C) then conformational transition or first broken up (C-D). These two steps are followed by nanoscale intermittent peeling (D-E) before the broken of the contact between the probe and the NWs, nanowire by nanowire. The steps from F to J, is the same that D to F with an uncertainty on the jump G-H which is either a conformational transition, or a broken link. In order to confirm exactly the phenomena, the experiments should be performed in-situ inside a SEM but the incertitude on the mechanism does not modify the conclusion on the NW using for the micromanipulation. The schema of Figure 6 resumes the different steps of the Figure 4a with no conformational transition.

Non-closed pack of PS spheres

The pull-off force measured between a sphere and a PS structured surface by AFM can be simulated by van der Waals monospheres model established by Dejeu et al. 1,8 because the radius of the PS sphere is superior to 100 nm. The authors modeled the pull-off force between a structured surface and a sphere whatever the nature and the size of both spheres respectively deposited on the surface and glued on the tipless cantilever extremity. 1 We suggest to modify and to expand the model in order to determine the pull-off force for a non-closed pack of PS spheres. The typical model of van der Waals force on smooth spheres is reported below (equation 1):

1 2 1 2 1 2 vdw 2 0 1 2 (A A ) r λ r F = 6 z (r + d r ) (1)
where r 1 is the sphere radius glued on the tipless cantilever extremity, r 2 is the initial PS sphere radius respectively, λ is the PS radius sphere reduction coefficient due to the etching process, A i the Hamaker constants of materials, z 0 the contact distance.

The simulation results and the comparison with the experimental measurements are summarized in Table 2. The simulated pull-off force is superior to the experimental measurements for all the nonclosed packed PS spheres. This phenomenon can be attributed to the spheres roughness increasing after shrinkage process.

Actually whereas the polymer sphere size is reduced, oxygen plasma bombardment causes also roughness on the PS sphere surface. The roughness induced by plasma etching has been widely reported. 32 Figure 1e shows the etched PS spheres of 495 nm reduced from 900 nm. Although the initial spherical surface is smooth and clean (Figure 1a), nanoscale roughness appears on the spherical surface after the plasma etching. The roughness is more noticeable with longer etching process time. The surface roughness has been attributed to microscopically inhomogeneous etching. 33,34 Again when the roughness increases, the contact area between the two surfaces (probe and substrate) decreases 35 and in turns this decreases the contact surface and the van der Waals forces. All studies conducted to observe the influence of roughness on the pull-off force converge to the same conclusion: the higher the roughness, the smaller the pull-off force is. [36][37][38][39][40][41][42] Tormoen et al. 38,39 reported that if the size of the asperities is of the same order of magnitude as the particle, the contact area as well as the pull-off forces decreases. The roughness plays also an important role in the error sources for the pull-off force measurements. 43 This error increases with the roughness. 38,39 In conclusion, the surface roughness would decrease the bond strength; nevertheless it would be more difficult to predict the forces since it becomes very noisy.

The modeling of the roughness could be done in different ways from fractal modeling to simplified geometries 1,8,[START_REF] Savia | Ieee In Ieee/Rsj 2010 International Conference on Intelligent Robots and Systems[END_REF][START_REF] Tam | [END_REF] . The first modeling is based on a complete definition of the roughness profile, enabling the construction of a precise model whose parameters are still difficult to identify in a practical case. The second modeling based on simplified geometry is based on assumptions on the roughness geometry but contains few parameters (or at least only one). In order to discuss the impact of roughness on the adhesion forces, we chose to simulate simulate the roughness by a simplified geometry consisting in a collection of nanospheres (radius, r 3 ) on the PS spheres periphery. The relative position between the probe and the spheres respectively the roughness r 3 , on the sphere r 2 , is described in Figure 7. As the pull-off force is a direct consequence of van der Waals force between both objects, the experimental pull-off measurements can be compared with van der Waals models. In the previous papers, we have shown that when the radius r 2 is greater than 100 nm, the pull-off force can be simulated by the interaction between the probe and only one or three spheres on the PS sphere depending on the initial position of the probe against the substrates. So four interaction cases between the probe and the PS spheres are possible (Figure 7):

_ Case 1: The probe is aligned and in contact with one asperity on one PS sphere, Figure 7a _ Case 2: The probe is in the middle of three asperities on one PS sphere and in contact with them,

_ Case 3: The probe is in the middle on the three PS spheres and in contact with one asperity on each PS sphere, Figure 7c _ Case 4: The probe is in the middle on the three PS spheres and in contact with three asperities on each PS sphere, Figure 7d.

The four total forces F between a PS sphere with roughness r 3 and the probe is described below:

2 min min 3 0 1 12 1 2 case 1 2 , 3 1 3 1 r + z + r A r r F = . 6 (r + r ) r + z + r i j ij ij z ∑ ℤ (2)
This interaction model between a spherical probe and a rough sphere has been simulated using the Matlab Simulink software. The evolution of F as a function of the interaction case and of the etching time is drawn in Figure 8a,b respectively.

In Figure 8, the first points on the right-hand of the Figure are the pull-off force without roughness (when r 2 = r 3 ). The result deals with the determination of a minimum of the interaction force which represents an optimum of adhesion reduction in the applicative field of micromanipulation. In our experimental case, the optimum radius r 3 in order to minimize the adhesion is between 6 nm and 10 nm.

This optimum is reached for radius 10 times smaller than in a case of a structured surface 1 with a force also ten times smaller. This value depends (i) on the interaction case between the probe and the asperities; (ii) on the borosilicate sphere radius glued to the cantilever (nature and size) and (iii) on the initial radius r 2 of the sphere. If the radius r 3 is lower than this optimum (10 nm), more and more spheres should be considered in the sum (eq 5), thus increasing the force.

Particular asymptotic convergences can be observed. In the right-hand part of the Figure, cases 2 and 3 are converging to the same values. In this case, the probe is in interaction with only the sphere(s) at distance z 0 . In the cases 1, 2, 3 and 4, the sphere(s) number in interaction with the probe are 1, 3, 3 and 9

respectively. So the adhesion force is the same for the cases 2 and 3. In the left-hand part of the Figure case 1 and 2 and case 3 and 4 are converging to the same values respectively. In this case, the radius r3 is lower than this optimum, and more and more spheres should be considered in the sum and the arrangement of the spheres is near to a plane surface, thus increasing the force. The sphere number on the sum is so important (more than 300 for r 3 =1 nm) that the number of spheres r 3 in contact has no influence, and only the number of sphere r 2 should be taken into consideration. So the cases 1 and 3 are similar to the cases 2 and 4 respectively.

In Figure 8, the force decreases from 2 to 15 by an appropriate roughness of the PS sphere. This modeling range is in agreement with the experimental measurement, Table 2 with a decrease from 3 to 7. An important roughness is also observed on organized Si NWs. However, it is due to the chemical etching process. It has been reported that metal induced chemical etching results in roughness along and on the top of Si NWs. 27,46 This roughness on the silicon NWs leads to a small pull-off force which is similar to one on the PS sphere after etching.

Potential application on micromanipulation

As all the experiments and the models have been established on micrometer spheres (r 1 =5µm), the application of the structurations of the surface in micromanipulation can be directly discussed.

The multi-contact release observed in non-organized NWs is able to disturb the micromanipulation.

Indeed it appears difficult to predict the distance required to release of all the contact. Despite the fact, that the adhesion is sensibly reduced, it seems consequently not applicable in micromanipulation application framework. The uncertainty on the distance required to the release is linked to the nonorganization of NWs. Organized NWs or spheres could be another approach.

So, for manipulation, the two processes can be used to decrease the adhesion and improve the object release, with a preference for the Si NWs because of their best stability. Indeed, the PS spheres are deposited by spin coating and could left the surface during the micromanipulation tasks whereas the Si NWs are engraved on the silicon wafer.

CONCLUSION

In conclusion, experimental pull-off forces have been measured by AFM with a borosilicate sphere glued on the tipless cantilever, on both non-closed pack of PS spheres and organized Si NWs produced by metal induced chemical etching. The adhesion values obtained for both are very low, close to the detection limit of the AFM. Thus, the nanostructures used can be seen as non-adhesive surfaces. The adhesion of these non-closed packed materials has been compared to non-organized NWs. A new model has been developed to predict these new properties. So, for manipulation, the two processes can be used to decrease the adhesion and improve the object release, with a preference for the Si NWs because of their best stability. A wide range of applications, in the field of telecommunications, bioengineering, and more generally speaking MEMS can be envisaged for these non-adhesive substrates. Experiments are in progress in order to investigate other organized and non-organized 1D nanostructures such ZnO, 25 BN, [47][48][49] SiC, 50 SiC@CNTs, 50

and Si 3 N 4 51 ACKNOWLEDGMENT.

 51 This work was partially supported by the EU under FAB2ASM, Contract FoF-NMP-2010-260079 (Efficient and precise 3D integration of heterogeneous microsystems from fabrication to assembly); and by the French National Agency (ANR) under NANOROL, Contract ANR-07-ROBO-0003, (Nanoanalyse for micromanipulation).

FIGURES.Figure 1 .Figure 2 .

 12 FIGURES.

Figure. 3 Figure 4 .Figure 5 .

 345 Figure.3 SEM images of different non-closed pack organized Si NWs used for the adhesion measurements synthesized used PS spheres of 200 nm (a) and 900 nm (b-c); (scale bars = 1 µm)

Figure 6 .

 6 Figure 6. Schema of the adhesion between several NWs and a sphere probe with the part of the

Figure 7 .

 7 Figure 7. Description of the different interactions (a) case 1, b) case 2, c) case 3, d) case 4) between the

Figure 8 :

 8 Figure 8: Theoretical pull-off forces as a function of PS sphere roughness, r 3 for the different interaction

TABLETable 1 .

 1 Adhesion force values of different type of Si NWs arrays.

N° sample Sample Name Spatial density [cm -2 ] Si NWs diameter [nm] Si NWs lengths [nm] Pull-off [nN]

	3	Si NWs	1.43 x 10 8	320	850	-9 ± 7
	4	Si NWs	1.43 x 10 8	460	1200	-10 ± 8
	5	Si wafer		-	-	-250 ± 30
	1	Si NWs	2.89 x 10 9	70	100	-3 ± 2
	2	Si NWs	1.43 x 10 8	300	520	-7 ± 5

Table 2 .

 2 Experimental and simulated pull-off force of nonclosed packed PS spheres.

	PS spheres diameter	Experimental pull-off force (nN)	Simulated pull-off force (nN)	Pull off F -	simulated	Pull off F -	exp	erimental
	800 nm	26 ± 14	91			3.5
	710 nm	12 ± 5	81			7
	560 nm	10 ± 6	65			6
	495 nm	13 ± 6	58			4

r + z + r A r r F = . 6 (r + r ) r + z + r

Where z ij min and z ij max are:

( )

Where d ij and D ij are the projection of the distance between the center of the sphere (0;0) and the sphere (i;j) on the z axis and r ij or R ij on the plan (xOy) depending on the case. So these distances are done by: ( )

( )


Anuradha Chauhan

Stephan Lorenzen

Hanspeter Herzel

Samuel Bernard
email: bernard@math.univ-lyon1.fr

Regulation of mammalian cell cycle progression in the regenerating liver

Keywords: liver regeneration, mitosis, bandpass filter, G2 delay, Cyclin B, Cyclin

The process of cell division in mammalian cells is orchestrated by cell-cycle-dependent oscillations of cyclin protein levels. Cyclin levels are controlled by redundant transcriptional, post-translational and degradation feedback loops. How each of these separate loops contributes to the regulation of the key cell cycle events and to the connection between the G1-S transition and the subsequent mitotic events is under investigation. Here, we present an integrated computational model of the mammalian cell cycle based on the sequential activation of cyclins. We validate the model against experimental data on liver cells (hepatocytes), which undergo one or two rounds of synchronous circadian-clock gated cell divisions during liver regeneration, after partial hepatectomy (PH). The model exhibits bandpass filter properties that allow the system to ignore strong but transient, or sustained but weak damages after PH. Bifurcation analysis of the model suggests two different threshold mechanisms for the progression of the cell through mitosis. These results are coherent with the notion that the mitotic exit in mammalian cells is bistable, and suggests that Cdc20 homologue 1 (Cdh1) is an important regulator of mitosis. Regulation by Cdh1 also explains the observed G2/M phase prolongation after hepatocyte growth factor (HGF) stimulation during S phase.

Introduction

The cell division cycle is a highly organized and structured mechanism in mammals. Because of its dynamical nature, it has been subject to several computational models. The first computational models pertained to the cell cycle in amphibian embryos and yeast (Tyson, 1991;Goldbeter, 1991;Novak and Tyson, 1993;Sha et al., 2003;Pomerening et al., 2003;Chen et al., 2004). More recently, computational models were proposed for specific parts of the mammalian cell cycle, particularly the G1/S phase transition and the restriction point (Aguda and Tang, 1999;Qu et al., 2003a,b;Swat et al., 2004;Novak and Tyson, 2004). A generic model for the eukaryotic cell cycle has also been presented (Csikasz-Nagy et al., 2006).

Few detailed, integrative models coupling the main events of the cell cycle, DNA synthesis and mitosis have been proposed. Models of this sort have been proposed for yeast, in which growth of cell mass is playing a key role in regulating mitosis (Chen et al., 2004), and more recently for mammals (Gerard and Goldbeter, 2009). In mammals, however, cell mass is not a major determinant for the control of cell cycle. Rather, mammalian cells possess multiple control mechanisms that prevent them from proliferating outside specific conditions. Mammalian cells vary in type and degree of differentiation, each one having different proliferative potential and mechanism. This makes the task of modelling mammalian cells challenging, as there is no universal experimental model. In vitro systems of dividing cells provide a way to study synchronized cell populations, which are suitable for modelling cell cycle events. However, cell cycle specific events are obtained under variable experimental conditions and different cell cycle synchronization procedures, which are difficult to reproduce on models. During liver regeneration, liver cells (hepatocytes) divide in a highly synchronized manner. We have taken advantage of the wealth of experimental data on the cell cycle during liver regeneration to construct the first integrative model of the mammalian cell cycle in the regenerating livers.

Liver regeneration is a process by which the liver can recover its normal mass and function following injury. After the surgical removal of two-third of the liver (partial hepatectomy), 90 % of the hepatocytes re-enter the cell cycle synchronously to complete one or two divisions and restore the liver mass. Partial hepatectomy (PH) triggers the release of pro-inflammatory cytokines (PICs). In the presence of PICs, hepatocytes leave a quiescent state, denoted G0, to enter a primed state, a process denoted priming. Primed cells, which are in the early G1 phase, are then driven by growth factors (GFs) to cross the restriction point, a point of no return beyond which they are irreversibly engaged in the cell cycle and do not require growth factors any more to complete mitosis (Taub, 2004). Progression within the cell cycle is controlled by the sequential activation of a family of cyclin dependent kinases (cdks) that allow an ordered succession of the cell cycle phases: G1, S, G2, and M [START_REF] Morgan | The cell cycle: principles of control[END_REF]. The cdk proteins are only active when they form a complex with their corresponding cyclin proteins. Cyclin D/Cdk4-6, Cyclin E/Cdk2, Cyclin A/Cdk2, and Cyclin B/Cdk1 complexes promote, respectively, progression in G1 phase, the transition from G1 to S phase, progression in S and G2 phase, and finally the G2/M phase transition, allowing entry into mitosis [START_REF] Morgan | The cell cycle: principles of control[END_REF]. Cyclin/cdk complexes are the workhorses of the cell cycle machinery, and as such, they need regulators. Cyclin/cdk regulation, which we henceforth mention as cyclin regulation for simplicity, is achieved through a variety of mechanisms. These include transcriptional regulation [E2F transcription factor 1 (E2F)], association with protein inhibitors [cyclin dependent kinase inhibitor (CKI)], phosphorylation/dephosphorylation [Wee1 and cell division cycle 25 (Cdc25)] and cyclin degradation [complexes formed with anaphase promoting complex (APC) and Cdc20 homologue 1 (Cdh1) or Cell division cycle 20 (Cdc20) (APC Cdh1 , APC Cdc20 ), Skp1/cullin/F-box protein related complexes (SCF)] (Morgan, 2007)(Fig. 1).

The model

DNA synthesis and mitosis. After both PH and CCl 4 treatment, liver regeneration exhibits well-synchronized DNA synthesis and mitosis (Deguchi et al., 2002). The temporal proliferation pattern of the cells in regenerating rodent livers consists of two waves of mitosis (Fabrikant, 1968). During the first round, hepatocytes synchronously undergo DNA replication and mitosis. It is then followed by a second round of division of less synchrony (Faktor, 1971). In mice, the first peak of DNA synthesis occurs 36 hours after PH and the mitotic peak, 4-12 hours after the peak of DNA synthesis (Matsuo et al., 2003).

E2Fs regulating sequential activation of cyclins. E2F activity is tightly controlled by binding to the retinoblastoma protein (Rb). Rb sequesters E2F and inhibits its transcriptional activity. Cyclins sequentially phosphorylate Rb and activate E2F in a positive feedback loop manner. Only hyperphosphorylated forms of Rb-E2F transactivates Cyclin A and Cyclin B (Lundberg and Weinberg, 1998) (see Appendix A.2 for the derivation of the Rb/E2F equations).

CKI at the interphase of S phase and mitosis. CKI stoichiometrically inhibits Cyclin D, Cyclin E and Cyclin

A by forming complexes with their cdk partners (Harper et al., 1995). CKIs are predominantly transcriptionally regulated by cytokine-dependent signaling (Gartel and Tyner, 1999;Coller et al., 2000). For further details on modelling CKI activation refer to (Chauhan et al., 2008). Degradation of CKI is mediated by SCF and APC-Cdc20 (Bashir et al., 2004;Amador et al., 2007). For simplicity, we do not model SCF and/or APC-Cdc20 dependent degradation of CKI, but use a general degradation term for CKI.

Cyclins. The whole process of cell division is orchestrated by the sequential activation of cyclins. Cyclin D and Cyclin E are the G1-S cyclins that trigger DNA synthesis. G1-S cyclins are induced by cytokine-and growth factordependent pathways and their degradation is controlled by SCF. For further details on Cyclin D and Cyclin E modelling refer to (Chauhan et al., 2008). The rise of G1-S cyclins is accompanied by the appearance of Cyclin A during S phase. In the late G2 phase, a transient activation of Cyclin B is observed enabling swift G2-M transition. This transient activation is enabled by regulatory feedbacks imposed on Cyclin B by Cdc25 and Wee1. At metaphase, sufficiently high levels of active Cyclin B phosphorylates Cdc20 to form an active complex with APC, which in turn, degrades Cyclin B. Inactivation of Cyclin B allows mitotic exit and return to the G1 phase [START_REF] Morgan | The cell cycle: principles of control[END_REF]. Inactivation of Cyclin B also leads to dephosphorylated Cdh1 which then forms an active complex with APC to degrade Cyclin B and Cyclin A during G1 phase.

APC and SCF: the degradators controlling the cell cycle. APC is a proteasomal degradation machinery that needs to form complexes with subunits to be activated. During the cell cycle, APC is successively activated by two of its subunits, Cdc20 and Cdh1, which have separate windows of activity, each regulated by feedback loops. Cdc20 is activated by Cyclin B during mitosis through a negative feedback loop. Cdh1 becomes active at the end of mitosis through a positive feedback loop, when Cyclin B is degraded, and assembles with APC to further degrade Cyclin A, Cyclin B, Cdc20 and SCF (Wasch et al., 2010). Cdh1 also acts as a G1 regulator by destroying mitotic cyclins during G1 phase and maintaining the cell in G1 by the destruction of SCF (Bashir et al., 2004;Wei et al., 2004). Cdh1 controls Cyclin A and B at two levels: first by a direct cyclin-Cdh1 double feedback loop and second by a CyclinB-Cdc20 negative feedback. Cdh1 controls G1 cyclins, Cyclin D and E by degrading their degradator SCF. SCF can degrade G1-S cyclins, Cyclin D and Cyclin E (? [START_REF] Nakayama | [END_REF]. SCF is down-regulated during M and G1 phases and is induced by Cyclin E near the S phase when its degradator Cdh1 is degraded (Bashir et al., 2004). Cdh1 connects the SCF-driven G1-S proteolytic machinery with the APC-dependent mitotic and G1 proteolysis. Therefore, at the core of the cell cycle lies Cdh1-driven degradation that results in a proteolytic oscillator setting the pace of the cell cycle progression.

Using the existing knowledge about cyclin control mechanisms and their temporal organization during liver regeneration, we designed the network controlling the cyclin-dependent kinase activity at all stages of the cell cycle (Fig. 1). The model is a system of ordinary differential and algebraic equations describing the concentration of key players of the cell cycle (See Appendix A for a description of the model variables). To make the model as tractable as possible, the variables were sorted according to the time scale on which they evolve. The time-scale of reference is the duration of one cell cycle (in hours). Variables that change on a shorter time-scale were approximated by their steady state (quasi-steady state assumption). For instance, concentrations of phosphorylated proteins were approximated by their steady states. To limit the number of variables in the model, we considered explicitly protein concentrations but not their corresponding mRNA concentrations. The kinetic parameters (see Appendix B) were adjusted so that the model could represent the experimentally observed temporal dynamics of respective model species. The induction of gene expression by pro-inflammatory cytokines, growth factors and by Rb-E2F (here called E2F) was incorporated directly by time-dependent input functions modulating the synthesis of cyclins (see (Chauhan et al., 2008) and Appendix A.2). Post-translational regulation through phosphorylationdephosphorylation or through complex formation is explicitly included in the model. For simplicity, we do not distinguish between the nuclear and cytosolic compartments of the cell. The model describes the dynamics of the intracellular cyclin/cdk network and represents an average hepatocyte during liver regeneration. Our aim is to address one cycle of division during liver regeneration triggered by a decaying damage (Hayashi et al., 2003;Blindenbacher et al., 2003). All concentrations in the model are expressed in arbitrary units (a.u.) since for most of the regulatory proteins, the actual concentrations are not known.

Numerical simulations of the full model were performed with Matlab. Bistability analysis of the model was done with the numerical bifurcation continuation process of XPPAUT.

Results

Sequential activation of cyclins and proteolytic degradators

The computational model is able to reproduce the sequential activation of cyclins and proteolytic degradators along the cell cycle (Fig. 2). Damage ac-tivates cytokine-and growth factor-induced pathways which lead to the activation of Cyclin D during late G1 and the subsequent activation of Cyclin E, indicating entry into S phase (Chauhan et al., 2008).

Step-wise activation of E2F via Cyclin D and Cyclin E leads to the sequential transcription of Cyclin A in the S phase and Cyclin B during mitosis [START_REF] Morgan | The cell cycle: principles of control[END_REF] (for equations see Appendix A.2). CKI is activated in early G1 and is maximally elevated at late S phase (Behrens et al., 2002). E2F activity peaks at S phase (Chen et al., 2009). APC Cdh1 is active during late mitosis and early G1 phase, degrading Cyclin A and Cyclin B. APC Cdc20 , which is a degradator of mitotic substrates, is activated during late mitosis, once Cyclin B is sufficiently activated. Degradator of G1-S cyclins, SCF, becomes active at the end of S phase, leading to the degradation of Cyclin D and Cyclin E . Also, mitotic regulators Cyclin B active, Wee1, Cdc25 and APC Cdc20 exhibit a delayed switch-like activation [START_REF] Morgan | The cell cycle: principles of control[END_REF] (Fig. 2).

A switch in Cyclin B activity drives M phase progression

A switch-like increase in the activity of Cyclin B enables entry into mitosis, whereas subsequent Cyclin B proteolysis promotes exit from mitosis [START_REF] Morgan | The cell cycle: principles of control[END_REF]. This switch-like behaviour is made possible by the presence of controls on Cyclin B activation, which conserves the total amount of Cyclin B, and on total Cyclin B concentration.

Two feedback loops control Cyclin B activity: a double negative feedback loop from the inhibitory kinase Wee1, which is itself inhibited by Cyclin B, and a positive feedback loop from the activating phosphatase Cdc25, which is further activated by Cyclin B (Tyson et al., 2003). Recently, Potapova et al. (2009) have shown that the exit from mitosis was also a switch phenomenon, due to the Wee1 and Cdc25-dependent positive feedback loops acting during G1 phase.

Total Cyclin B concentration is controlled through synthesis, by E2F, and degradation, by APC Cdh1 and APC Cdc20 . E2F promotes Cyclin B synthesis. At low E2F activity, there is not enough Cyclin B for any of it to be activated. If E2F reaches a threshold value, Cyclin B gets activated. This threshold activation is achieved through a transcritical bifurcation (Fig. 3a) (Guckenheimer et al., 1997). At any E2F values, two steady states for Cyclin B exist, one zero and one either negative (below the threshold) or positive (above the threshold). The zero steady state is stable below the threshold and the positive steady state is stable above it. This transcritical bifurcation is possible because of the Cdc25 loop. When this loop is switched off by setting Cdc25 to a low constant level (set at 0.01), the transcritical bifurcation disappears. No effect was observed when Wee1 loop was removed (results not shown). Downregulation of APC Cdh1 is responsible for the activation of Cyclin B, while APC Cdc20 is mainly responsible for its degradation at the exit of mitosis. Varying the strength of the Cdh1-mediated Cyclin B degradation rate, d Cdh1 CB , revealed a bistable switch that allows rapid activation and inactivation of Cyclin B (Fig. 3). At high degradation rate, there is a small total amount of Cyclin B, and no active Cyclin B. When the degradation rate is reduced, a second stable steady state appears through a saddle-node bifurcation. These two stable steady states co-exist until the low level steady state becomes unstable, through a reverse pitchfork bifurcation (Guckenheimer et al., 1997). For degradation rates below that level, there is only high Cyclin B activity. This combination of bifurcations form a bistable switch whereby low and high levels of Cyclin B are possible.

Thus, the model includes two mechanisms for controlling Cyclin B activity: a transcritical activation by E2F and a bistable switch induced by APC Cdh1mediated Cyclin B degradation. The transcritical activation ensures active Cyclin B levels remain to zero whenever total Cyclin B level are low, therefore preventing accidental entry into mitosis. The bistable switch is responsible for the rapid activation of Cyclin B at the entry to mitosis, and its rapid elimination at the end of mitosis.

Liver regeneration and cell cycle progression are robust processes

Liver regeneration is a robust process. After PH, the liver grows back to its normal mass even in the absence of important cell cycle players. Therefore we expect Cyclin E and Cyclin B activity, as a measure of total DNA synthesis and mitosis respectively, to be robustly activated. We analysed the sensitivity of Cyclin E and Cyclin B activity with respect to changes in biochemical parameters. Both Cyclin E and Cyclin B were robust to parameter change except when parameters immediately downstream of the damage, such as degradation of PIC, were varied. In case of these sensitive parameters, Cyclin E and Cyclin B peaks showed a bell-shaped response to systematic variation in these sensitive parameters (results now shown).

We further went on to study the response of the model to systematic changes in the damage. We varied the damage from strong-transient to weak-sustained input keeping the the total amount of damage, i.e. the integral under the curve, constant (Fig. 4a). Both DNA synthesis (Cyclin E) and mitosis (Cyclin B) responded as a bandpass filter to such a systematic variation in damage (Fig. 4b-c, upper panels). Weak-sustained and strong-transient damages were rejected and the cells did not progress through both DNA synthesis and mitosis in both cases. The timing of cell cycle events was robust to varying nature of damage (Fig. 4b-c, lower panels).

Bandpass filtering properties have been studied previously in a G1/S model (Chauhan et al., 2008). The damage was transmitted through two pathways: 1) direct ultrasensitive induction of Cyclin E, which filtered out the weaksustained damages; and 2) Cytokine mediated delayed pathway, which consisted of nested feedforward loops. Incoherent feedforward loops via CKI and HGF filtered out sharp-transient damages. In the present model, changing the strength of feedback loops during G2/M/G1 phases did not change bandpass filtering behaviour of the model. However, the range of the bandpass filter was affected. Decreasing the positive feedback loop strength of the Cdh1-Cyclin A loop by reducing the two Cyclin A and Cdh1 mutual negative regulation parameters to one-half, one-fourth and zero respectively, had two distinct filtering effects: it decreased the range of the filter in case of Cyclin B (Fig. 5b), while for Cyclin E (Fig. 5a), it increased the range of filter. This opposite control of cell cycle events: DNA synthesis (Cyclin E) and mitosis (Cyclin B) with respect to varying damage, has not been systematically studied to our knowledge.

The circadian clock gates mitosis but not DNA synthesis

Cell division in many mammalian tissues is associated with the circadian clock, a ∼24 h rhythm regulating physiology and behaviour (Reppert and Weaver, 2002). In hepatocytes, Wee1 is the gatekeeper of this circadian control of cell cycle (Matsuo et al., 2003). Wee1 is a clock-controlled gene (its expression is modulated over 24 h), and regulates the timing and efficiency of the cell cycle. In mice hepatocytes, Wee1 synthesis peaks around ZT12 (ZT0 represents lights on and ZT12, lights off, in a 24 h light-dark cycle) (Matsuo et al., 2003). To simulate the circadian expression of Wee1, we used a sinusoidal-shaped synthesis rate of Wee1, with a maximal synthesis rate at ZT12 (Fig. 6a, See Appendix A.6). We simulated PHs performed on mice at ZT8 or ZT0, and used peak of activity in Cyclin B and E as markers of mitosis and S phase. We found that mitotic activity induced by Cyclin B at ZT8 or ZT0 changes from 44.5 to 51.5 hours (Fig. 6b). These results reproduce well the gating properties of Wee1 observed experimentally (Matsuo et al., 2003).

In our model, DNA synthesis is directly coupled to mitosis via a Cdh1-SCF loop. Nevertheless, Cyclin E activity, which is representative of DNA replication, was independent from Zeitgeber time PH (Fig. 6c). Imposition of circadian control on Wee1 maintains DNA synthesis as an independently controlled property of hepatocytes (Matsuo et al., 2003).

HGF treatment at S phase causes prolonged Cyclin A and Cyclin B activity

Growth factors are well known to promote the transition from G1 to S phase in cell cycle progression. Beyond G1-S transition though, growth factors have been considered non-essential during the cell cycle (Jones and Kazlauskas, 2001). However, when injected at S and G2 phases, hepatocyte growth factor induces a prolongation of G2. Increased ERK activation and delayed degradation of Cyclin A and Cyclin B are observed when the cells are induced with HGF at S and G2 phase (Park et al., 2007;Nam et al., 2008). It is not intuitive how HGF induction at S phase might lead to delayed degradation of Cyclin A and Cyclin B, but CKI up-regulation has been implicated (Dangi et al., 2006).

We went on to simulate growth factor induction at S phase by giving an extra HGF pulse during S phase between 36 and 40 hours (See Appendix A.7). Our model was able to reproduce the observed delay in the degradation of Cyclin A and Cyclin B (Fig. 7e-f). Our simulations suggest that HGF treatment at S phase prolongs CKI (Fig. 7b) and E2F-FoxM1 (Fig. 7c) expression and delays APC Cdh1 activation (Fig. 7d). Thus, growth factor leads to E2F-FoxM1 dependent Cyclin A up-regulation. As long as Cyclin A is active at G2, APC-Cdh1 can not be activated due to its antagonistic control by Cyclin A. This leads to delayed Cdh1 activation. Delayed Cdh1 activation leads to delayed degradation of Cyclin A and Cyclin B (Fig. 7b-d). More recently, CKI (Dangi et al., 2006), FoxM1 (Alvarez-Fernandez et al., 2010) and APC Cdh1 (Holt et al., 2010) have been directly implicated in G2 arrest. CKI and FoxM1 are activated through growth factor induced MAPK signaling at G2 (Dangi et al., 2006;Ma et al., 2005).

Discussion

Computational model for mammalian cell cycle

We have developed a computational model for the cell cycle of hepatocytes during liver regeneration that couples different cyclins and proteolytic degradators in a sequential manner. The model recapitulates the temporal profile of dividing hepatocytes following partial hepatectomy: the sequential activation of cyclins and proteolytic degradators; the effect of the circadian clock on the timing of mitosis; the G2 phase delay caused by HGF treatment during S phase and the effect of Cdh1 on cell cycle progression. Gerard and Goldbeter (2009) recently presented a mammalian cell cycle model describing the sequential cyclin-cdk activation. The model demonstrates repet-itive cell cycling in the presence of suprathreshold amounts of growth factors.

The emphasis was on the Rb-E2F network during G1 phase and Cyclin Adependent activation of Cyclin B during mitotic entry. Our model is also structurally able to generate sustained oscillations like many other cell cycle models (Csikasz-Nagy et al., 2006;Gerard and Goldbeter, 2009). However, the focus of the model presented here is to simulate one or two cycles of cell division observed during liver regeneration after injury.

Threshold activation of Cyclin B

Entry to and exit from M phase display a switch-like behaviour in Cyclin B levels. It is rapidly switched on at the G2-M phase transition and rapidly switched off at the end of mitosis. Threshold activation provides a robust mechanism for switch. We have shown that total Cyclin B concentration and activity is regulated by two threshold processes: a transcritical bifurcation, mediated by E2F and a bistable loop, mediated by APC Cdh1 . Both the transcritical bifurcation and the bistable loop allowed active Cyclin B to be kept at zero levels outside the specific time window of activity during G2/M phases. Although transcritical activation is qualitatively different from a bistable switch, it also provides a sharp activation threshold.

Bandpass Filter of Damage Input

The model exhibits bandpass filter properties which allows the system to ignore strong-transient and weak-sustained damages (Chauhan et al., 2008). Thus, if the damage is too transient (short) or too weak, cells will not enter the cell cycle events of DNA synthesis and mitosis. This is in agreement with observations that liver regeneration is proportional to the amount of PH, but does not occur if the PH is too small or too large (Lambotte et al., 1997). Also, varying the feedback strength of the Cdh1-Cyclin A positive feedback loop in the model tunes the range of filtering for Cyclin E and Cyclin B in an opposite manner. Such an opposite control of DNA synthesis and mitosis with varying strength of the Cdh1-Cyclin A positive feedback loop has not been systematically studied to our knowledge. Opposite control of filtering range for Cyclin E and Cyclin B can have some interesting physiological implications. Certain environmental conditions promote DNA synthesis while suppressing mitosis, inducing endoreplication and polyploidization (Nevzorova et al., 2009). Cdh1mediated control of cyclins during endoreplication has been widely reported in mammals, Drosophila and plants (Garcia-Higuera et al., 2008;Sorensen et al., 2000;Sigrist and Lehner, 1997;Kasili et al., 2010). It would be interesting to study Cdh1-mediated feedback loop control of cell cycle with respect to endoreplication and polyploidization, which are common forms of cell cycle progression during liver regeneration (Duncan et al., 2010).

Circadian control of Wee1

Our model is consistent with a circadian control of Wee1 synthesis, affecting the timing of mitosis but not of DNA synthesis. When the entrainment to light-dark cycle is simulated by varying Wee1 synthesis with a 24 h period, the time at which PH is performed has a impact on the timing of mitosis. In agreement with Matsuo et al (2003) results, we found that in PH performed at ZT0 and ZT8 (at lights on and 8 hours later respectively), cells enter mitosis roughly at the same time (around 60 h after ZT0). This shows that circadian regulation of Wee1 induces a gating process by which cells cannot divide at any time of the day. The timing of DNA synthesis was independent from the timing of PH, and occurred 36 hours after PH, coherent with experimental results. In hepatocytes, Wee1 is a major link between the circadian clock and the cell cycle and S phase is largely a circadian clock-independent process, despite the presence of a feedback loop linking G1/S and G2/M phases via Cdh1 and SCF. In continuously dividing cells this independence might not be preserved as gating mitosis might affect subsequent phases.

HGF treatment during S phase causes prolonged G2/M phase

HGF promotes and accelerates cell cycle progression but has been mostly assumed to be non-essential beyond the G1-S transition. However, HGF injection during S phase leads to a prolongation of G2 resulting in the delayed degradation of cyclins during G2-M phase. The model was able to reproduce the prolonged activation of Cyclin A and Cyclin B, caused by a delayed reactivation of APC Cdh1 during M phase. Moreover, activation of Cyclin A and Cyclin B was not delayed after HGF injection, consistent with experimental results (Nam et al., 2008). The delay in G2-M transition is explained by a delay in Cyclin B-associated kinase activity, due to a delay in nuclear translocation. Our model is not refined enough to reproduce intracellular compartimentalization results, although it is consistent with total Cyclin B time courses.

Erk signaling is known to mediate G2 delay via CKI (Han et al., 2005;Dangi et al., 2006;Park et al., 2007). Our simulations suggest that CKI, APC Cdh1 and E2F-FoxM1 contribute in delaying the degradation of Cyclin A and Cyclin B. This leads us to the conclusion that induction of growth factors during S phase might lead to prolonged transcriptional activity of E2F-FoxM1 resulting in increased Cyclin A activity. Since the antagonistic players Cyclin A and APC Cdh1 can not coexist, APC Cdh1 activation is delayed. Thus the APC Cdh1dependent degradation machinery might also be responsible for the delayed Cyclin A and Cyclin B degradation during G2 phase delay.

Conclusion

Mounting evidences demonstrate that Cdh1 plays an important role in the control of proliferation, differentiation and maintenance of genomic integrity by timely and coordinated degradation of several substrates (Wasch et al., 2010). Cdh1 maintains the stability of G1 phase by degrading the degradator of G1-S cyclins and SCF. Cdh1 is required for the G2 DNA-damage checkpoint (Wasch et al., 2010). Cdh1 is implicated in genomic instability and cancers, and is under investigation as a therapeutic target. Cdh1 also plays a role in specialised cell division cycles of endoreplication and polyploidy, widely observed during liver regeneration. These observations call for a better characterization of the role of Cdh1 in the control of G2 phase. It will be interesting to refine the model presented here around Cdh1 in order to understand the G2 delay and get a better understanding of the fate of the cell division cycle.

p 2 4 k 1 25 k 2 1 k 3 0.1 APC AP C T 1 k Cdh1 AP C 0.5 k Cdc20 AP C 0.1 Mitosis v E2F CA 1.75 d Cdh1 CA 1 d CA 0.1 v CKI CA 0.01 d CKI CA 0.1 k 1 0.03 k 2 5 k a1 60 v E2F CB 1.2 d CB 0.01 d Cdh1 CB 0.8 d Cdc20 CB 0.2 P W ee1
SCF G1 S G2 M G1 S G2 M G1 S G2 M G1 S G2 M G1 S G2 M G1 S G2 M G1 S G2 M G1 S G2 M G1 S G2 M G1 S G2 M G1 S G2 M G1 S G2 M G1 S G2 M G1 S G2 M G1 S G2 M G1 S G2 M G1 S G2 M G1 S G2 M G1 S G2 M G1 S G2 M G1 S G2 M G1 S G2 M G1 S G2 M G1 S G2 M

Fig. 1 .

 1 Fig. 1. Damage induced model of the cell cycle during liver regeneration based on sequential cyclin activation and degradation. The PH-induced damage primes quiescent cells via cytokine signaling. Primed cells then activate growth factor-induced signaling cascades. Cytokine-and growth factor-induced pathways act in concert to provide mitogenic signals for Cyclin D activation in early G1 phase. Cyclin D initiates a chain of sequential phosphorylation of Rb/E2F which leads to sequential transcription of further G1-S (Cyclin E), S (Cyclin A) and M (Cyclin B) phase cyclins. As early as G1 phase, stoichiometric inhibitors of cyclins, CKI, are also activated by cytokines. The G1-S proteolytic degradation machinery, SCF, is activated by Cyclin E. Entry to mitosis is initiated by the abrupt activation of Cyclin B by Cdc25 and Wee1 in a positive feedback loop fashion at G2-M transition. Cdc25and Wee1-mediated positive feedback loops also regulate the rapid exit from mitosis by swiftly inactivating Cyclin B at M-G1 in coordination with a positive feedback loop from the M-G1 degradator APC Cdh1 . Mitotic degradator APC Cdc20 exerts a negative feedback loop. All cyclin-cdk complexes are depicted by their respective activating cyclin partner for simplicity. E2F in the scheme represents all three forms of E2F, viz, unphosphorylated, hypophosphorylated and hyperphosphorylated.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Sequential activation and degradation of cyclins and their key regulators. Cyclin D appears at the late G1, Cyclin E at the G1-S transition, Cyclin A in the S phase and Cyclin B in the late S phase and mitosis. At the G2-M interface, transient activation of Cyclin B active, Cdc25 and inactivation of Wee1, leads to abrupt mitotic entry. At the M-G1 interface, Cdc25 and Wee1 are respectively rapidly inactivated and activated, leading to swift exit from mitosis. APC Cdc20 is also transiently activated once enough Cyclin B has accumulated at mitosis. APC Cdh1 is active from late mitosis to G1 phase keeping Cyclin A and Cyclin B degraded in a positive feedback loop fashion during late mitosis and G1. SCF is activated by Cyclin E during G1-S phase leading to degradation of G1-S cyclins. All concentrations are normalized by their maximum concentrations.

Fig. 4 .

 4 Fig. 4. Bandpass filter characteristics of the model. Both DNA synthesis (Cyclin E) (b, top) and mitosis (Cyclin B) (c, top) exhibit bandpass filtering properties when the damage (a) is changed systematically from weak-sustained damage to strong-transient damage, keeping the total amount of damage constant. Timing remains relatively constant (b and c, bottom).

Fig. 5 .Fig. 6 .Fig. 7 .

 567 Fig. 5. Opposite bandpass filtering effect on DNA synthesis (a) and mitosis (b) with respect to changing feedback strength of Cdh1-Cyclin A loop.

Acknowledgements

AC was supported by the German Ministry Federal Ministry of Education and Research (BMBF, Hepatosys, 0313078A) and by SFB 618 grant of the Deutsche Forschungsgesellschaft (DFG). SL was supported by the German Ministry Federal Ministry of Education and Research (BMBF, Hepatosys, 0313078A). SB was supported by the Project-Team INRIA Dracula (France), the French National Research Agency (ProCell, ANR-09-JCJC-0100-01), and acknowledges support from Rhône-Alpes Complex Systems Institute (IXXI, France). We thank two anonymous reviewers for their useful suggestions.

Appendix

A Model equations

The model contains 25 dynamical variables and 84 parameters. The 25 variables are the concentrations of the following players involved in the cell cycle control: Damage (D) induced pro-inflammatory cytokine pathways (PIC and PGE); proinflammatory cytokine induced growth factor pathways (PAI, HGF, HB-EGF); Cyclins D, E, A, and B; the inhibitor CKI and its complexes with the active cyclin-cdk complexes; the active and inactive forms of phosphatase Cdc25 and of kinase Wee1; the SCF proteasomal ligases involved in the degradation of cyclin D, Cyclin E and CKI; and the active and inactive forms of the proteins Cdh1 and Cdc20 involved in the degradation of cyclins A and B

A.1 G1-S model

A.2 E2F steady state

Rb phosphorylation is initiated by Cyclin D, releasing E2F in sufficient quantity for Cyclin E activation. Cyclin E further phosphorylates Rb (Rb P ) resulting in hyperphosphorylated Rb pp , releasing higher amounts of E2F required for transcriptional activation of further Cyclins. Phosphorylation/dephsophorylation reactions of Rb and association/dissociation of its three phosphorylated forms with E2F are fast enough to be in the steady state equilibrium.

Substituting the steady state values, we solve the equation for E2F as

Substituting the steady state values, we solve the equation for APC as (2005) with the simplification that the phosphorylation/dephosphorylation of the three inactive forms of Cyclin B viz transcribed Cyclin B (CycB), Tyrosine phosphorylated (CycB Y ), and Tyrosine and Threonine phosphorylated (CycB Y T ) is fast enough to be always in equilibrium; so that equation for CycB Y T can be algebraically solved.

[