Julien Letesson

Bastien Goin

Thibaut Cachon

Eric Viguier

Biomechanical analysis of an extra-articular stabilization using a synthetic implant for caudoventral hip luxation in a feline cadaver model: preliminary results

Keywords: Biomechanical analysis, caudoventral hip luxation, UHMWPE implant, ex-vivo, cat

Biomechanical analysis of an extra-articular stabilization using a synthetic implant for caudoventral hip luxation

in a feline cadaver model: preliminary results

Introduction

Coxofemoral luxation is a common problem in small animal practice. Caudoventral hip luxation may occur in 3.2% of cases [START_REF] Brinker | Handbook of small animal orthopedics and fracture treatment[END_REF]. It is mostly observed after minor trauma, such as a jump or fall with a flexed hindlimb in internal rotation and abduction. Surgical management of caudoventral hip luxation consists in a per-operative adaptation of the hip stabilization technique by a cranio-lateral approach initially described to treat craniodorsal luxation, i.e. intra [START_REF] Flynn | Biomechanical Evaluation of a Toggle Pin Technique for Management of Coxofemoral Luxation[END_REF] or extra-articular [START_REF] Johnson | A Retrospective Study of Prosthetic Capsule Technique for the Treatment of Problem Cases of Dislocated Hips[END_REF] synthetic ligament implantation by modifying the implantation sites of the anchoring systems, such as toggle pins, screws and anchors [START_REF] Brinker | Handbook of small animal orthopedics and fracture treatment[END_REF][START_REF] Johnson | A Retrospective Study of Prosthetic Capsule Technique for the Treatment of Problem Cases of Dislocated Hips[END_REF][START_REF] Flynn | Biomechanical Evaluation of a Toggle Pin Technique for Management of Coxofemoral Luxation[END_REF]. The postoperative failure rate is estimated between 11 and 25% of cases [START_REF] Helmick | Evaluation of Modified Techniques for Toggle Pin Repair of Coxofemoral Luxation in Dogs: A Cadaveric Study[END_REF]. To the authors' knowledge, there is no study published on this rare type of hip luxation and its surgical management in cats particularly by ventral approach, whether in the clinical or biomechanical fields. The aim of this study was to compare the biomechanical strength of caudoventral hip luxation for (i) physiologic hip and (ii) stabilized hip with an extraarticular UHMWPE implant secured by an interference screw in a feline cadaver model.

Methods

Sample preparation protocol

Six right hip joints were harvested from six feline cadavers weighing between 3.9 and 4.6 kg. These cats had died from reasons unrelated to this study. Soft tissues were removed, except the hip articular capsule and the femoral head ligament. Femoral parts were transected at the level of distal metaphysis to allow their normalized inclusion into a metallic mold (15x100 mm). Normalized drilling (Ø4 mm) was performed through the distal part of each mold in latero-medial direction, at 115 ± 1.2 mm from the great trochanter.

Implantation of the UHMWPE ligament

After performing a quasi-static traction tensile test on the physiologic hips following biomechanical caudoventral luxation, the cadaveric hip joints were stabilized with a UHMWPE implant (Novalig 2000 Platine, Novetech Surgery, Monaco). A 4-mm wide tunnel was drilled in the center of the pubis, from the ventral to the dorsal direction. The cortical button of the UHMWPE implant was then inserted in the first bone tunnel to fix the pubis (Figure 1a). A second 2.5mm femoral tunnel was drilled at 45° in the frontal plane of the femur through the trochanter, from the medial part of the lesser trochanter to the insertion of the deep gluteal muscle. The second tunnel was tapped diameter 3 mm. The UHMWPE implant was then inserted into the femoral tunnel and tensioned in order to prevent femoral abduction, while maintaining very mild external rotation of the femur. Finally, the implant was secured with an interference screw (3x11mm), (Novetech Surgery, Monaco) implanted mediolaterally through the second tunnel (Figure 1b).

Biomechanical testing

Based on 3D reconstruction from post-mortem CT scans of cat cadavers, six biomechanical bases adapted to cadaveric pelvises were made by 3D printing. This computational design planification facilitated the correct placement and simplified fixation of the hemipelves onto the biomechanical bases in physiologic weight-bearing condition (Figure 2). A metal cable (working length: 73.5 cm) was then inserted and secured through the femoral mold drilling as far as the upper mechanical grip. No pre-loading was performed. The correct initial position of the biomechanical setup was established manually when the femoral mold was parallel to the table of the testing machine (Figure 2). The speed of the traction tensile test was set at 100 mm/min in order to induce coxofemoral luxation [START_REF] Flynn | Biomechanical Evaluation of a Toggle Pin Technique for Management of Coxofemoral Luxation[END_REF]). The traction pull-out test was manually stopped by a veterinary surgeon (E.V.) as soon as caudoventral hip luxation was observed. Given the low angle formed by the metal wire compared to the gravity axis, its impact on the load recorded not taken into account.

Data processing

Linear stiffness was assessed by calculating the slope of the load displacement curve in its linear interval for each tensile test. Yield load was defined as the load at which the first deviation from linearity in the load displacement curve was visually observed. Failure was designed as the maximum force measured during each test. Statistical analyses were performed using nonparametric paired t-tests. A p-value of <0.05 was considered significant. No significant difference was found between the physiologic and stabilized joint groups for yield, failure load and linear stiffness output (Table 1). The use of this type of implant was first described by Goin and colleague in 2019 as part of the reconstruction of the cranial cruciate ligament in dogs [START_REF] Goin | Biomechanical analysis of a ligament fixation system for CCL reconstruction in a canine cadaver model[END_REF].

Results and discussion

The specificity of this surgical technique lies in the use of an interference screw fixation system flush with the bone surfaces, unlike the anchors [START_REF] Brinker | Handbook of small animal orthopedics and fracture treatment[END_REF]. This choice of fixation system would make it possible to avoid possible interference between the fixation system and the periarticular soft tissues in this area of high mobility. The imagined setup could lack stiffness, the specific biomechanical bases could have been performed by titanium 3D printing in order to overcome this potential weakness.

Conclusions

The biomechanical strength of extra-articular hip joint stabilization using a UHMWPE implant secured by an interference screw was not statistically different from the biomechanical strength of a physiologic hip joint tested biomechanically in conditions of caudoventral luxation. The biomechanical fixation strength of the UHMWPE implant should be suitable for extraarticular stabilization by ventral approach of the hip joint after caudoventral hip luxation in cats.

Figure 1 :

 1 Figure 1: Diagram of the extra-articular hip joint stabilization using a UHMWPE implant for caudoventral hip luxation.

Figure 2 :

 2 Figure 2: Biomechanical setup to simulate caudoventral hip luxation in a feline cadaver model.

Table 1 :

 1 Results of the twelve quasi-static tensile tests.

		Linear
	Hip	stiffness	Yield load (N)	Failure load (N)
	joint	(N/mm)
		PHY	STA	PHY	STA	PHY	STA
	N°1	28	21	163	130	163	142
	N°2	25	40	213	203	213	203
	N°3	27	38	181	129	195	129
	N°4	22	36	113	129	113	130
	N°5	32	49	126	199	126	262
	N°6	31	29	156	176	156	176
	Mean	28	36	159	161	161	174
	SD	4	10	36	36	39	52
	P-value	0.107	0.902		0.667
	Abbreviations: PHY, Physiologic; STA, Stabilized
	Biomechanical tests performed on physiologic hip
	joints demonstrated the same failure mode by
	caudoventral hip luxation, causing rupture of the
	femoral head ligament and complete tearing of the hip
	joint capsule at its ventral area (Table 1). For the
	stabilized hip joint group, no rupture of the UHMWPE
	implant was observed. Progressive slippage of the
	UHMWPE implant through the femoral tunnel at the
	bone/interference screw interface was observed during
	each test before failure occurred by caudoventral hip


J Letesson

B Goin

T Cachon

E Viguier

Biomechanical analysis of an extra-articular stabilization using a synthetic implant for craniodorsal hip luxation repair in a feline cadaver model: preliminary results

Keywords: Biomechanical analysis, craniodorsal hip luxation, UHMWPE implant, ex-vivo, cat

Introduction

Coxofemoral luxation is a common traumatic lesion in small animals. The surgical management of hip luxation consists in various hip stabilization techniques, including capsulorrhaphy, extra-articular ilio-femoral suture [START_REF] Meij | Results of extra-articular stabilisation following open reduction of coxofemoral luxation in dogs and cats[END_REF]) secured with anchors [START_REF] Spranklin | Comparison of a Suture Anchor and a Toggle Rod for Use in Toggle Pin Fixation of Coxofemoral Luxations[END_REF]) and intra-articular replacement of the femoral head ligament with a synthetic ligament secured by toggle pins [START_REF] Flynn | Biomechanical Evaluation of a Toggle Pin Technique for Management of Coxofemoral Luxation[END_REF]. The choice of surgical treatment depends on the type of hip luxation, which occurs usually in a craniodorsal direction and more rarely in a caudodorsal or caudoventral direction [START_REF] Brinker | Handbook of small animal orthopedics and fracture treatment[END_REF]). These stabilization techniques and the physiologic hip joint strength that they confer have already been tested biomechanically in dogs but data on cats is scarce [START_REF] Flynn | Biomechanical Evaluation of a Toggle Pin Technique for Management of Coxofemoral Luxation[END_REF][START_REF] Baltzer | Biomechanical analysis of suture anchors and suture materials used for toggle pin stabilization of hip joint luxation in dogs[END_REF]. The aim of this study was to evaluate the biomechanical strength of an extra-articular stabilization technique using a UHMWPE implant secured by an interference screw for craniodorsal hip luxation in a feline cadaver model.

Methods

Sample preparation protocol

Four left hip joints were harvested from four feline cadavers weighing between 3.5 and 4.5 kg. The cats had died of causes unrelated with this study. Soft tissues were removed, except the hip articular capsule and the femoral head ligament. Femoral parts were transected at the level of the distal metaphysis to facilitate their normalized inclusion into a metallic mold (15x100 mm), already assembled with a universal joint (20 mm) in its distal part [START_REF] Flynn | Biomechanical Evaluation of a Toggle Pin Technique for Management of Coxofemoral Luxation[END_REF].

Implantation of the UHMWPE ligament

After performing a controlled 5-mm long incision of the hip joint capsule in the craniodorsal area and transecting the femoral head ligament, cadaveric hip joints were stabilized with a UHMWPE implant (Novalig 2000 Platine, Novetech Surgery, Monaco). A 4-mm wide ilium tunnel was drilled at 10 o'clock dorsally to the origin of the articular coxal muscle and the rectus femoris muscle, close to the joint. The cortical button of the UHMWPE implant was inserted latero-medially and secured to the inner cortical part of the ilium (Figure 1a). A second 2.5-mm bone tunnel was drilled from the level of the greater trochanter in a lateral (center of the origin of the vastus lateralis) to medial (dorsal and close to the neck) direction. A third 2.5-mm bone tunnel was drilled in a latero-caudal to cranial direction of the femoro-proximal metaphysis. The entry point was defined as the caudal part of the vastus lateralis origin, next to the gluteal tuberosity (on the linea aspera), and the exit point was centered on the sagittal plane of the cranial part of the diaphysis (Figure 1b). The third tunnel was tapped (∅3 mm). The UHMWPE implant was passed through the second and third tunnels and tightened with a zero version angle to the hip (mild internal rotation and abduction of the femur) and secured with a 3x11-mm interference screw (Novetech Surgery, Monaco) (Figure 1c).

Biomechanical testing

Cadaveric hemipelves were fixed on specific biomechanical bases made by 3D printing. A 3D reconstruction of cadaveric pelvises from CT scan imaging was used to perform the hip joint placement onto the biomechanical bases in physiologic weightbearing condition (hip joint flexor angle of 120) [START_REF] Guillot | Coxofemoral joint kinematics using video fluoroscopic images of treadmill-walking cats: development of a technique to assess osteoarthritis-associated disability[END_REF]. Femoral diaphyses were placed in adduction (15° angle) to reproduce the condition of physiologic craniodorsal luxation (Figure 2). A 10 mm/min pre-load was applied to reach 10 N before starting the compression test to failure at 100 mm/min, in order to reproduce coxofemoral luxation [START_REF] Flynn | Biomechanical Evaluation of a Toggle Pin Technique for Management of Coxofemoral Luxation[END_REF].

Data processing

Linear stiffness was assessed by calculating the slope of the load displacement curve in its linear interval for each tensile test. Yield load was defined as the load at which the first deviation from linearity in the load displacement curve was observed visually. Failure load was defined as the maximum force measured during each test corresponding to dorsal hip luxation. No rupture of the UHMWPE implant was observed.

Results and discussion

The biomechanical tests led to different outcomes in failure mode (Table 1). There are two possible explanations for this: (i) the universal joint preassembled with the distal femoral mold allowed rotation of the femoral part during the quasi-static tensile compression test; (ii) owing to the location of the UHMWPE implant after extra-articular hip joint stabilization, its placement acted as a non-return pulley in the craniodorsal direction. The femoral heads were therefore expelled out of the acetabulum during the compression tests, in the direction where the mechanical strength was the weakest, i.e. the caudal direction, thus causing caudodorsal hip joint luxation.

Our stabilized hip joint results may be interpreted as the maximum load prior to the re-occurrence of hip luxation in the craniodorsal or caudodorsal direction during the immediate postoperative period. This hip joint stabilization technique can avoid reoccurrence of hip joint luxation with a compression load equivalent to up to four or five times the cat's weight. New biomechanical tests will be performed in order to validate these preliminary findings.

Conclusions

Extra-articular hip joint stabilization for craniodorsal hip luxation in feline cadavers, using an UHMWPE implant secured by an interference screw, produces a biomechanical failure load equivalent to four to five times mean cat weight (171 ± 34 N) prior to the reoccurrence of hip luxation. The biomechanical fixation strength of the UHMWPE implant should be suitable for extra-articular stabilization of the hip joint after craniodorsal hip luxation in cats.

Figure 1 :

 1 Figure 1: Diagram of the extra-articular hip joint stabilization using a UHMWPE implant for craniodorsal hip luxation.

Figure 2 :

 2 Figure 2: Biomechanical setup for stabilized hip with an UHMWPE implant before compression test.

Table 1 :

 1 Results of the four quasi-static tensile compression tests.

	Sample	Linear stiffness (N/mm)	Yield load (N)	Failure load (N)	Failure mode
	N°1	79	185	208	CaD luxation
	N°2	46	138	139	CaD luxation
	N°3	60	192	192	CaD luxation
	N°4	57	146	146	CrD luxation
	Mean	61	165	171
	SD	14	27	34

Abbreviations: CaD, Caudodorsal; CrD, Craniodorsal


B Goin

V Morvan

P Buttin

Y Lafon

M Massenzio

E Viguier

T Cachon

Biomechanical comparison of two femoral fixation methods for synthetic cranial cruciate ligament reconstruction in canine cadavers

Keywords: Biomechanical analysis, cranial cruciate ligament reconstruction, UHMWPE implant, ex-vivo, dog

Introduction

Cranial cruciate ligament rupture (CCLr) is the most common cause of hindlimb lameness in dogs [START_REF] Witsberger | Prevalence of and risk factors for hip dysplasia and cranial cruciate ligament deficiency in dogs[END_REF]. Currently, surgical management of CCLr is mostly performed using tibial osteotomy techniques to modify the biomechanical conformation of the affected stifle [START_REF] Kim | Tibial Osteotomies for Cranial Cruciate Ligament Insufficiency in Dogs[END_REF]). These surgical techniques have a significant complication rate, associated with persistent instability of the stifle which may lead to chronic postoperative pain. Over the last decade, studies have been published on various techniques of anatomical CCL reconstruction in veterinary practice, using physiological autografts or woven synthetic implants. In most techniques, the latter are secured to the bone by interference screws. High fixation strength is mandatory to achieve satisfactory clinical outcomes. A study published by Blanc and al. in 2019 reported no statistical difference in maximum pull-out strength between a physiological CCL and an ultra-highmolecular-weight polyethylene (UHMWPE) implant secured by four interference screws (two in the femoral part and two in the tibial part), highlighting the biomechanical potential of this synthetic CCL reconstruction technique [START_REF] Blanc | Effect of the number of interference screws for the fixation of an intra-articular cranial cruciate ligament prosthesis in dogs: Biomechanical study[END_REF]. Recently, Rafael et al. have shown that fixation with only two interference screws (one in the femoral part and another one in the tibial part), associated with a new surgical implantation technique, provided a level of biomechanical strength compatible with synthetic CCL reconstruction in dogs [START_REF] Rafael | Comparison of two methods of fixation with interference screw for cranial cruciate ligament reconstruction in canine cadaver model[END_REF]. The weakest point reported in these two studies is the tibial fixation part, with slippage of the implant at the bone / UHMWPE implant / interference screw interface. Owing to this slippage observed on complete assembly (femur and tibia), the mechanical pull-out strength of the femoral fixation could not be defined [START_REF] Goin | Biomechanical analysis of a ligament fixation system for CCL reconstruction in a canine cadaver model[END_REF][START_REF] Rafael | Comparison of two methods of fixation with interference screw for cranial cruciate ligament reconstruction in canine cadaver model[END_REF]. The aim of this study was to compare the pull-out strength of two femoral fixation methods used in CCL reconstruction with an UHMWPE implant on canine cadavers.

Methods

Sample preparation protocol

Eight femurs were obtained from four large-breed adult dogs weighing between 35 and 45 kg and were dissected to leave only the femoral part intact. Each femur was transected at the diaphysis level to facilitate its placement into a metallic mold (7x3x3 cm) filled with resin [START_REF] Goin | Biomechanical analysis of a ligament fixation system for CCL reconstruction in a canine cadaver model[END_REF].

Implantation of the UHMWPE ligament

For each femoral sample, a 4-mm wide tunnel was drilled from the caudo-lateral femoral insertion of the physiologic CCL to the distolateral femoral metaphysis. The empty tunnels were pre-formed by an interference screw (Ø5 x 20 mm). The eight femoral samples were then randomly assigned to two femoral fixation groups (n=4/group). Group A: A standard UHMWPE implant (Novalig 4000, Novetech Surgery, Monaco) was inserted through the pre-formed tunnel and secured by an interference screw (Ø5x20 mm) (Novetech Surgery, Monaco), implanted following the "In-Out" surgical technique (Figure 1a) [START_REF] Rafael | Comparison of two methods of fixation with interference screw for cranial cruciate ligament reconstruction in canine cadaver model[END_REF]. Group B: A UHMWPE implant pre-assembled with a cortical button (Novalig 4000 Platine, Novetech Surgery, Monaco) was secured by an interference screw in the same way as in group A (Figure 1b).

Biomechanical testing

Eight pull-out quasi-static tensile tests were performed following the same mechanical protocol [START_REF] Goin | Biomechanical analysis of a ligament fixation system for CCL reconstruction in a canine cadaver model[END_REF]. The samples were pre-loaded at 10N (20mm/min) before starting the pull-out failure test at 1 mm/min.

Data processing

Linear stiffness was assessed by calculating the slope of the load displacement curve in its linear interval for each tensile test. Yield load was defined as the load at which the first deviation from linearity in the load displacement curve was observed. Failure load was defined as the maximum force measured during each test. Statistical analyses were performed using nonparametric paired t-tests. A p-value of <0.05 was considered significant. No rupture of the fixation system was observed. Failure load outcome corresponds to the loss of functionality of the implants and not to their rupture. Progressive slippage of the UHMWPE implant occurred at the bone / UHMWPE implant / interference screw interface in group A, associated with 6.2 ± 2.2 mm of displacement for the failure load output parameter. In group B, progressive slippage associated with damage of the UHMWPE implant and the cortical button was observed around 800 N during pull-out tests, before failure occurred at 12.2 ± 5.2 mm. Yield, failure load and linear stiffness output parameters were higher in group B than in group A. A significant difference was observed for the failure load output parameter (P-value = 0.018). Interestingly, the use of an UHMWPE implant pre-assembled with a cortical button secured by an interference screw increased the mechanical strength of the femoral fixation. Based on physiologic CCL failure load results reported by Blanc et al. with the same biomechanical protocol (Blanc et al. 2019), no statistical difference was found with the failure load output parameter of group B (Wilcoxon rank sum test, P-value = 0.412), while a statistical difference was observed with group A (P-value = 0.015). According to Rafael et al., the best way to maximize the fixation strength of synthetic CCL reconstruction in dogs is to associate an UHMWPE implant pre-assembled with a cortical button, secured by a first interference screw implanted following the "In-Out" surgical technique through the femoral part, and a second interference screw implanted following the same surgical procedure in the tibial part [START_REF] Rafael | Comparison of two methods of fixation with interference screw for cranial cruciate ligament reconstruction in canine cadaver model[END_REF]. Biomechanical pull-out tensile tests will be performed to confirm these findings.

Results and discussion

Conclusions

The two femoral fixations tested in this study provide satisfactory pull-out strength compatible with synthetic CCL reconstruction in dogs. This study shows that the use of a UHMWPE implant preassembled with a cortical button secured with an interference screw increases mechanical strength compared with a standard UHMWPE implant secured by only one interference screw. No statistical failure load difference was found between the femoral fixation of the UHMWPE implant preassembled with a cortical button secured with an interference screw and the physiologic canine CCL.

Figure 1 :

 1 Figure 1: Diagram of the two femoral fixation groups tested

Figure 2 :

 2 Figure 2: Biomechanical setup of an implanted femoral sample placed in the testing machine.

Table 1 :

 1 Results of the eight quasi-static tensile tests

	Dog	Linear stiffness (N/mm)	Yield load (N)	Failure load (N)
		Gr A	Gr B	Gr A	Gr B	Gr A	Gr B
	N°1	144	156	372	620	376	675
	N°2	124	92	284	1224	394	1242
	N°3	116	213	358	631	576	1071
	N°4	90	124	616	795	617	1122
	Mean	119	146	408	818	491	1028
	SD	22	52	144	283	123	246
	P-value	0.377		0.104		0.018


N Giraud

B Goin

Y Lafon

M Massenzio

E Viguier

T Cachon

Research of a new synthetic implant fixation interface material used for cranial cruciate ligament repair in dogs: The choice of Nylon?

Keywords: Biomechanical analysis, cranial cruciate ligament reconstruction, UHMWPE implant, ex-vivo, dog

Introduction

The cranial cruciate ligament (CCL) rupture is one of the most common orthopedic affections in dogs [START_REF] Hayashi | Cranial Cruciate Ligament Pathophysiology in Dogs With Cruciate Disease: A Review[END_REF]). Nowadays, the surgical management of this orthopedic disorder concerns a tibial osteotomy to modify the biomechanical conformation of the stifle [START_REF] Von Pfeil | Results of a survey of Veterinary Orthopedic Society members on the preferred method for treating cranial cruciate ligament rupture in dogs weighing more than 15 kilograms (33 pounds)[END_REF]). However, these last surgical treatments do not respect the physiological insertions of the CCL which could be one of the explanations for the long-term complications of these techniques [START_REF] Krotscheck | Long Term Functional Outcome of Tibial Tuberosity Advancement vs. Tibial Plateau Leveling Osteotomy and Extracapsular Repair in a Heterogeneous Population of Dogs: Long Term Outcome of TTA in Dogs[END_REF]. During last decade, a renewed interest has been observed in the field of veterinary surgery concerning intra-articular reconstruction of CCL techniques, either by allograft, or by synthetic implant. The challenge of these two types of CCL reconstructions relies on finding a fixation technique respecting the stabilization function of the stifle and its biocompatibility over the long term. In particular, the synthetic CCL reconstructions using Ultra-High Molecular Weight Polyethylene (UHMWPE) implants yields to excellent results regarding the biocompatibility [START_REF] Smith | Intra-Articular Biocompatibility of Multistranded, Long-Chain Polyethylene Suture Tape in a Canine ACL Model[END_REF]. And the recent published studies on improving the fixation system underline promising results [START_REF] Blanc | Effect of the number of interference screws for the fixation of an intra-articular cranial cruciate ligament prosthesis in dogs: Biomechanical study[END_REF][START_REF] Goin | Biomechanical analysis of a ligament fixation system for CCL reconstruction in a canine cadaver model[END_REF][START_REF] Rafael | Comparison of two methods of fixation with interference screw for cranial cruciate ligament reconstruction in canine cadaver model[END_REF]. Following the same research field, authors hypothesize that the use of a fixation interface between the interference screw and the bone could increase the pull-out strength of the fixation system. The aim of this ex-vivo biomechanical study was to test this hypothesis with a titanium interference screw inserted through a nylon dowel as femoral fixation used for synthetic reconstruction of the CCL in dogs.

Methods

Sample preparation protocol

Eight stifles were harvested from 4 cadaveric dogs between 25 and 35 kg. All dogs were of similar breed and died from reasons unrelated with this study. Femoral parts were disarticulated from the tibias and all soft tissues were removed to let intact only the femur bones. Then, femoral parts were transected at the level of diaphysis to allows its inclusion into a metallic mold (7x3x3 cm) filled with resin [START_REF] Goin | Biomechanical analysis of a ligament fixation system for CCL reconstruction in a canine cadaver model[END_REF].

Implantation of the UHMWPE ligament

A femoral tunnel (Ø6 mm) was drilled from the caudolateral femoral insertion of the physiologic CCL. The UHMWPE implant (Novalig 8000, Novetech Surgery, Monaco) was passed through the tunnel. Then, a nylon dowel (Ø6 mm, 30 mm-long) (Fischer, Strasbourg, France) was inserted into the tunnel from the distolateral femoral metaphysis to intra-articular area. Finally, an interference screw (Ø5mm, 20 mm-long) was implanted inside the nylon dowel following the same insertion protocol than the nylon dowel (Fig. 1).

Biomechanical testing

The static tensile test protocol and biomechanical setup were inspired by [START_REF] Goin | Biomechanical analysis of a ligament fixation system for CCL reconstruction in a canine cadaver model[END_REF]. The samples were pre-loading at 10N (20 mm/min), before started the traction pull-out test to failure at 1 mm/min. The femoral samples were maintained proximally into a stirrup by an 8-mm pin through the metallic mold, while the UHMWPE implants was inserted into a mechanical grip mounted on the mobile traverse of the traction machine (Fig. 1).

Figure 1: Biomechanical setup of an implanted femoral sample and placed into the testing machine.

Data processing

Linear stiffness was defined by calculating the slope of the load displacement curve in its linear interval for each tensile test. Failure load was designed as the maximum force measured during each test. Load for 3 mm of displacement has been chosen as the output parameter according a previous study [START_REF] Rafael | Comparison of two methods of fixation with interference screw for cranial cruciate ligament reconstruction in canine cadaver model[END_REF]. No rupture of the UHMWPE implant was reported.

Results and discussion

The failure mode was the same among each test: progressive slippage of the UHMWPE implant through the femoral tunnel at the bone, UHMWPE implant, nylon dowel (dilatated by an interference screw) interface. The dispersion of the results for the output parameter L3mm shows that the femoral fixation system does not reported constant and homogeneous biomechanical strengths for all performed tests. Results for each biomechanical output parameter studied (Table 1) were lower than previous studies [START_REF] Blanc | Effect of the number of interference screws for the fixation of an intra-articular cranial cruciate ligament prosthesis in dogs: Biomechanical study[END_REF][START_REF] Rafael | Comparison of two methods of fixation with interference screw for cranial cruciate ligament reconstruction in canine cadaver model[END_REF]. A strict comparison with these results was not possible because tensile tests were performed on a complete fixation system of the UHMWPE implant (femoral and tibial part) [START_REF] Blanc | Effect of the number of interference screws for the fixation of an intra-articular cranial cruciate ligament prosthesis in dogs: Biomechanical study[END_REF][START_REF] Rafael | Comparison of two methods of fixation with interference screw for cranial cruciate ligament reconstruction in canine cadaver model[END_REF], while in the present study, only femoral fixation was tested. However, the failure mode reported in these studies [START_REF] Blanc | Effect of the number of interference screws for the fixation of an intra-articular cranial cruciate ligament prosthesis in dogs: Biomechanical study[END_REF][START_REF] Rafael | Comparison of two methods of fixation with interference screw for cranial cruciate ligament reconstruction in canine cadaver model[END_REF]) was in agreement with our results. We can classified the utilization of an nylon dowel dilatated by an interference screw for synthetic LCC fixation system as the lower pull-out strength than only interference screws secured the UHMWPE implant following "Out-In" surgical technique i.e. maximum strength reported (335 ± 59 N) [START_REF] Blanc | Effect of the number of interference screws for the fixation of an intra-articular cranial cruciate ligament prosthesis in dogs: Biomechanical study[END_REF]. Finally, the higher pull-out strength results were obtained with interference screws secured the UHMWPE implant following "In-Out" technique ie maximum strength reported (509 ± 124 N) [START_REF] Rafael | Comparison of two methods of fixation with interference screw for cranial cruciate ligament reconstruction in canine cadaver model[END_REF].

Conclusions

This study shows that the utilization of a nylon dowel interface dilatated by an interference screw against an UHMWPE implant does not allow to obtain satisfactory pull-out strength compatible with LCC synthetic reconstruction for dogs.

Table 1 :

 1 Results of the eight quasi-static tensile tests.

		Load for 3	Maximum strength
	Sampl e	mm of displacem -ent (N) (L3mm)	Load (N)	(MS) Displacem-ent (mm)	Linear stiffness (N/mm)
	N°1	124	508	9,6	60
	N°2	41	517	10,2	88
	N°3	159	325	5,2	75
	N°4	125	279	6,3	47
	N°5	59	336	8,2	79
	N°6	221	563	6	134
	N°7	191	226	4,3	45
	N°8	82	411	8,4	70
	Mean	125	396	7,3	75
	SD	63	123	2,1	28


Hakki Ulaş Ünal
email: huunal@eskisehir.edu.tr

Marc R Roussel
email: roussel@uleth.ca

Islam Boussaada
email: islam.boussaada@l2s.centralesupelec.fr

Silviu-Iulian Niculescu
email: silviu.niculescu@l2s.centralesupelec.fr

Insights from a qualitative analysis of a gene expression model with delays

Keywords: Delay systems, biochemical systems, gene-regulatory systems

Delays appear in the dynamics of many systems due to non-vanishing reaction times of control systems. In biochemical systems, long sequences of repeated steps, especially in biopolymerization processes, can be modeled by delays. However, modelling systems with delays is often complicated by physical constraints, such as the requirement that solutions representing concentrations of chemical species remain positive. In this work, we consider a model for a detoxifying enzyme whose synthesis is controlled by its substrate. The model includes bindingsite clearance delays, caused by the time required for an RNA polymerase or ribosome to clear its binding site before another such machine can bind. The existence of a positive equilibrium and the positivity and boundedness of solutions of the corresponding delay-differential equations are proven. In addition, the stability of the model is studied using the "small-gain" theorem.

INTRODUCTION

Delays appear in a wide variety of models, from engineering to biological systems, due to transport and propagation phenomena (see [START_REF] Niculescu | Delay Effects on Stability: A Robust Control Approach[END_REF] and references therein). In addition to "physical" delays, delays may be introduced into a model to reduce the number of variables required to describe the system, sometimes leading to insights regarding the dynamical roles of some of the variables [START_REF] Epstein | Differential delay equations in chemical kinetics. Nonlinear models: The cross-shaped phase diagram and the Oregonator[END_REF][START_REF] Roussel | The use of delay differential equations in chemical kinetics[END_REF][START_REF] Ünal | A delay based sustained chemical oscillator: Qualitative analysis of Oregonator based models[END_REF]. However, since systems with time delays are in the class of infinite-dimensional systems [START_REF] Curtain | An Introduction to Infinite-Dimensional Linear Systems Theory[END_REF], their analysis is not an easy task. For the sake of simplicity, "sufficiently small" delays are often neglected. Since delays may however induce unexpected behaviours, neglecting small delays is not without risk [START_REF] Niculescu | Delay Effects on Stability: A Robust Control Approach[END_REF].

The law of mass-action, which states that the rate of an elementary reaction is proportional to the product of the reactant concentrations [START_REF] Waage | Studies concerning affinity[END_REF], is the foundation for modeling biochemical systems. This leads to models expressed by polynomial ordinary differential equations (ODEs) with positive solutions for posi-MRR's work is supported by the Natural Sciences and Engineering Research Council of Canada.

tive initial conditions [START_REF] Horn | General mass action kinetics[END_REF]Vol'pert, 1972). However, when modeling gene expression, delaydifferential equations (DDEs) are often used to take into account the significant time required for transcription and translation. It is well known that delays can have important qualitative and quantitative effects on the dynamics. However, it remains to be seen if it is necessary to include some of the smaller delays associated with the clearance of binding sites for the molecular machines that carry out biopolymerizations (RNA polymerase, ribosome) found in some models [START_REF] Roussel | Validation of an algorithm for delay stochastic simulation of transcription and translation in prokaryotic gene expression[END_REF]Zhu and Salahub, 2008). In a recent study, it was found that small bindingsite delays can have dramatic effects on bifurcation diagrams of gene expression models that cannot be compensated for by, e.g., adjusting the rate constant for initiation [START_REF] Trofimenkoff | Small binding-site clearance delays are not negligible in gene expression modeling[END_REF]. As these delays are almost always neglected in gene expression models, it seems worthwhile to explore these effects further, using a different model with a different bifurcation structure.

Delayed mass-action kinetics provides a convenient framework for modeling gene expression delays [START_REF] Roussel | The use of delay differential equations in chemical kinetics[END_REF] and forms the basis for this work. This formalism was originally formulated to avoid the unphysical negative solutions that plague some DDE models. A formal proof of the positivity of solutions of delayed mass-action models was however only recently obtained [START_REF] Lipták | Semistability of complex balanced kinetic systems with arbitrary time delays[END_REF].

In this work, a model for an inducible gene is considered to understand the effects of both standard synthesis (transcription and translation) and binding-site clearance delays. First, the occurrence of a single positive equilibrium for the model will be proven. Then, positivity and boundedness of the solutions of the corresponding DDEs will be discussed. Using methods of asymptotic stability analysis, a complete stability picture of the model will be presented. Numerical examples will illustrate the results, which suggest a stabilizing effect of binding-site clearance.

Notation and Preliminaries

Throughout the paper, C, R + , and Z + are, respectively, the sets of complex numbers, of positive real numbers, and of the positive integers. R n ≥0 represents the non-negative orthant of R n , n ∈ Z + . For x ∈ R n , x is called positive (non-negative) and bounded if x l > 0(≥ 0) and

x l ≤ M , for some M , l = 1, 2, . . . , n. C o := {λ ∈ C | R(λ) > 0},
where R(λ) is the real part of λ, and

C + := {λ ∈ C | R(λ) ≥ 0} is called the rhp (right half-plane), for short.
H ∞ is the space of analytic and bounded functions in

C o . A function Φ(λ; τ ) ∈ H ∞ , τ ∈ R n ≥0 , called strictly proper in C o , if lim ω→∞ |Φ(iω; τ )| = 0.

INDUCIBLE-GENE MODEL AND RELATED ASSUMPTIONS

We study a model of a gene induced by its own substrate. This is a common regulatory relationship in gene expression systems. Examples include: the lac operon, in which several genes related to the metabolism of lactose are induced, albeit indirectly, by lactose [START_REF] Beckwith | The lactose operon[END_REF]; and Hmp, the enzyme chiefly responsible for detoxifying nitric oxide (NO) in bacteria [START_REF] Robinson | A kinetic platform to determine the fate of nitric oxide in Escherichia coli[END_REF], whose synthesis is induced is a fairly direct way by the presence of NO in the cell [START_REF] Poole | New functions for the ancient globin family: Bacterial responses to nitric oxide and nitrosative stress[END_REF][START_REF] Cruz-Ramos | NO sensing by FNR: Regulation of the Escherichia coli NO-detoxifying flavohaemoglobin, Hmp[END_REF]. The model described here is based on an earlier, highly simplified model [START_REF] Roussel | Approximating state-space manifolds which attract solutions of systems of delaydifferential equations[END_REF] of a gene induced by its own substrate. Here we formulate a delayed-mass action version of the model in order to study the effects of the binding-site clearance delays for both the promoter and ribosome binding site (RBS):

κin --→ S(t) (C1) E(t) + S(t) κ1 ----- κ-1 C(t) κ-2 --→ E(t) + Q(t) (C2) E(t) κ d --→ (C3) n S(t) + A(t) κ3 ----- κ-3 P(t) (C4) P(t) κ4 --→ P(t + τ 1 ) + R(t + τ 2 ) (C5) R(t) κ5 --→ R(t + τ 3 ) + E(t + τ 4 ) (C6) R(t) κ6 --→ (
C7) In this model, t denotes time. S is a substrate to be metabolized to a product Q by the enzyme E. E is synthesized by the usual gene expression pathway, namely transcription (C5) and translation (C6). The active gene promoter, P, is cleared and the RNA, R, appears τ 1 and τ 2 time units after transcription initiation respectively. Similarly, the RBS is cleared and E appears τ 3 and τ 4 time units after translation initiation, respectively. The delays τ 1 and τ 3 are associated with the clearance of, respectively, the RNA polymerase binding site in the promoter region of the gene, and the RBS of the messenger RNA by the corresponding biomolecular machines. These binding-site clearance delays would usually be short, for transcription in prokaryotes of the order of a few seconds [START_REF] Lutz | Dissecting the functional program of Escherichia coli promoters: The combined mode of action of Lac repressor and AraC activator[END_REF][START_REF] Lloyd-Price | Dissecting the stochastic transcription initiation process in live Escherichia coli[END_REF] but sometimes up to tens of minutes [START_REF] Hsu | Promoter clearance and escape in prokaryotes[END_REF]Aye-Han and Hsu, unpublished), and in eukaryotes of the order of ten minutes [START_REF] Kugel | A kinetic model for the early steps of RNA synthesis by human RNA polymerase II[END_REF]Darzacq et al., 2007), which is still a short time compared to the overall transcription time, particularly in mammals due to the very long genes [START_REF] Venter | The sequence of the human genome[END_REF] and slow splicing [START_REF] Singh | Rates of in situ transcription and splicing in large human genes[END_REF].

The activation of the gene by the substrate would normally involve binding or modification of a regulatory protein by the substrate, although we do not represent this binding explicitly, nor do we differentiate between the case in which binding of S to the regulatory protein derepresses the gene and the case in which S binds to a protein that activates the promoter, resulting in the effective activation reaction (C4). The cooperativity implied by reaction (C4) likewise can be thought of as a simple model of any of a variety of processes that result in sigmoidal activation kinetics [START_REF] Ferrell | Tripping the switch fantastic: how a protein kinase can convert graded inputs into switchlike outputs[END_REF][START_REF] Ferrell | How responses get more switchlike as you move down a protein kinase cascade[END_REF][START_REF] Ferrell | How regulated protein translocation can produce switch-like responses[END_REF].

For simplicity, and because we want to particularly focus on the control system, we treat the transformation of S to Q as a simple Michaelis-Menten reaction (C2), although it will usually be the case that metabolizing S will involve other substrates.

We define the following dimensionless variables:

c( t) := C(t)/K M , e( t) := E(t)/K M , s( t) := S(t)/K M , a( t) := A(t)/K M , p( t) := P (t)/K M , r( t) := R(t)/K M , t := γt,
where K M := γ/κ 1 is the Michaelis constant, and

γ := (κ -1 + κ -2 ).
(1)

We also define the dimensionless parameters

k l = κ l /γ, where l ∈ {-1, -3, 4, 5, 6}, k in = κ in /(K M γ), k d = κ d /γ, k 3 = κ 3 K n M /γ, τ i := γτ i , i ∈ {1
, 2, 3, 4}. Note that, physically, all of the dimensionless parameters must be positive quantities. Applying the delayed mass-action construction yields DDEs for the concentrations which, using the dimensionless variables and parameters defined above, become

d dt c(t) = e(t)s(t) -c(t), (2a)
d dt e(t) = -e(t)s(t)-k d e(t) + c(t) + k 5 r(t -τ 4 ), (2b)
d dt s(t) = k in -e(t)s(t)+k -1 c(t)-nk 3 s(t) n a(t) +nk -3 p(t), (2c) d dt a(t) = -k 3 s(t) n a(t) + k -3 p(t), (2d)
d dt p(t) = k 3 s(t) n a(t) -(k -3 + k 4 )p(t) + k 4 p(t -τ 1 ), (2e)
d dt r(t) = -(k 5 + k 6 )r(t) + k 4 p(t -τ 2 ) + k 5 r(t -τ 3 ),
(2f) where the tildes have been dropped for notational simplicity. The solutions of these equations obey the following dimensionless conservation equation

z = a(t) + p(t) + k 4 t t-τ1 p(θ) dθ.
(3)

We can see that z is a conserved quantity by differentiating equation ( 3) with respect to time and applying the fundamental theorem of calculus, which gives

dz dt = da dt + dp dt + k 4 p(t) -k 4 p(t -τ 1 ).
Subtitution of equations ( 2d) and (2e) into the latter gives dz/dt = 0, thus proving that z is a conserved quantity, namely the dimensionless equivalent of the total concentration of promoter. Note that for any physically sensible initial function, since concentrations must be positive, z must be a positive parameter.

QUALITATIVE ANALYSIS

In this section, first, dynamic constraints such as presence of a positive equilibrium, positivity and boundedness of solutions under the positive and bounded initial functions will be considered. Then, asymptotic stability properties of the model will be presented.

Physical Constraints

In open systems governed by the law of mass-action, there are no guarantees with respect to the existence or uniqueness of positive equilibria. Indeed, it is easy to write down models that have runaway solutions. These issues are even trickier in delay systems. Let x(t) := [ c(t) e(t) s(t) a(t) p(t) r(t) ]

T and x * be the corresponding equilibrium point of the model. Then, from (2) and (3),

x * = β β/S * S * K 3 βα/(S * ) n+1 βα/S * k d β/(k 5 S * ) T , (4) where
α := k 6 k d /(k 4 k 5 ), β := k in /(1 -k -1 ), K 3 = k -3 /k 3 , and S * is the positive solution of (z/(αβ)) (S * ) n+1 -(k 4 τ 1 + 1)(S * ) n -K 3 = 0.
(5) Remark 1. Note that the equilibrium point depends on the value of the promoter clearance delay τ 1 . Thus, even the position of the equilibrium point depends on this small delay. Remark 2. Since the coefficients of the polynomial (5) in variable S * are real and positive, and the number of sign changes is 1, by Descartes' rule, the positive solution of ( 5) is unique. Proposition 3. Let M := (K 3 + k 4 τ 1 + 1)αβ/z. Then, the positive solution of ( 5) is bounded as follows:

For

M > 1, 1 < S * < M ; (6a) for M < 1, M < S * < 1; (6b) for M = 1, S * = 1. ( 6c
)
Proof. Equation ( 5) can be rearranged to

(S * ) n (S * -M ) = K 3 (αβ/z)(1 -(S * ) n ).
(7) Since (5) has a positive solution S * and M > 0, then the left-hand side of ( 7) is negative iff S * < M , and the right-hand side is negative iff S * > 1, which proves (6a). Similarly, the left-hand side of ( 7) is positive iff S * > M , and the right-hand side is positive iff S * < 1, which proves (6b). Finally, the left-hand side passes through zero when S * = M and the right-hand side passes through zero when S * = 1, which proves (6c). Theorem 4. Let us consider the following DDE

d dt x(t) = -α 0 x(t) + N l=1 α l x(t -τ l ) + Q(t), (8)
which is supposed to have non-negative solutions for nonnegative initial functions defined on (-max l=1,...,N τ l , 0], with α 0 > 0, α l ≥ 0 and τ l ≥ 0, l = 1, . . . , N , and Q(t) non-negative and bounded. Then, the solution of ( 8) is bounded if

α 0 > N l=1 α l .
Proof. Note that (8) can be written as

d dt x(t) + N l=1 α l t t-τ l x(s)ds = -α 0 + N l=1 α l x(t) + Q(t). (9)
Now, suppose that α 0 > N l=1 α l and that the solution is unbounded. Then, there exists a time-sequence

{t n } → ∞ as n → ∞ such that d dt x(t) + N l=1 α l t t-τ l x(s)ds t=tn > 0,
which implies by (9),

Q(t n ) > α 0 - N l=1 α l x(t n ). ( 10
)
However, (10) holds only if the solutions are non-negative. Thus, since Q(t) is assumed to be non-negative and bounded, non-negative solutions cannot be unbounded. Proposition 5. The solutions of the DDEs (2) are nonnegative and bounded whenever the initial history functions are non-negatively valued and bounded.

Proof. Non-negativity follows from the proof of [START_REF] Lipták | Semistability of complex balanced kinetic systems with arbitrary time delays[END_REF] that the solutions of delayed mass-action systems are non-negative when the initial functions are non-negative. By (3), since z is constant, a(t) and p(t) are bounded. Then, by Theorem 4, r(t) is bounded. Moreover, from (2e) it can be shown that

p(t) = -k 4 t t-τ1 p(ν)dν -k -3 t to p(ν)dν + p(t o ) +k 4 to to-τ1 p(ν)dν + t to k 3 s(ν) n a(ν)dν.
Then, since p(t) is bounded, s(t) cannot be unbounded. Now, assume that c(t) is unbounded. Then, there exist a sequence

{t k } → ∞ as k → ∞ such that c(t k ) <c(t k+1 ), k ∈ Z + .
(11) Then, since p(t), s(t) and a(t) are bounded and nonnegative, and e(t) is also non-negative, ( 11) and (2c) together would imply that d dt s(t k ) > 0 for k > N , where N is sufficiently large. However, this contradicts the boundedness of s(t), so c(t) must be bounded. Now, assume that e(t) is unbounded. Then, there exists a sequence

{t k } → ∞ as k → ∞ such that d dt e(t k ) > 0. Then, from (2b), c(t k ) + k 5 r(t k -τ 4 ) > e(t k )(k d + s(t k )), k ∈ Z + .
However, since c(t) and r(t) are non-negative and bounded, by the above inequality, e(t k ), k ∈ Z + , cannot be unbounded.

Asymptotic Stability Properties

Let δx(t) := x(t) -x * , where x * is given in (4). Then, linearization of (2a)-(2f) around x * results in

δ x(t) = Aδx(t) + 4 i=1 A i δx(t -τ i ), ( 12
)
where

A =       -1 S * β/S * 0 0 0 1 -k d -S * -β/S * 0 0 0 k -1 -S * -β/S * -ng -nk 3 S * n nk -3 0 0 0 -g -k 3 S * n k -3 0 0 0 g k 3 S * n -k 4 -k -3 0 0 0 0 0 0 -k 5 -k 6       , A 1 =   0 4×4 0 0 k 4 0   , A 2 = 0 5×4 0 0 k 4 0 , A 3 = 0 5×5 0 0 k 5 , A 4 =   0 0 k 5 0 4×5 0   , and g = αk -3 nβ/S * .
Then, the characteristic function of ( 12) can be written as

∆(λ; τ ) = λ (∆ 1 (λ)Q 2 (λ; τ 1 ) + k -3 ∆ 1s (λ)) Q 3 (λ; τ 3 ) + (k 6 k d k -3 βn/S * )λ (λ + 1 -k -1 ) e -τ T λ (13) where τ = (τ 1 , τ 3 , τ T ) ∈ R 3 ≥0 with τ T = τ 2 + τ 4 , (14) and ∆ 1 (λ) = (λ + S * n k 3 )∆ 1s (λ) + (gn/S * )λQ 1 (λ), (15a) ∆ 1s (λ) = λQ 1 (λ) + β(λ + k d )(λ + 1 -k -1 )/S * , (15b) Q 1 (λ) = k d + λ(1 + S * + k d ) + λ 2 , (15c) Q 2 (λ; τ 1 ) = 1 + (k 4 /λ) 1 -e -λτ1 , (15d)
Q 3 (λ; τ 3 ) = λ + k 6 + k 5 (1 -e -τ3λ
). (15e) Remark 6. Note that the stability of the equilibrium point depends on the values of the clearance delays τ 1 and τ 3 as well as on the total expression delay τ T .

Theorem 7. Let f (λ; τ ) ∈ H ∞ , where τ ∈ R n ≥0 and sup ω∈R |f (iω; τ )| < 1 for all τ ∈ R n ≥0 . Then (1 + f (λ; τ )) has no zeros in C + .
Proof. It is a consequence of the "small-gain" theorem, see [START_REF] Translated | Delay stochastic simulation of single-gene expression reveals a detailed relationship between protein noise and mean abundance[END_REF]. Proposition 8. Let Φ(λ; τ ) be a strictly proper function in

C o and suppose that |Φ(0; τ )| > 1, τ ∈ R m ≥0 .
Then, there exists some positive τ T such that ∆(λ; τ, τ T ) := 1 + e -λτ T Φ(λ; τ ) has a zero at iω o , where ω o > 0.

Proof. Note, ∆(iω; τ, τ T ) = 0 for some positive ω iff e -iωτ T Φ(iω; τ ) = -1. Then, since |Φ(jω; τ )| → 0 as ω → ∞ and it is supposed that |Φ(0; τ )| > 1, τ ∈ R m ≥0 , there always exists a finite positive ω o such that

|Φ(iω o ; τ )| = 1. Then, ∆(iω o ; τ, τ T ) = 0 for τ T = ((2l -1)π -arg(Φ(iω o ; τ ))) /ω o )
, where l is a some positive integer ensuring the positivity of τ T . Proposition 9. The rightmost real zero of the characteristic function is λ = 0 and it is simple.

Proof. Note, since λ = 0 is removable singularity of (1 -e -τ1λ )/λ, from (13), it is obvious that ∆(0; τ ) = 0, τ ∈ R m ≥0 . In addition, since d dρ ∆(ρ; τ ) is strictly positive for ρ ≥ 0, 0 is simple and the right-most real zero of ∆(λ; τ ), τ ∈ R m ≥0 . Proposition 10. Assume that 2k 4 τ 1 < 1 and

k -3 / k 3 S * n 1 -2k 4 τ 1 < 1. ( 16
)
Then,

G(λ; τ 1 ) = ∆ 1 (λ)Q 2 (λ; τ 1 ) + k -3 ∆ 1s (λ) (17)
has no zeros in the rhp.

Proof. From (15a) and (15b), ∆ 1 (λ) and ∆ 1s (λ) are respectively 4 th and 3 rd order polynomials with positive real coefficients. Then, by the Hurwitz-criterion, it can be shown that ∆ 1 (λ) has no zeros in C + , therefore

H(λ) := ∆ 1s (λ)/∆ 1 (λ) is analytic in C o . In addition, since it can be shown that ∂ ∂ω |H(iω)| 2 < 0, ω ∈ R, then, sup ω∈R |H(iω)| = |H(0)| = (S * n k 3 ) -1 . Thus, H(λ) ∈ H ∞ . As discussed above, Q 2 (λ; τ 1 ) := (1 -e -τ1λ )/λ is analytic in C o . In addition, since sup ω∈R Q 2 (iω; τ 1 ) = τ 1 and k 4 τ 1 < 1/2, by Theorem 7 and from (15d), Q 2 (λ; τ 1 ) has no zeros in C o . Therefore, G 1 (λ; τ 1 ) := H(λ)/Q 2 (λ; τ 1 ) is analytic in C o . Since |Q 2 (iω; τ 1 )| 2 = 1 + k 4 τ 1 sin(ωτ 1 ) (ωτ 1 ) 2 + 2k 4 τ 1 sin(ωτ 1 ) (ωτ 1 ) + k 4 τ 1 1 -cos(ωτ 1 ) ωτ 1 2 ≥ 1 -2k 4 τ 1 , (18) then sup ω∈R |G 1 (iω; τ 1 )| ≤ sup ω∈R |H(iω)| 1 inf ω∈R |Q 2 (iω; τ 1 )| ≤ k -3 √ 1 -2k 4 τ 1 S * n k 3 , (19)
which ensures that G 1 (λ; τ 1 ) ∈ H ∞ . Furthermore, since it is assumed that k -3 / k 3 S * n √ 1 -2k 4 τ 1 < 1, then, by Theorem 7, 1 + G 1 (λ; τ 1 ) has no zeros in the rhp for all non-negative τ 1 . Then, since (17) can be written as

G(λ; τ 1 ) = Q 2 (λ; τ 1 )∆ 1 (λ) (1 + G 1 (λ; τ 1 )
) , G(λ; τ 1 ) has no zeros in the rhp for all non-negative τ 1 . Proposition 11. Suppose that k -3 / k 3 S * n √ 1 -2k 4 τ 1 < 1 and 2k 4 τ 1 < 1. Then, ∆(λ; τ ) has a zero on the imaginary axis, except λ = 0, for some τ T , if (n+1) .

k -3 nαβ > zk 3 S *
(20)

Proof. Note, since k 5 and k 6 are positive, Q 3 (λ; τ 3 ) := k 5 e -λτ3 /(λ

+ k 5 + k 6 ) is analytic in C o , τ 3 ≥ 0. In addition, since sup ω∈R Q 3 (iω; τ 3 ) = k 5 /(k 6 + k 5 ) < 1, Q 3 (λ; τ 3 ) ∈ H ∞ .
In addition, by Theorem 7 and (15e)

(λ + k 5 + k 6 ) 1 -Q 3 (λ; τ 3 ) ,
has no zeros in C + . Note, by Proposition 10, G d (λ; τ 1 , τ 3 ) = G(λ; τ 1 )Q 3 (λ; τ 3 ), where G(λ; τ 1 ) is defined as in ( 17), has no zero in the rhp. Then,

Φ(λ; τ 1 , τ 3 ) = k 6 k d k -3 βn S * λ + 1 -k -1 G d (λ; τ 1 , τ 3 ) is analytic in C o . Furthermore, since sup ω∈R |Φ(jω; τ 1 , τ 3 )| ≤ k 6 k d k -3 βn S * sup ω∈R iω + 1 -k -1 Q 3 (iω; τ 3 ) × 1 inf ω∈R |G(iω; τ 1 )| < ∞, (21)
Φ(λ; τ 1 , τ 3 ) ∈ H ∞ . Then, by ( 5), since

|Φ(0; τ 1 , τ 3 )| = k -3 n S * n k 3 (1 + k 4 τ 1 ) + k -3 = k -3 nαβ k 3 zS * n+1
which is assumed to be strictly greater than 1, by Proposition 8, there exists a positive τ T such that ∆(λ; τ, τ T ) = 1 + e -τ T λ Φ(λ; τ 1 , τ 3 ) ( 22) has a zero at some iω o , ω o > 0. Note, from (13), since ∆(λ; τ ) = λG d (λ; τ 1 ) ∆(λ; τ, τ T ), ∆(iω o ; τ ) = 0.

NUMERICAL RESULTS

By using the QPmR software package [START_REF] Vyhlídal | QPmR -quasipolynomial root-finder: Algorithm update and examples[END_REF], zeros of ∆(λ; τ ) were computed as shown in Fig. 1(a). As seen in the figure, ∆(λ; τ ) has imaginary axis zeros (in addition to the zero at the origin) for τ T ≈ 40, using parameters that satisfy both ( 16) and (20). A limitcycle solution of equations (2) for the same parameter set and a value of τ T above the Andronov-Hopf bifurcation value is shown in Fig. 1(b) [computed using the stiff integrator in Xppaut [START_REF] Ermentrout | Simulating, Analyzing, and Animating Dynamical Systems[END_REF], and reducing the step size until solutions of consistent amplitude were obtained]. The model displays sustained oscillations, as was the case for the simpler single-delay model [START_REF] Roussel | Approximating state-space manifolds which attract solutions of systems of delaydifferential equations[END_REF]. When the value of the promoter clearance delay τ 1 is halved, from 0.04 to 0.02, the limit cycle becomes substantially larger [Fig. 1(c)]. Note that, relative to τ T , τ 1 is a small delay that might have been thought to have been negligible. We see here however that it has a dramatic effect on the oscillations. In fact, since the limit cycle shrinks as τ 1 increases, we see that the promoter clearance process has a stabilizing effect. The RBS clearance delay τ 3 has a similar effect (not shown). The stabilizing effect of the binding-site clearance delays is likely related to what MacDonald (1987) called an "interference effect" of independent delays.

CONCLUSION

In this work, a model for a gene regulatory system incorporating both conventional production delays and bindingsite clearance delays was presented. We analyzed these equations to show that they have bounded solutions for all positive values of the parameters provided the initial functions are non-negative. We also studied the stability properties of the model, developing the conditions for an Andronov-Hopf bifurcation. We found that the bindingsite clearance delays, usually neglected in gene expression models, in fact have a significant effect on the dynamics. In both this and a previous paper [START_REF] Trofimenkoff | Small binding-site clearance delays are not negligible in gene expression modeling[END_REF], binding-site clearance delays were found to have a stabilizing effect. It is an interesting question whether this will generally be the case. If so, one wonders if the lengthy clearance times observed in some systems [START_REF] Hsu | Promoter clearance and escape in prokaryotes[END_REF]Aye-Han and Hsu, unpublished) are an evolved mechanism for stabilizing steady states.

These studies also suggest that it would be worth revisiting some classic gene expression models with delays to see if similarly large effects are seen with modest clearance delays in a wider range of models than we have so far examined. It would also be interesting to examine the effects of clearance delays in models implementing the network motifs with delays recently studied by [START_REF] Glass | Nonlinear delay differential equations and their application to modeling biological network motifs[END_REF]. The authors of this study suggested, for example, that the response delays encoded in biological networks may have evolved to suppress chaos. If so, the stabilizing tendency of binding-site clearance delays may play an important role in avoiding dynamical regimes that imperil a cell.

Fig. 1 .

 1 Fig. 1. Characteristic function zeros and numerical solutions of the model for the parameter setS 1 = {k in = 2, k -1 = 0.001, k 3 = 3, k -3 = 1, k 4 = 20, k 5 = 80, k 6 = 0.1, k d = 3, K M = 16 × 10 -7 , τ 1 = 0.04, τ 3 = 0.005, n = 3}. (a)Zeros of the characteristic function while 5 ≤ τ T ≤ 100. (b) Limit cycle for τ 2 = 35, τ 4 = 20 (τ T = 55) and initial conditions c = e = s = m = a = 0, p = z = 0.001. (c) Same as (b), except τ 1 = 0.02.

ACKNOWLEDGEMENTS

MRR would like to thank Professor Lilian Hsu for kindly sharing her unpublished work.