Ophelia Godin

Marion Leboyer

Yassin Mazroui

Bruno Aouizerate

Jean- Michel M Azorin

Raoul Belzeaux

Frank Bellivier

Mircea Polosan

B Etain

C Henry

E Olié

E Haffen

P M Llorca

; Face-Bd

V Barteau

S Bensalem

M Laaidi

H Laouamri

K Souryis

S Hotier

A Pelletier

N Drancourt

J P Sanchez

E Saliou

C Hebbache

J Petrucci

L Willaume

E Bourdin

GH ; Ap-Hp

Saint-Louis-Lariboisière-Fernand Widal

M Carminati

E Marlinge

J Meheust

Centre Hôpital C Perrens

A Antoniol

S Desage

A Gard

K Jutant

I Mbailara

L Minois

Zanouy

C Abettan

L Bardin

A Cazals

P Courtet

B Deffinis

D Ducasse

M Gachet

A Henrion

E Martinerie

F Molière

B Noisette

G Tarquini

N Correard

J L Consoloni

F Groppi

L J Lescalier

M Montant

N Rebattu

Viglianese

R Cohen

J P Kahn

M Milazzo

O Wajsbrot-Elgrabli

B Fredembach

S Garçon

P Grignon

A Perrin

A M Galliot

I Grévin

A S Cannavo

N Kayser

C Passerieux

P Roux

V Aubin

I Cussac

M A Dupont

J Loftus

I Medecin

; Ahph

N Dubertret

C Mazer

Portalier

Trajectories of functioning in bipolar disorders: A longitudinal study in the FondaMental Advanced Centers of Expertise in Bipolar Disorders cohort

Keywords: bipolar disorder, functioning, trajectories, prediction, longitudinal

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

Funding and sponsor: This research has been supported by the Foundation FondaMental, the Institut National de la Santé et de la Recherche Médicale (INSERM), Assistance Publique des Hôpitaux de Paris (AP-HP), and the Investissements d'Avenir program managed by the Agence national pour la Recherche (ANR) under reference ANR-11-IDEX-0004-02 and ANR-10-COHO-10-01. This funding sources had no role in the study design, data collection, analysis, preparation of the manuscript, or decision to submit the manuscript for publication.

Abstract

Objective: We aimed at identifying distinct trajectories of functioning and at describing their respective clinical characteristics in a cohort of individuals with bipolar disorders (BD).

Methods:

We included a sample of 2351 individuals with BD who have been followed-up to 3 years as part as the FACE-BD cohort. Global functioning was measured using the Functioning Assessment Short Test. We used latent class mixed models to identify distinct longitudinal trajectories of functioning over three years. Multivariable logistic regression models were used to identify the baseline factors that were associated with the membership to each trajectory of functioning.

Results: Three distinct trajectories of functioning were identified: (i) a majority of individuals (72%) had a stable trajectory of mild functional impairment, (ii) 20% of individuals had a stable trajectory of severe functional impairment, and (iii) 8% of individuals had a trajectory of moderate functional impairment that improved over time. The membership to a trajectory of stable severe versus stable mild functional impairment was associated with unemployment, a higher number of previous hospitalizations, childhood maltreatment, a higher level of residual depressive symptoms, higher sleep disturbances, a higher BMI and a higher number of psychotropic medications being prescribed at baseline. The model that included these seven factors led to an AUC of 0.85.

Conclusions:

This study enabled to stratify individuals with BD according to three distinct trajectories of functioning. The results regarding the potential determinants of the trajectory of severe functional impairment needs to be replicated in independent samples. Nevertheless, these potential determinants may represent possible therapeutic targets to improve the prognosis of those patients at risk of persistent poor functioning.

Introduction

Bipolar Disorder (BD) is characterized by a heavy functional burden since a large number of individuals with BD (30-60%) will experience social impairment, both in occupational and social domains, in between mood episodes [START_REF] Macqueen | A review of psychosocial outcome in patients with bipolar disorder[END_REF] The degree of functional impairment in BD is heterogeneous and has consistently been associated with residual depressive symptoms and/or cognitive impairment in cross-sectional studies [START_REF] Baş | The impact of cognitive impairment, neurological soft signs and subdepressive symptoms on functional outcome in bipolar disorder[END_REF][START_REF] Samalin | Residual symptoms and specific functional impairments in euthymic patients with bipolar disorder[END_REF][START_REF] Samalin | Residual depressive symptoms, sleep disturbance and perceived cognitive impairment as determinants of functioning in patients with bipolar disorder[END_REF][START_REF] Sanchez-Moreno | Factors associated with poor functional outcome in bipolar disorder: sociodemographic, clinical, and neurocognitive variables[END_REF]. In addition, a lower level of functioning has been associated to male gender, older age, unemployment, a higher number of previous mood episodes and of hospitalizations, psychiatric comorbidities, sleep disorders or higher emotional reactivity [START_REF] Fekih-Romdhane | Factors predicting functioning in type 1 bipolar patients during remission period[END_REF][START_REF] Murru | Modifiable and non-modifiable factors associated with functional impairment during the inter-episodic periods of bipolar disorder[END_REF][START_REF] Sanchez-Moreno | Factors associated with poor functional outcome in bipolar disorder: sociodemographic, clinical, and neurocognitive variables[END_REF].

Only a few longitudinal studies have used functioning as an outcome and found that both subsyndromal depressive symptoms at baseline and cognitive impairment were associated with worse long-term functional outcome [START_REF] Bonnin | Functional remediation in bipolar disorder: 1year follow-up of neurocognitive and functional outcome[END_REF][START_REF] Demmo | Neurocognitive functioning, clinical course and functional outcome in first-treatment bipolar I disorder patients with and without clinical relapse: A 1-year follow-up study[END_REF][START_REF] Frías | Longitudinal cognitive trajectories and associated clinical variables in youth with bipolar disorder[END_REF][START_REF] Martino | Neurocognitive and symptomatic predictors of functional outcome in bipolar disorders: a prospective 1 year follow-up study[END_REF][START_REF] Tabarés-Seisdedos | Neurocognitive and clinical predictors of functional outcome in patients with schizophrenia and bipolar I disorder at one-year follow-up[END_REF]. However, these studies were performed in small samples of individuals (n=32-100) and without any formal modeling of the trajectories of functioning. To our knowledge, only Frias and collaborators modeled the longitudinal trajectories of cognitive functioning in 135 young individuals with BD aged 7-17 years.

In a cohort of 2351 individuals with BD who have been followed-up to 3 years, we used a minimally-supervised clustering method, called the Latent Class Mixed Model (LCMM), to identify homogeneous trajectories of functioning. We also aimed at characterizing each trajectory of functioning according to the baseline clinical factors.

Material and methods

Study population

The cohort consisted in individuals with BD who were clinically assessed within the French network of FondaMental Advanced Centers of Expertise in Bipolar Disorders (FACE-BD). This network has been developed under the aegis of the FondaMental Foundation and is supported by the French Ministry of Health. The network provides support to general practitioners and psychiatrists for the diagnosis and management of individuals with BD. All outpatients, aged 16 years or above and diagnosed with BD according to DSM-IV criteria (all BD subtypes [I, II, and not otherwise specified]), were assessed in one of the 10 centers. All centers used the same package of thorough and standardized clinical assessments, described in details elsewhere [START_REF] Henry | A French network of bipolar expert centres: a model to close the gap between evidence-based medicine and routine practice[END_REF]. These individuals were re-assessed at 6 months, one year, two years and up to 3 years.

The follow-up protocol was approved by the institutional review board (Comité de Protection des Personnes Ile de France IX; January 18, 2010). In accordance with the French laws for non-interventional studies and according to the Ethical Committee, the study participation requires only an information letter since the clinical data were collected as part as the usual medical care system in France.

Data collected

At baseline, a specialized team (psychiatrist, psychologist and neuropsychologist) interviewed each individual using the SCID (Structured Clinical Interview for DSM-IV Disorders) and systematically recorded information related to socio-demographic characteristics, onset and course of BD and psychiatric comorbidities. The current use of psychotropic treatments (firstgeneration neuroleptics, atypical antipsychotics, mood stabilizers (including lithium and anticonvulsants), and antidepressants) was recorded. The adherence to medication was assessed using the French version of the Medication Adherence Rating Scale (MARS) [START_REF] Misdrahi | Therapeutic adherence and schizophrenia: the interest of the validation of the French translation of Medication Adherence Rating Scale (MARS)][END_REF]. Current depressive and (hypo)manic symptoms were assessed with the Montgomery Asberg Depression Rating Scale [START_REF] Montgomery | A new depression scale designed to be sensitive to change[END_REF]) and the Young Mania Rating Scale [START_REF] Young | A rating scale for mania: reliability, validity and sensitivity[END_REF] respectively.

Global functioning was assessed using the Functioning Assessment Short Test (FAST) [START_REF] Rosa | Validity and reliability of the Functioning Assessment Short Test (FAST) in bipolar disorder. Clinical practice and epidemiology in mental health[END_REF]. The FAST is a clinician-based interview specifically developed to assess the level of functioning among individuals with BD. It includes 24 items in six sub-domains (cognition, autonomy, occupational functioning, financial issues, leisure time and interpersonal relationships). A higher score indicates a worse functioning. [START_REF] Sanchez-Moreno | Factors associated with poor functional outcome in bipolar disorder: sociodemographic, clinical, and neurocognitive variables[END_REF] further described that FAST scores from 0 to 11 included patients with no impairment.

Scores from 12 to 20 represented the category of mild impairment. Moderate impairment comprised scores from 21 to 40. Finally, scores above 40 represent severe functional impairment [START_REF] Bonnín | Thresholds for severity, remission and recovery using the functioning assessment short test (FAST) in bipolar disorder[END_REF]. The individuals were also assessed using the Global Assessment of Functioning scale (GAF).

Childhood maltreatment was assessed using the Childhood Trauma Questionnaire (CTQ) [START_REF] Bernstein | Development and validation of a brief screening version of the Childhood Trauma Questionnaire[END_REF]. Subjective sleep quality was assessed with the Pittsburgh Sleep Quality Index (PSQI) [START_REF] Buysse | The Pittsburgh Sleep Quality Index: a new instrument for psychiatric practice and research[END_REF].

A blood sample was collected at baseline and the presence of a metabolic syndrome was defined according to the criteria of the International Diabetes Federation [START_REF] Alberti | The metabolic syndrome--a new worldwide definition[END_REF].

This definition requires the presence of three or more of the following five criteria: high waist circumference (>94 cm for men and >80 cm for women), hypertriglyceridemia (1.7 mM or on lipid lowering medication), low HDL cholesterol level (<1.03 mM in men and <1.29 mM in women), high blood pressure (130/85 mmHg or on antihypertensive medication), or high fasting glucose concentration (5.6 mM or on glucose-lowering medication). Body Mass Index (BMI) was also recorded.

Statistical analyses

Baseline socio-demographic and clinical characteristics are presented using means and Standard Deviation (SD) for continuous data and frequencies for categorical variables. Intervals (CI) was used to investigate the accuracy of risk factors to predict the membership to each trajectory. Sensitivity analyses were performed in the subsample of individuals seen at each visit and in individuals who were euthymic at baseline (i.e. with MADRS scores <8 and YMRS scores <8). Separate LCMM analyses were then performed for each sub-domain of functioning in order to determine which factors might be associated with any specific subdomain. We focused only on the four main domains of functioning (cognition, occupational functioning, personal interrelationship, and autonomy) that have been previously reported as persistently altered in euthymic individuals with BD as compared to healthy controls [START_REF] Rosa | Clinical predictors of functional outcome of bipolar patients in remission[END_REF]. Statistical analyses were performed with SAS (release 9.4; SAS Statistical Institute, Cary, NC) and R Statistical Software version 3.4.4. All statistical tests were two-tailed.

Results

Among

Trajectories of functioning

The LCMM model identified three trajectories of global functioning as displayed in Figure 1.

The criteria for the selection of the optimal number of trajectories are described in supplementary table 1. The average posterior probability for individuals being assigned to each class is reported as a measure of goodness of discrimination and described in supplementary table 2. A first trajectory of stable mild functional impairment (72.5% of patients) was characterized by a relatively low baseline scores on the FAST (indicating good functioning to mild functional impairment) which remained stable during the follow-up. A second trajectory of functional improvement (8.4% of patients) was characterized by an initial high FAST score (moderate functional impairment), but this score decreased steadily during the 3 years of follow-up, indicating a progressive functional improvement. A third trajectory of stable severe functional impairment (19.1% of patients) was characterized by a high FAST score at baseline that remained stable during the follow-up. The respective FAST and GAF scores for the three trajectories are described in supplementary table 3.

Sensitivity analyses in individuals who have been assessed at each follow-up visit (n=375) showed that the same three trajectories were observed in this subsample (see supplementary fig 1). Moreover, the mean number of visits completed by the individuals did not differ between the three trajectories (p=0.57), meaning that individuals with missing visits were not likely to contribute to one trajectory more than to another.

Characteristics of individuals belonging to the trajectories of stable severe versus stable mild functional impairment

We focused on the differences between the two trajectories with the most individuals, i.e. stable severe versus stable mild functional impairment because of the small sample size of the trajectory of functional improvement. The differences (univariable analyses) of baseline characteristics between the trajectories of severe versus mild functional impairment are presented in Table 1. The individuals belonging to the trajectory of stable severe functional impairment had lower education level, were more frequently unemployed, and were characterized by a longer duration of illness, a higher number of lifetime hospitalizations, more depressive symptoms at baseline, a higher frequency of rapid cycling, a previous history of suicide attempt and of childhood trauma. Individuals with stable severe functional impairment also experienced more frequent psychiatric comorbidities (anxiety disorders and substance use disorders). These individuals were also more likely to present with a metabolic syndrome, a higher BMI, and higher sleep disturbances as compared to individuals with a stable mild functional impairment. Individuals with a stable severe functional impairment had a lower adherence to psychotropic medications and they were more likely to currently use atypical antipsychotics, antidepressants and anxiolytics. As a whole, they received a higher number of psychotropic medications at baseline. Multivariable analyses (see table 2) were performed by including all variables identified by the univariable analyses with p-value <0.20. The membership to the trajectory of stable severe functional impairment was associated with unemployment (OR=4.02 95%CI= 2.52-6.61), a higher number of lifetime hospitalizations (OR=1.1 95%CI=1.0-1.2), a higher level of depressive symptoms (OR=1.12 95%CI=1.09-1.16) and of sleep disturbances (OR=1.07 95%CI=1.00-1.15) at baseline, an history of childhood traumatic experiences (OR=1.03 95%CI=1.01-1.04), a higher BMI (OR=1.08 95%CI=1.03-1.12) and a higher number of psychotropic medications prescribed at baseline (OR=1.3 95%CI=1.08-1.50). The inclusion of these seven factors in the model led to an AUC of 0.85 (see figure 2).

Sensitivity analyses performed in the subgroup of individuals who were euthymic at baseline showed that the membership to the trajectory of stable severe functional impairment was associated with a higher number of hospitalizations, a higher BMI, the presence of childhood traumatic experiences and higher sleep disturbances at baseline. The inclusion of these four factors in the model led to an AUC of 0.83 (see supplementary figure 2).

Trajectories of subdomains of functioning and associated factors

For each of the 4 main sub-domains of functioning (cognition, occupational functioning, personal interrelationship, and autonomy), only two trajectories were observed (mild versus severe functional impairment). Most individuals with a stable mild (global) functional impairment (76.7%) also belong to a trajectory of stable mild functional impairment for these 4 subdomains (data not shown, available on request). We then tested for the associations between the previously identified baseline factors and trajectories of severe functional impairment in these 4 sub-domains to highlight any specificity (supplementary Table 4).

Higher baseline depressive symptoms and unemployment were both associated with a higher risk of membership to the trajectories of severe impairment in the four subdomains. The number of psychotropic medications used at baseline was associated with most trajectories of severe impairment (for autonomy, cognition, personal interrelationship, but not for occupational functioning). Childhood trauma was associated with the trajectory of severe impairment in terms of occupational functioning and personal interrelationship. BMI and number of previous hospitalizations were associated only with the trajectory of severe impairment in occupational functioning. Sleep disturbances were associated only with the trajectory of severe impairment in terms of personal interrelationship. A summary of these associations is presented as a heatmap in Figure 3.

Discussion

This is the first study that has modeled the longitudinal trajectories of functioning in a large sample of adult individuals with BD. Three trajectories were observed with a majority of individuals (72%) having a stable mild functional impairment, 20% of individuals having a stable severe functional impairment and 8% of individuals having a trajectory of functional improvement. As compared to individuals with a stable mild functional impairment, those with a stable severe functional impairment were characterized by childhood trauma, unemployment, a higher number of lifetime hospitalizations, more depressive symptoms, overweight/obesity, more sleep disturbances, and more psychotropic drugs used at baseline. The factors that characterized the trajectory of stable severe functional impairment were consistent with those obtained in cross-sectional studies. Indeed, several cross-sectional studies reported associations between impaired functioning and persistence of (syndromal) depressive symptoms [START_REF] Altshuler | Subsyndromal depressive symptoms are associated with functional impairment in patients with bipolar disorder: results of a large, multisite study[END_REF][START_REF] Bonnin | Functional remediation in bipolar disorder: 1year follow-up of neurocognitive and functional outcome[END_REF][START_REF] Frías | Longitudinal cognitive trajectories and associated clinical variables in youth with bipolar disorder[END_REF][START_REF] Martino | Neurocognitive and symptomatic predictors of functional outcome in bipolar disorders: a prospective 1 year follow-up study[END_REF], between obesity and cognitive impairment [START_REF] Mora | The Impact of Obesity on Cognitive Functioning in Euthymic Bipolar Patients: A Cross-Sectional and Longitudinal Study[END_REF], however not replicated in an independent study [START_REF] Bond | Lower cognitive functioning as a predictor of weight gain in bipolar disorder: a 12-month study[END_REF]. The association between sleep disturbances and stable severe functional impairment is also consistent with previous crosssectional studies [START_REF] Boland | Associations between sleep disturbance, cognitive functioning and work disability in Bipolar Disorder[END_REF][START_REF] De La Fuente-Tomás | Sleep disturbances, functioning, and quality of life in euthymic patients with bipolar disorder[END_REF][START_REF] Gruber | Sleep functioning in relation to mood, function, and quality of life at entry to the Systematic Treatment Enhancement Program for Bipolar Disorder (STEP-BD)[END_REF][START_REF] Pinho | The association between biological rhythms, depression, and functioning in bipolar disorder: a large multi-center study[END_REF], showing that both short and long durations of sleep were associated with poor functioning [START_REF] Gruber | Sleep functioning in relation to mood, function, and quality of life at entry to the Systematic Treatment Enhancement Program for Bipolar Disorder (STEP-BD)[END_REF] and that higher biological rhythms disturbances correlated with lower functioning [START_REF] Rosa | Biological rhythm disturbance in remitted bipolar patients[END_REF]. Finally, the previous number of hospitalizations, which may be considered as a proxy of the severity and recurrence of mood episodes, was also associated with a stable worse functioning, this being consistent with the staging model of BD that postulates that 'patients with multiple episodes had a worse prognosis on functioning' [START_REF] Kapczinski | Staging systems in bipolar disorder: an International Society for Bipolar Disorders Task Force Report[END_REF].

The results of this study should be interpreted with caution, since we have tested for associations between factors and trajectories, and we cannot infer any definitive conclusions about causal links. The factors that are associated with a stable severe functional impairment during follow-up, may be themselves the consequences of a poor functioning during the years before the baseline assessment. For example, depressive symptoms at baseline are associated with poor functioning at three years but might also be the consequence of a previous period of poor functioning that maintains depressive symptoms at a significant level. Moreover, some of the identified factors may be inter-related such as depressive symptoms and sleep disturbances, or depressive symptoms and unemployment for instance. Some other limitations deserve some comments. The representativeness of this sample that has been assessed in the FACE-BD network might be questioned. We have previously suggested that even if these individuals with BD are moderately severe, our sample might be under-represented in terms of highly severe patients (i.e. individuals with chronic hospitalization for example) or individuals with a more favorable course of BD (i.e. individuals not being referred to the network because stabilized and with a good functioning).

From a statistical perspective, latent class mixed models do not require that all individuals were observed the same number of times or at the same times and are not likely to be sensitive to missing data. Nevertheless, in order to ensure that the identified trajectories were not biased due to drop-out or missing data, we performed sensitivity analyses in individuals who have been assessed at each study wave (complete-case sample) and the results were unchanged (Supplementary Figure 1). We also observed that the number of visits being completed by the individuals was not associated to the membership to one specific trajectory and no difference of functioning at baseline was observed between individuals with or without assessment at all time. The investigation of the associations between factors and trajectories of subdomains of functioning tended to highlight certain specificities. Since this part of the study was exploratory, the results should be replicated in independent samples. We also restricted these analyses to 4 out 6 subdomains of functioning (not exploring financial and leisure time domains) that were considered as those being altered in euthymic patients as compared to healthy controls. Future studies should extend this exploratory part to all the subdomains of functioning.

Nevertheless, this study has several strengths. This is the first longitudinal study that explored trajectories of functioning in a large cohort of individuals with BD. We were able to study a large number of potential associated factors in the multivariate analyses given the exhaustive clinical characterization at baseline. Moreover, the sample is relatively large thus reducing potential false negative results. Sensitivity analyses also help ensuring that most positive results were not explained only by the level of depressive symptoms that is a major determinant of functional impairment. We also further explored main subdomains of functioning even if this part of the analyses was exploratory.

Altogether, our results suggested that some of the factors associated with a stable trajectory of severe functional impairment might be modifiable. In the future, these possible therapeutic targets should be part of a personalized care plan for individuals with BD in order to observe whether modifying them may result in changes in trajectories of functioning. For instance, depressive symptoms might be improved by Mindfulness Based Cognitive Therapy as suggested in a review of 13 studies in BD [START_REF] Bojic | Mindfulness-Based Treatment for Bipolar Disorder: A Systematic Review of the Literature[END_REF]. A recent meta-analysis of RCTs supported the efficacy of the Clubhouse Model in promoting employment, reducing hospitalization(s), and improving quality of life [START_REF] Mckay | A Systematic Review of Evidence for the Clubhouse Model of Psychosocial Rehabilitation[END_REF] in individuals with mental disorders. An Integrated Risk Reduction Intervention study demonstrated its efficacy in reducing BMI during a RCT including 122 individuals with BD [START_REF] Frank | An Integrated Risk Reduction Intervention can reduce body mass index in individuals being treated for bipolar I disorder: results from a randomized trial[END_REF]. A longitudinal clinical trial including 116 individuals with BD, showed that interpersonal and social rhythm therapy (a treatment combining a behavioral approach to increasing the regularity of daily routines with interpersonal psychotherapy) tended to improve occupational functioning [START_REF] Frank | The role of interpersonal and social rhythm therapy in improving occupational functioning in patients with bipolar I disorder[END_REF]. Finally, polypharmacy, a common phenomenon in BD [START_REF] Fornaro | Prevalence and clinical features associated with bipolar disorder polypharmacy: a systematic review[END_REF], was associated with lower functioning in our study and should be carefully revised if possible. Even though these potentially modifiable factors can be targeted by specific interventions, it is speculative to firmly suggest that any modification of one (or several) factor(s) would necessarily result in an improvement of the trajectory of stable severe functional impairment. Prospective intervention studies would be required to test this hypothesis. Nevertheless, in a previous study, we have suggested that the improvement of functioning observed during a two-year follow-up period was associated with a reduction of depressive and manic symptoms, of comorbidities (anxiety disorders and substance use disorders) and of BMI, and with an improvement of the adherence to medication. This suggests that acting on some modifiable factors being associated to the trajectory of stable severe functional impairment might be a therapeutically lever to modify the functioning [START_REF] Henry | Outcomes for bipolar patients assessed in the French expert center network: A 2-year follow-up observational study (FondaMental Advanced Centers of Expertise for Bipolar Disorder [FACE-BD])[END_REF].

Finally, even though not modifiable by definition, childhood maltreatment might also be considered as an important therapeutic target in patients with BD. Individuals with BD reported severe and frequent traumatic experiences in childhood [START_REF] Aas | The role of childhood trauma in bipolar disorders[END_REF][START_REF] Palmier-Claus | Relationship between childhood adversity and bipolar affective disorder: systematic review and meta-analysis[END_REF], that increase the severity of the clinical expression of the disorder and alter the cognitive impairment [START_REF] Jiménez | Impact of childhood trauma on cognitive profile in bipolar disorder[END_REF]. For example, some study protocols have been published for the use of EMDR (Eye Movement Desensitization and Reprocessing) in individuals with BD, with planned secondary outcomes concerning functioning [START_REF] Moreno-Alcázar | Eye movement desensitization and reprocessing therapy versus supportive therapy in affective relapse prevention in bipolar patients with a history of trauma: study protocol for a randomized controlled trial[END_REF].

In conclusion, the use of longitudinal data from the FACE-BD cohort provides a unique opportunity to stratify individuals according to their trajectories of functioning and to disentangle the potential associated risk factors. We identified here two main trajectories of functioning (stable mild versus stable severe functional impairment). Moreover, seven factors (unemployment, depressive symptoms, sleep disturbances, BMI, childhood maltreatment, number of prescribed psychotropic medications, number of previous hospitalizations) accurately predict the membership to a trajectory of stable severe functional impairment. As such, even if replications are required in independent longitudinal samples, the identified factors may provide some potential targets for psychosocial interventions that should be included as part as a personalized care plan.

Access to data: Due to ethical and legal restrictions, data involving clinical participants cannot be made publicly available. All relevant data are available upon request to the Fondation FondaMental for researchers who meet the criteria for access to confidential data.

  . Longitudinal studies of functioning in individuals with BD are crucial since they can provide new elements about the dynamics and evolution of functioning. In this context, the use of longitudinal data provides a unique opportunity to analyze the heterogeneity of trajectories of functioning and the respective associated factors. Such trajectories and potential determinants remained poorly described because of the paucity of longitudinal studies. The identification of potential trajectory-associated factors would help personalizing treatment design and setting-up concrete interventions to improve the functional prognosis of BD.

  To identify distinct trajectories of functioning, we used a Latent Class Mixed Model (LCMM) based on a maximum likelihood framework with a modified Marquardt iterative algorithm and a Newton-Raphson-like algorithm[START_REF] Proust-Lima | Joint modeling of repeated multivariate cognitive measures and competing risks of dementia and death: a latent process and latent class approach[END_REF]. LCMM extends growth modeling such as Linear Mixed Model (LMM) or Latent Class Growth Analysis (LCGA) by considering sample heterogeneity and individual variability. LCMM allows to handle latent continuous or ordinal processes and person-specific processes derived from multiple measurements over time. The use of LCMM is based on the assumption of Missing At Random (MAR) for missing data and dropout. The choice of the optimal number of trajectories was based both on the Bayesian Information Criterion (BIC) fit index, and on the number of individuals in each trajectory. The lowest BIC suggests the best fit. The membership of an individual to a specific class (trajectory) is determined by calculating the posterior probability of belonging to one class and assigning the individual to a given class with the highest probability. The average posterior probability for individuals assigned to each class is reported as a measure of goodness of discrimination. To test whether the trajectories differed in terms of demographic, clinical, dimensional characteristics and current medication use, we performed Chi2 square, Student or Mann-Whitney Wilcoxon tests, as appropriate. Given the number of tests performed, we used a Bonferroni correction for multiple testing. The level of significance was set up at p = 0.0001. All baseline variables with p-values < 0.20 identified in univariable analyses were included in a multivariable logistic regression model with backward selection, to identify predictors of membership to the distinct trajectories of functioning. Receiver Operating Characteristic (ROC) curves were plotted, and the Area Under the Curve (AUC) with 95% Confidence

  the 2351 individuals with BD included in the FACE-BD cohort, 1550 (66%) were assessed at least two times during the 3 years of follow-up (1055 were assessed at month 6, 1091 at month 12, 704 at month 24 and 375 at month 36). Individuals had a mean follow-up duration of 16.4 months (SD: 13.3). The sample consisted in 61.2% of females and 38.8% of males. The median age at inclusion was 40.4 years (InterQuartile: IQR = 31.4-50.4). Most individuals were diagnosed with BD type I (47.6% (n=1148)), 40.8% with BD type II (n=985) and 11.6% with BD not otherwise specified (n=281). The median duration of illness was 15.0 years (IQR=8.3-24.0).

Acknowledgments:

We thank the FondaMental Foundation (www-fondationfondamental.org), that promotes scientific cooperation in mental health. The FondaMental Foundation is developing a new model for translational research in psychiatry in France and supports the infrastructure of Centers of Expertise for Bipolar Disorders. We thank the patients who participated to the present study. We thank Hakim Laouamri, PhD and his team (Seif Ben Salem, Msc, Karmène Souyris, PhD, Victor Barteau, Msc and Mohamed Laaidi, PhD) for the development of the FACE-BD computer interface, data management, quality control and regulatory aspects.

Trajectories of functioning in bipolar disorders: a longitudinal study in the FACE-BD cohort

Running title: Functioning in bipolar disorders Ophelia Godin 1,3 , Marion Leboyer 1,3 , Yassin Mazroui 4 , Bruno Aouizerate 1,5 , Jean-Michel Azorin 1,6 ,Belzeaux Raoul 1,6 Conflicts of interest: All authors reported no biomedical financial interests or potential conflicts of interest. Dr. Costagliola reports HIV grants from Janssen (2017)(2018)(2019)(2020) and MSD France (2015-2017), personal fees from Janssen (2016, 2018), MSD France (2015, 2017) and Gilead (2018) for lectures, personal fees from Innavirvax (2016) and Merck Switzerland (2017) for consultancy, outside the submitted work. All authors have approved the manuscript.

Ethical standards:

The authors assert that all procedures contributing to this work comply with the ethical standards of the relevant national and institutional committees on human experimentation and with Helsinki Declaration of 1975, as revised in 2008.


Agnès Pellissier-Tanon

Gabriel Morgado

Annie Lemarchand

Ludovic Jullien

Jullien Ludovic

Quasi

Quasi-steady-state and partial-equilibrium approximations in chemical kinetics: One stage beyond the

come

Introduction

The laws of chemical kinetics lead to systems of differential equations for the concentrations of the reactive species. Depending on the fluxes of matter and energy, a reactive fluid maintained far from equilibrium may present qualitative changes of behaviors called bifurcations, such as the emergence of regular or chaotic temporal oscillations. [START_REF] Irving R Epstein | An introduction to nonlinear chemical dynamics: oscillations, waves, patterns, and chaos[END_REF] Following the dynamical systems theory, all systems of equations exhibiting a typical bifurcation can be reduced to a normal form involving a small number of variables and specific nonlinearities. [START_REF] Ali | The method of normal forms[END_REF] Knowing these nonlinearities is therefore sufficient to predict the behavior of a complex system. For example, this approach has been successfully applied to the modeling of biological rhythms such as glycolytic oscillations. [START_REF] Goldbeter | Biochemical oscillations and cellular rhythms: the molecular bases of periodic and chaotic behaviour[END_REF] Even if systems with a large number of variables can be solved numerically, the approximations used to eliminate variables and determine reduced systems of equations remain crucial.

Two approximations based on timescale analysis are usually applied in chemical kinetics. [START_REF] Miles | Simplification of mathematical models of chemical reaction systems[END_REF] The quasi-steady-state approximation (QSSA) is used to eliminate a chemical species which evolves faster than the others. The partial-equilibrium approximation (PEA) is employed to remove a reaction step associated with an extent reaching a steady state faster than the other extents. [START_REF] Wha | Steady-state and equilibrium approximations in chemical kinetics[END_REF] These approximations have been extensively studied over the last century. [START_REF] Godfried | Elimination of fast variables[END_REF][START_REF] Alexander N Gorban | Method of invariant manifold for chemical kinetics[END_REF][START_REF] Alexander N Gorban | Asymptotology of chemical reaction networks[END_REF][START_REF] Goussis | Quasi steady state and partial equilibrium approximations: their relation and their validity[END_REF][START_REF] Alexander N Gorban | Model reduction in chemical dynamics: slow invariant manifolds, singular perturbations, thermodynamic estimates, and analysis of reaction graph[END_REF][START_REF] Rae | Pre-equilibrium approximation in chemical and photophysical kinetics[END_REF][START_REF] Rae | A generalized pre-equilibrium approximation in chemical and photophysical kinetics[END_REF] Both methods are specific examples of the elimination of a fast variable, also called adiabatic elimination. [START_REF] Viossat | A test of the validity of steady state and equilibrium approximations in chemical kinetics[END_REF][START_REF] Viossat | Validity of the quasi-stationary-state approximation in the case of two successive reversible first-order reactions[END_REF][START_REF] Richard | The quasi-steady-state approximation: Numerical validation[END_REF] Nevertheless QSSA and PEA do not simply consist in eliminating a fast variable from a system of rate equations. The eliminated variable must be either a concentration for QSSA [START_REF] Morgado | Elimination of fast variables in stochastic nonlinear kinetics[END_REF] or an extent for PEA. [START_REF] Wha | Steady-state and equilibrium approximations in chemical kinetics[END_REF] Both methods are singular perturbation methods introducing a small parameter in the rate equations. [START_REF] Robert | Singular perturbation methods for ordinary differential equations[END_REF][START_REF] Noethen | Tikhonov's theorem and quasi-steady state[END_REF][START_REF] Goeke | A constructive approach to quasi-steady state reductions[END_REF][START_REF] Goeke | Determining "small parameters" for quasi-steady state[END_REF][START_REF] Eilertsen | The quasi-steady-state approximations revisited: Timescales, small parameters, singularities, and normal forms in enzyme kinetics[END_REF] However QSSA and PEA are based on the zeroth order of the expansion, so that the small parameter does not explicitly appear in the reduction procedure. Consequently the validity of the expansion is not usually checked. In addition the higher-order expansion does not help, since it reintroduces the fast variable. Both QSSA and PEA exploit the existence of different timescales: [START_REF] Shoffner | Approaches for the estimation of timescales in nonlinear dynamical systems: Timescale separation in enzyme kinetics as a case study[END_REF][START_REF] Eilertsen | Characteristic, completion or matching timescales? an analysis of temporary boundaries in enzyme kinetics[END_REF] The trajectories in the space of the concentrations or extents are quickly attracted by a lower-dimensional invariant manifold in which the evolution continues more slowly. [START_REF] Fenichel | Geometric singular perturbation theory for ordinary differential equations[END_REF][START_REF] Simon | The steady state and equilibrium approximations: A geometrical picture[END_REF][START_REF] Eilertsen | Phase-plane geometries in coupled enzyme assays[END_REF] In the two approximations the fast evolution is considered instantaneous and only the slow evolution on the invariant manifold is described.

In other words the fast variable, concentration or extent, is supposed to instantaneously adapt to the evolution of the slow variables. Quasi-steady state does not mean that the fast variable has reached a steady state but that it keeps evolving at a rate imposed by the slow variables. For this reason, the fast variable is sometimes called a slave variable. [START_REF] Haken | Synergetics: Introduction and advanced topics[END_REF] Within the framework of the approximation the fast variable is not independent and can be ignored in the reduced dynamics.

In this article we present necessary and sufficient conditions for eliminating the concentration of a chemical species or the extent of a reaction step in the rate equations in the linear domain around a steady state. These conditions are valid in the entire space of the variables for first-order kinetics. The conditions apply to any reaction scheme and do not require the prior knowledge of the rate constant values but they can be checked when the rate constants are known.

The manuscript is organized as follows. In section 2 the rate laws associated with a general reaction scheme are written when choosing either concentrations or extents as dynamical variables. Section 3 is devoted to the quasi-steady-state approximation and section 4 to the partial-equilibrium approximation. Section 5 gives conclusion.

ν ij C i k j ---- k -j ns i=1 ν ij C i (1)
where ν ij and ν ij are the left and right stoichiometric coefficients associated with species C i and reaction j. The forward and backward rate constants of reaction j are denoted k j and k -j for j = 1, .., n r . The system can be described by two sets of variables, either the concentrations or the extents. In the following we present the basics of the resolution of linearized rate equations around a steady state using matrix methods. [START_REF] Mn Berberan-Santos | The integration of kinetic rate equations by matrix methods[END_REF] The eigenvalues and eigenvectors of the Jacobian matrix are involved in the elimination conditions presented in section 3.

Evolution of the concentrations

The system of ordinary differential equations governing the evolution of the n s concentrations C i is a coupled system of equations

dC i dt = nr j=1 ν ij -ν ij k j ns l=1 C ν lj l -k -j ns l=1 C ν lj l (2)
If n c conservation laws can be written, only n = n s -n c concentrations are independent.

The system is written in matrix form for the n independent variables

dC dt = f (C) (3)
where

C =   C 1 ... C n 
 is the concentration vector and f is a possibly nonlinear function of the variables. The solution of Eq. ( 3) is highly dependent on the nonlinearities of the function f . The steady state obeys the system of equations dC i dt = 0 for i = 1, ..., n. The small deviation from the steady-state vector C 0 is

C = C -C 0 (4)
Substituting for the concentration vector from Eq. ( 4) into Eq. ( 3) and linearizing around the steady state, we have

d C dt = J C (5)
where the Jacobian matrix is

J = ∂f (C) ∂C C=C 0 (6)
Equation ( 5) is a linear system of n equations approximately accounting for the dynamics around the steady state. In the case of a linear function f , the Jacobian matrix J is independent of the steady state and the resolution of Eq. ( 5) gives the exact evolution of the concentrations even far from the steady state. We consider the case where the Jacobian matrix J has n eigenvalues denoted λ i for i = 1, .., n. The eigendecomposition of the Jacobian matrix J leads to

J = PΛP -1 ( 7
)
where Λ is a diagonal matrix whose diagonal terms are the eigenvalues and P is the change-of-basis matrix. In the eigenbasis, the rate equations are uncoupled and can be written as

dγ i dt = λ i γ i (8)
for i = 1, .., n. The vector of the variables in the eigenbasis

Γ =   γ 1 ... γ n   is related to C through C = PΓ (9)
The solution of Eq. ( 8) is

γ i = γ ini i exp (λ i t) (10)
for i = 1, .., n, where γ ini i is the initial condition of the variable γ i . At a given time, the state of the system is represented by a point in the n-dimension space of the concentrations. In the same space, the evolution of the system is a curve named trajectory. If the real parts Re(λ i ) of all eigenvalues λ i , i = 1, .., n, are negative the steady state C 0 is stable. A stable steady state is an attractor, i.e. for well-chosen initial conditions the concentrations eventually reach the stable steady concentrations. The trajectory converges towards the steady state. If Re(λ i ) > 0 for at least one eigenvalue λ i , the steady state is unstable.

The time τ i = 1/ | Re(λ i ) | associated with the variable γ i characterizes the evolution along the i th eigendirection, a large value of τ i being associated with a slow variable and a small value with a fast variable. Equation ( 9) written for t = 0 gives the relation between the initial conditions γ ini i for the variables in the eigenbasis and the initial conditions C ini i for the concentrations, leading to

Γ ini = P -1 Cini (11)
The evolution of the concentration C i is then obtained using Eqs. (4,9-11)

C i = n j=1 n k=1 P ij P -1 jk Cini k exp (λ k t) + C 0 i ( 12
)
where P ij and P -1 ij are elements of the matrices P and P -1 , respectively. The concentrations are linear combination of the variables in the eigenbasis. Equation (12) shows how each variable γ i contributes to the evolution of the concentrations.

Evolution of the extents of the reaction steps

The law governing the evolution of the extent ξ j of step j given in Eq. ( 1) is

dξ j dt = k j ns l=1 C ν lj l -k -j ns l=1 C ν lj l (13)
Substituting for the right-hand side from Eq. ( 13) into Eq. ( 2) and integrating on the interval [t, ∞[, we obtain a linear transformation between the two sets of variables, concentrations and extents. We choose that each extent ξ j vanishes at the steady state, leading to

C i -C 0 i = nr j=1 ν ij -ν ij ξ j (14)
Substituting C i from Eq. ( 14) into Eq. ( 13), we find the ordinary differential equations governing the evolution of the extents

dξ j dt = k j ns i=1 nr k=1 (ν ik -ν ik ) ξ k + C 0 i ν ik -k -j ns i=1 nr k=1 (ν ik -ν ik ) ξ k + C 0 i ν ik (15)
The method of resolution used for concentrations can be straightforwardly implemented for the extents. The concentrations and the extents exhibit linear relationships. Consequently the numbers of independent concentrations and independent extents are equal.

In addition, the concentrations and the extents have the same characteristic times.

3 Quasi-steady-state approximation (QSSA)

The quasi-steady-state approximation (QSSA) is a well-known method used to eliminate a fast concentration. [START_REF] Bodenstein | Eine theorie der photochemischen reaktionsgeschwindigkeiten[END_REF][START_REF] Turanyi | On the error of the quasi-steady-state approximation[END_REF][START_REF] Lee | On the validity of the steady state assumption of enzyme kinetics[END_REF][START_REF] Lee | The quasi-steady-state assumption: a case study in perturbation[END_REF] For example, when one of the chemical species C i is much more reactive than the other chemical species, the fast variation of C i is over at the timescale at which the other chemical species are still evolving. The approximation consists in writing dC i dt = 0 for a single species C i . According to Eq. ( 2) the concentration C i can then be written as a function of the other concentrations, which reduces by one the number of independent variables in the dynamics. QSSA is used to eliminate the fast concentration C i in the rate equations and we make precise the validity domain of the approximation in the following.

Fast and slow variables are defined in the eigenbasis of the linearized system, consequently in the vicinity of the steady state. The elimination of C i can be performed if two conditions are fulfilled. First a variable γ k in the eigenbasis must be faster than the other variables, i.e.

τ k max l =k (τ l ) (16)
Second the concentration C i must vary as the fast variable γ k during the time interval [0, τ k ]. Using the normalized change-of-basis matrix P and Eq. ( 9) we write the condition as

| Re (P ik ) | max j =k (| Re (P ij ) |) (17)
According to Eq. ( 17), the slow eigendirections are close to the hyperplan defined by

C i = C 0 i .
According to QSSA, the relaxation with the short characteristic time τ k along the γ k -axis can be considered instantaneous. The conditions given in Eqs. (16,17) involve the rate constants. They are necessary but not sufficient to ensure the validity of the elimination of the variable C i in the entire space of the concentrations since they result from a linearization of dynamics around the steady state.

Even if these conditions are met, the approximation is a priori not valid at the beginning of the evolution of the concentrations. It is valid only after the fast variable γ k has relaxed. Between 0 and τ k the trajectories in the space of the concentrations are close to the k th fast eigendirection which is close to the C i -axis. After the relaxation of the fast variable γ k the trajectories evolve on an (n -1)-dimensional space called the slow manifold and defined by the single equation

dC i dt = 0 (18)
At the fixed point corresponding to the steady state, the slow manifold is tangent to the vector space spanned by the n -1 eigendirections associated with the slow variables. On the slow manifold, the concentration C i slowly evolves and adapts to the evolution of the slow concentrations.

After the elimination of the concentration C i , the system of differential equations governing the evolution of the other concentrations can be written as a system of dimension n -1. If this system of differential equations involves polynomials compatible with the rate laws of reaction steps, the evolution of the remaining concentrations can be associated with a reduced set of reaction steps. The existence of a reduced reaction scheme is not always guaranteed as illustrated by Michaelis-Menten kinetics which leads to rational functions of the concentrations in the rate equations. [START_REF] Lee | The quasi-steady-state assumption: a case study in perturbation[END_REF] Approximate initial conditions adapted to the elimination of a concentration can be defined as the intersection of the slow manifold and the fast direction passing through the initial conditions before elimination.

If the remaining characteristic times have different orders of magnitude, an analogous procedure of elimination may be performed to further reduce the number of variables.

For illustration we consider two reaction schemes differing by one reaction step and associated with either linear or nonlinear rate equations. Using a chemostat, i.e. a reservoir of some species, we limit the number of variables without the need to introduce laws of conservation.

Linear dynamics

We consider the first-order reaction scheme

X k 1 ---- k -1 Y (19) Y k 2 ---- k -2 A ( 20
)
where A is a reservoir of A species also called chemostat, implying that the concentration A is constant. Only the concentrations X and Y associated with chemical species X and Y are independent variables obeying the linear system of ordinary differential equations

dX dt = -k 1 X + k -1 Y (21) dY dt = k 1 X -(k -1 + k 2 ) Y + k -2 A ( 22
)
where k 1 , k -1 , k 2 , and k -2 are rate constants. The steady state is

(X 0 , Y 0 ) = ( k -1 k -2 A k 1 k 2 , k -2 A k 2 )
. Following the method detailed in section 2 for the Jacobian matrix

J = -k 1 k -1 k 1 -(k -1 + k 2 ) (23)
we obtain the analytical expressions of the concentrations, which are valid in the entire concentration space (X, Y ) due to the linear nature of the equations given in Eqs. (21,22). The eigenvalues of J are

λ 1 = 1 2 -(k 1 + k -1 + k 2 ) + (k 1 + k -1 + k 2 ) 2 -4k 1 k 2 ( 24
)
λ 2 = 1 2 -(k 1 + k -1 + k 2 ) -(k 1 + k -1 + k 2 ) 2 -4k 1 k 2 (25)
and the change-of-basis matrix is

P =   P 11 = k -1 √ k 2 -1 +(k 1 +λ 1 ) 2 P 12 = k -1 √ k 2 -1 +(k 1 +λ 2 ) 2 P 21 = k 1 +λ 1 √ k 2 -1 +(k 1 +λ 1 ) 2 P 22 = k 1 +λ 2 √ k 2 -1 +(k 1 +λ 2 ) 2   (26)
The eigenvalues are real negative numbers regardless of the values of the rate constants.

The steady state is thus always stable. Interestingly the rate constant k -2 does not appear in the Jacobian matrix and consequently in the eigenvalues and the change-of-basis matrix. The conditions given in Eqs. (16,17) applied to the elimination of species Y read

τ 2 ≤ 10τ 1 (27)
| P 21 | ≤ 10 | P 22 | (28)
where

τ 1 = 1/ | λ 1 | and τ 2 = 1/ | λ 2 |
are deduced from Eqs. (24,25) and P 21 and P 22 are given in Eq. (26). Figure 1a displays the domain in the rate constant space in which the conditions given in Eqs. (27,28) are met. The two conditions are numerically found to be equivalent to

k 1 k -1 + k 2 .
The usual condition for the application of QSSA is recovered: Y can be considered a very reactive intermediate, slowly formed and rapidly consumed. [START_REF] Wha | Steady-state and equilibrium approximations in chemical kinetics[END_REF] The variable γ 2 associated with the smallest characteristic time τ 2 is the fast variable and evolves like Y . The variable γ 1 associated with the largest characteristic time τ 1 is the slow variable.

According to Eqs. (18,22) the equation of the slow manifold is dY /dt = 0, i.e.

k 1 X -(k -1 + k 2 ) Y + k -2 A = 0 (29)
QSSA does not mean that the fast variable Y has reached a steady state but that its evolution depends on X and follows the slow manifold. Substituting for the concentration Y from Eq. ( 29) into Eq. ( 21), we obtain the rate equation governing the evolution of the concentration of species X after elimination of Y

dX dt = - k 1 k 2 k -1 + k 2 X + k -1 k -2 k -1 + k 2 A (30)
This equation can be associated with the reduced reaction scheme

X k 1 k 2 k -1 +k 2 -------- k -1 k -2 k -1 +k 2 A ( 31
)
Using an expansion in power of k 1 /(k -1 + k 2 ), we check that the characteristic time τ 1 associated with the slow variable converges toward the characteristic time τ = (k

-1 + k 2 )/k 1 k 2 of the reduced dynamics.
Approximate initial conditions associated with the reduced reaction scheme can be deduced from the initial conditions X ini and Y ini associated with linear dynamics. Using the second column of the change-of-basis matrix given in Eq. ( 26), we find that the straight line parallel to the fast eigendirection and passing through the point (X ini , Y ini ) is given by

(k 1 + λ 2 ) X -X ini -k -1 Y -Y ini = 0 (32)
Solving the system given in Eqs. (29,32), we obtain the initial condition associated with the reduced reaction scheme

X ini = (k 1 + λ 2 ) X ini -k -1 Y ini (k -1 + k 2 ) + k -1 k -2 A (k 1 + λ 2 ) (k -1 + k 2 ) -k 1 k -1 (33)
Before studying the behavior of the trajectories in the space (X, Y ) in subsection 3.3, we illustrate QSSA in the case of nonlinear dynamics.

Nonlinear dynamics

We consider the following reaction scheme

2X k 1 ---- k -1 Y (34) Y k 2 ---- k -2 A ( 35
)
where A is a chemostat.

Following the laws of chemical kinetics, the rate equations for the concentrations X and Y are nonlinear and given by

dX dt = -2k 1 X 2 + 2k -1 Y (36) dY dt = k 1 X 2 -(k -1 + k 2 ) Y + k -2 A (37) The steady state is (X 0 , Y 0 ) = ( k -1 k -2 A k 1 k 2 , k -2 A k 2
). Linearizing Eqs. (36,37) around the steady state and following the same method as previously for the Jacobian matrix

J =   -4κ k -1 k 2 2k -1 2κ k -1 k 2 -(k -1 + k 2 )   (38)
with κ = k 1 k -2 A, we find the eigenvalues

λ 1 = 1 2 -4κ k -1 k 2 + k -1 + k 2 + √ ∆ ( 39
)
λ 2 = 1 2 -4κ k -1 k 2 + k -1 + k 2 - √ ∆ (40) with ∆ = 4κ k -1 k 2 + k -1 + k 2 2 -16κ k -1 k 2 .
The change-of-basis matrix is

P =        P 11 = 2k -1 4k 2 -1 + 4κ k -1 k 2 +λ 1 2 P 12 = 2k -1 4k 2 -1 + 4κ k -1 k 2 +λ 2 2 P 21 = 4κ k -1 k 2 +λ 1 4k 2 -1 + 4κ k -1 k 2 +λ 1 2 P 22 = 4κ k -1 k 2 +λ 2 4k 2 -1 + 4κ k -1 k 2 +λ 2 2        (41)
The steady state is stable since the eigenvalues are always real and negative regardless of the values of the rate constants.

The conditions for the elimination of Y are given in Eqs. (27,28) where λ 1 and λ 2 are given in Eqs. (39,40) and P 21 and P 22 , in Eq. ( 41). The variable Y can be eliminated if the parameters meet the conditions given in Eqs. (27,28) as shown in Fig. 1b. The variables γ 2 and γ 1 are the fast and slow variables, respectively.

According to Eqs. (18,37), the slow manifold obeys

k 1 X 2 -(k -1 + k 2 ) Y + k -2 A = 0 (42)
Substituting for the concentration Y from Eq. (42) into Eq. ( 36), we have

dX dt = -2 k 1 k 2 k -1 + k 2 X 2 + 2 k -1 k -2 k -1 + k 2 (43)
The associated reduced reaction scheme is

2X k 1 k 2 k -1 +k 2 -------- k -1 k -2 k -1 +k 2 A ( 44
)
Linearizing Eq. ( 43) around the steady state X 0 = k -1 k -2 A k 1 k 2 , we find that the variable X obeying the reduced reaction scheme evolves with the characteristic time

τ = k -1 + k 2 4κ k -1 k 2 (45)
Approximate initial conditions associated with the reduced reaction scheme given in Eq. ( 44) can be deduced from the initial conditions X ini and Y ini associated with the initial reaction scheme. Using the second column of the change-of-basis matrix given in Eq. ( 41), we obtain the equation of a straight line parallel to the fast eigendirection and

passing through (X ini , Y ini ) 4κ k -1 k 2 + λ 2 X -X ini -2k -1 Y -Y ini = 0 ( 46
)
Solving the system given in Eqs. (42,46), we obtain the initial condition X ini of the reduced reaction scheme

X ini = ζ + √ ∆ 2k -1 (47)
where

∆ = ζ 2 -4 κ 2 + k 1 ζX ini -k 1 (k -1 + k 2 ) Y ini and ζ = (k -1 +k 2 ) 2k -1 4κ k -1 k 2 + λ 2 .
The linear or nonlinear nature of the reaction schemes induces differences in the elimination of the variable Y that are analyzed in the next subsection.

Comparison of QSSA application to linear and nonlinear dynamics

As shown in Fig. 1, the linear and nonlinear dynamics lead to very different parameter domains in which the elimination of Y is valid. Figure 2 displays the eigendirections given in Eq. ( 23) (Eq. (38), resp.) and the slow manifold given in Eq. ( 29) (Eq. ( 42), resp.) for linear (nonlinear, resp.) dynamics.

Trajectories in the space (X, Y ) deduced from the numerical integration of Eqs. (19,20) for linear dynamics and Eqs. (20,34) for nonlinear dynamics are also shown for rate constant values in the validity domain of Y elimination. For linear dynamics, the slow manifold is a straight line identical to the slow eigendirection. In this case, the slow manifold is a vector space. For nonlinear dynamics, the slow manifold is curved but, as expected, is tangent to the slow eigendirection at the steady state. For both dynamics, the fast eigendirections are quasi-parallel to the Y -axis. The trajectories are first parallel to the fast eigendirection, then close to the slow manifold, and eventually reach the steady state in the slow eigendirection. The elimination of the variable Y is valid as soon as the trajectories have reached the vicinity of the slow manifold. The rate constants are expressed in arbitrary units.

The variation of X at large timescales (t τ 2 ) is correctly described by the reduced reaction schemes with the initial conditions given in Eqs. (33,47).

Figure 3a displays the variation of the absolute characteristic times deduced from Eqs. (39,40) with the rate constant k 2 for nonlinear dynamics. The characteristic time τ given in Eq. ( 45) associated with the reduced reaction scheme is also displayed. In agreement with Fig. 1b, the variation of τ is similar to the variation of τ 1 for sufficiently large k 2 .

Figure 3b shows the variation of the change-of-basis matrix elements given in Eq. ( 41

Parallel with the Born-Oppenheimer approximation

As shown in Fig. 3a for the chemical scheme given in Eqs. (34,35), the characteristic times τ 1 and τ 2 display a typical avoided crossing in the sense of quantum chemistry. A parallel between QSSA and the Born-Oppenheimer approximation can be drawn. Both approximations are adiabatic eliminations. [START_REF] Godfried | Elimination of fast variables[END_REF] The Born-Oppenheimer approximation consists in neglecting the fast movements of the electrons and considering only their response to the slower movement of the nuclei. For degenerate energy states, i.e. identical eigenvalues of the Hamiltonian, the Born-Oppenheimer approximation is not valid. Introducing a perturbation splits the degenerate energy states and leads to an avoided crossing. Similarly, the introduction of a fast variable in a reaction scheme can be considered a perturbation leading to an avoided crossing for the characteristic times. The trajectories are deduced from the numerical integration of Eqs. (21,22) for linear dynamics and Eqs. (36,37) for nonlinear dynamics. The initial conditions (X ini , Y ini ) are (0, 0.

partial-equilibrium approximation (PEA)

The partial-equilibrium approximation (PEA) offers another way to eliminate a fast variable, systemically leading to a reduced reaction scheme. [START_REF] Simon | The steady state and equilibrium approximations: A geometrical picture[END_REF][START_REF] Michaelis | The kinetics of the inversion effect[END_REF] When a reaction step is much faster than the others, the fast variation of its extent ξ i can be eliminated at the timescale of the evolution of the other extents. The approximation consists in writing dξ i dt = 0. This approximation is similar to QSSA, with the only difference that the eliminated variable is an extent and not a concentration.

The same comments as for the elimination of a concentration can be made for the elimination of an extent. In summary, the extent ξ i can be eliminated in the vicinity of the steady state if the following conditions are met. First a variable γ k in the eigenbasis of the Jacobian matrix associated with the extents must be faster than the other variables, leading to the condition given in Eq. ( 16). Second the variables γ k and ξ i must evolve similarly on the time interval [0, τ k ], leading to the condition given in Eq. ( 17). The elimination of the extent ξ i is valid when the variable γ k associated with the smallest characteristic time τ k has relaxed. In the space of the extents, the trajectories are first parallel to the fast k th eigendirection, i.e. the ξ i -axis, and then follow the slow manifold defined by

dξ i dt = 0 (48)
The slow manifold is tangent to the hyperplane defined by the slow eigendirections.

Contrary to QSSA, PEA always leads to a reduced chemical scheme straightforwardly obtained by removing the eliminated reaction step from the initial scheme.

We illustrate PEA using the following reaction scheme

2X k 1 ---- k -1 Y (49) X + Y k 2 ---- k -2 A ( 50
)
where A is a chemostat. The rate laws for the concentrations are given by

dX dt = -2k 1 X 2 + 2k -1 Y -k 2 XY + k -2 A (51) dY dt = k 1 X 2 -k -1 Y -k 2 XY + k -2 A (52)
The steady state is

X 0 , Y 0 = k -1 k -2 A k 1 k 2 1 3 , k 1 k 2 -2 A 2 k -1 k 2 2 1 3 (53)
The extents ξ 1 and ξ 2 are associated with the steps given in Eq. (49) and Eq. (50), respectively. Using Eq. ( 14) we obtain the relationships between the extents and the concentrations

X -X 0 = -(2ξ 1 + ξ 2 ) (54) Y -Y 0 = ξ 1 -ξ 2 (55)
Following Eq. ( 13), we find

dξ 1 dt = k 1 X 2 -k -1 Y (56) dξ 2 dt = k 2 XY -k -2 A (57)
Substituting for X and Y from Eqs. (54,55) into Eqs. (56,57) we obtain the rate laws for the extents

dξ 1 dt = k 1 (2ξ 1 + ξ 2 ) 2 -4k 1 X 0 + k -1 ξ 1 + -2k 1 X 0 + k -1 ξ 2 (58) dξ 2 dt = k 2 -2ξ 2 1 + ξ 2 2 -ξ 1 ξ 2 + X 0 -2Y 0 ξ 1 -X 0 + Y 0 ξ 2 (59)
where X 0 and Y 0 are given in Eq. ( 53). We follow the resolution method detailed in section 2. The system given in Eqs. (58,59) is linearized leading to the Jacobian matrix

J = -(4k 1 X 0 + k -1 ) -2k 1 X 0 + k -1 k 2 (X 0 -2Y 0 ) -k 2 (X 0 + Y 0 ) ( 60
)
The eigenvalues of J are given by

λ 1 = 1 2 -k 2 X 0 + Y 0 + 4k 1 X 0 + k -1 + √ ∆ (61) λ 2 = 1 2 -k 2 X 0 + Y 0 + 4k 1 X 0 + k -1 - √ ∆ (62) with ∆ = (k 2 (X 0 + Y 0 ) + 4k 1 X 0 + k -1 ) 2 -36k 2 k -1 Y 0 .
The change-of-basis matrix is

P =   P 11 = -2k 1 X 0 +k -1 √ (-2k 1 X 0 +k -1 ) 2 +(4k 1 X 0 +k -1 +λ 1 ) 2 P 12 = -2k 1 X 0 +k -1 √ (-2k 1 X 0 +k -1 ) 2 +(4k 1 X 0 +k -1 +λ 2 ) 2 P 21 = 4k 1 X 0 +k -1 +λ 1 √ (-2k 1 X 0 +k -1 ) 2 +(4k 1 X 0 +k -1 +λ 1 ) 2 P 22 = 4k 1 X 0 +k -1 +λ 2 √ (-2k 1 X 0 +k -1 ) 2 +(4k 1 X 0 +k -1 +λ 2 ) 2   (63)
The eigenvalues of J are negative for all values of the rate constants. Consequently the steady state (X 0 , Y 0 ) is stable. The conditions given in Eqs. (16,17) applied to the elimination of the extent ξ 1 are written as According to Eqs. (48,56,58), the slow manifold is defined by

τ 2 ≤ 10τ 1 (
k 1 (2ξ 1 + ξ 2 ) 2 -4k 1 X 0 + k -1 ξ 1 + -2k 1 X 0 + k -1 ξ 2 = 0 (66)
in the space of the extents and by

k 1 X 2 -k -1 Y = 0 (67)
in the space of the concentrations.

The elimination of the fast extent ξ 1 leads to the reduced reaction scheme

X + Y k 2 ---- k -2 A ( 68
)
The mere knowledge of the reduced chemical scheme is not sufficient to correctly describe the evolution of the concentrations X and Y after the elimination of the extent ξ 1 . The concentrations X and Y are not independent but obey Eq. (67). Equations (67,68) are necessary to write the rate laws associated with the reduced reaction scheme.

Trajectories in the spaces (ξ 1 , ξ 2 ) of the extents and (X, Y ) of the concentrations are shown in Fig. 5 with rate constant values for which the elimination of the extent ξ 1 is valid. The eigendirections are obtained using Eq. ( 63) and the slow manifold using Eq.

(66) in the space (ξ 1 , ξ 2 ) and Eq. (67) in the space (X, Y ). The trajectories are obtained by numerical integration of Eqs. (58,59) in the space (ξ 1 , ξ 2 ) and Eqs. (51,52) in the space (X, Y ). As expected the fast eigendirection is parallel to the ξ 1 -axis in the space (ξ 1 , ξ 2 ). Both eigendirections are not parallel to any axis in the space (X, Y ) since the evolution of X and Y depends on both characteristic times τ 1 and τ 2 . In both spaces, the slow manifold is tangent to the slow eigendirection. In addition, the trajectories are first parallel to the fast eigendirection and evolve toward the steady state along the slow manifold. The reduced reaction scheme is valid once the trajectory have reached the vicinity of the slow manifold.

Conclusion

Chemical kinetics is usually described by the evolution of two sets of variables, either the concentrations of the chemical species or the extents of the reaction steps. The concentrations and the extents are linked by linear relationships. Laws of conservation are used to reduce dynamics to the evolution of independent variables. The quasi-steady-state and partial-equilibrium approximations require more hypotheses than the mere elimination of a fast variable in rate equations since the eliminated variable must have a physical meaning. It must be a chemical species in the case of QSSA and an extent in the case of PEA.

We have proposed a general framework to make precise the minimum number of conditions on the rate constants for which either approximation is valid in a domain of the space of the variables. The knowledge of the values of all rate constants is not necessary to apply one or the other approximation. We exploit the eigendecomposition into eigenvalues and eigenvectors of the Jacobian matrix deduced from the linearization of the rate equations around a steady state. The absolute real part of the eigenvalues correspond to the inverse of the characteristic times. Eliminating a fast variable is possible as soon as one characteristic time is one order of magnitude smaller than the others. We write a second condition on the eigenvectors imposing that the eliminated variable is the concentration C i for QSSA or the extent ξ i for PEA. The two conditions are necessary and sufficient in the linear domain around the steady state, i.e. in a sufficiently small neighborhood of the steady state in the space of the concentrations or extents.

Both approximations are zeroth-order perturbation methods consisting in considering that the relaxation of the fast variable is instantaneous leading to dC i /dt = 0 or dξ i /dt = 0. Either of these equations defines the slow manifold, i.e. a relationship between the fast variable and the slow variables. After the elimination of the fast variable, dynamics occurs on the slow manifold. Substituting for the fast variable from the slow manifold equation into the rate equations of the slow variables leads to a reduced dynamics. For some chemical schemes QSSA and PEA may lead to the same reduced rate equations when the conditions of validity of the two approximations overlap. Far from being stationary, the evolution of the eliminated variable occurs at the timescale on which the reduced rate equations focus. To point out this essential feature, we suggest to use the name quasi-steady-state instead of steady-state approximation and partial-equilibrium instead of equilibrium approximation. It is to be noted that QSSA does not always provide a reduced mechanism since the elimination of the fast variable may lead to non polynomial rate equations for the slow variables as illustrated by the familiar Michaelis-Menten equations. In the case of PEA, the reduced mechanism is straightforwardly obtained but may not be sufficient to account for dynamics. The equation of the slow manifold may be necessary.

Figure 1 :

 1 Figure 1: Domains of validity of the elimination of species Y for linear (a) and nonlinear (b) dynamics in the parameter space. The colored volumes are defined by Eqs. (27,28). The rate constants are expressed in arbitrary units.

) with k 2

 2 Figure 3b shows the variation of the change-of-basis matrix elements given in Eq. (41) with k 2 for nonlinear dynamics. For large values of k 2 , the concentration X essentially varies as γ 1 and Y as γ 2 . For small values of k 2 , both characteristic times are necessary to account for the evolution of X and Y . Whereas the condition τ 2 τ 1 for the elimination of Y is obeyed for all values of k 2 , the condition | P 21 | | P 22 | is fulfilled only for large enough k 2 . The condition on the elements of the change-of-basis matrix is more restrictive than the condition on the characteristic time.

1 YFigure 2 :

 12 Figure2: Trajectories (black solid lines) in the space (X, Y ), steady state (red disk), eigendirections associated with the fast variable (blue dashed line) and the slow variable (red dashed line) in the eigenbasis, and slow manifold (green dotted line) for linear (a) and nonlinear (b) dynamics. The trajectories are deduced from the numerical integration of Eqs.(21,22) for linear dynamics and Eqs. (36,37) for nonlinear dynamics. The initial conditions (X ini , Y ini ) are (0, 0.1), (0.3, 0), (0.7, 0.1), (1, 0) and (0, 0.1), (0.3, 0), (0.9, 0.1), (1.4, 0) in arbitrary units (a.u.) for linear and nonlinear dynamics, respectively. The parameter values are k 1 = 0.1 a.u., k -1 = 1 a.u., k 2 = 2 a.u., and k -2 A = 0.1 a.u.

Figure 3 :

 3 Figure2: Trajectories (black solid lines) in the space (X, Y ), steady state (red disk), eigendirections associated with the fast variable (blue dashed line) and the slow variable (red dashed line) in the eigenbasis, and slow manifold (green dotted line) for linear (a) and nonlinear (b) dynamics. The trajectories are deduced from the numerical integration of Eqs.(21,22) for linear dynamics and Eqs. (36,37) for nonlinear dynamics. The initial conditions (X ini , Y ini ) are (0, 0.1), (0.3, 0), (0.7, 0.1), (1, 0) and (0, 0.1), (0.3, 0), (0.9, 0.1), (1.4, 0) in arbitrary units (a.u.) for linear and nonlinear dynamics, respectively. The parameter values are k 1 = 0.1 a.u., k -1 = 1 a.u., k 2 = 2 a.u., and k -2 A = 0.1 a.u.

  64) | P 11 | ≤ 10 | P 12 | (65) where τ 1 = 1/ | λ 1 | and τ 2 = 1/ | λ 2 | are deduced from Eqs. (61,62) and P 11 and P 12 are given in Eq. (63).

Figure 4

 4 displays the domain in the rate constant space in which the conditions given in Eqs. (64,65) are met. In the eigenbasis the variables γ 1 and γ 2 are the fast and slow variables, respectively.

Figure 4 :Figure 5 :

 45 Figure 4: Domains of validity of the elimination of extent ξ 1 in the parameter space. The colored volumes are defined by Eqs. (64,65). The rate constants are expressed in arbitrary units and k -2 A = 1.