De Martini

Daniele De Martini
email: daniele.demartini@unimib.it

Robustness and corrections for sample size adaptation strategies based on effect size estimation

Keywords: structural bias, conservativeness, sample size estimation structural bias, conservativeness, launch threshold, Overall Power, postulated correction, sample size estimation

The robustness of the adaptation of the sample size for a phase III trial based on phase II data is studied --when phase III is lower than phase II effect size. Conservative sample size estimation strategies are compared. When the rate between phase III and phase II effect size is greater than 0.8, Calibrated Optimal Strategy provides acceptable results. A correction for balancing the structural bias is introduced, based on a postulation of the bias. When the postulated correction is right, or even smaller than necessary, COS works well. A higher than necessary correction should be avoided.

Introduction

Sample size estimation (SSE) consists in estimating the sample size required by a statistical test for achieving a predefined power. In the parametric framework the power is a function of the true effect size (ES), so that SSE can be accomplished through the estimation of this ES. In practice, the true ES is unknown (being this latter the object of the research) and it can be estimated on the basis of data from a pilot study. Subsequently, the sample size for the further study is estimated. For simplicity, and to establish SSE in the context of clinical trials, we assume that the pilot and the subsequent experiments are a phase II and a phase III trial, respectively. SSE is an intuitive practice that has been adopted by many authoritative authors (see for example Rosner, 2005, Ch.8;Efron and Tibshirani, 1993, Ch.25).

In the last decade, the conservative approach to SSE (i.e. CSSE), which accounts for the variability of phase II data, has been proposed under both the frequentist (Shao and Chow, 2002, Sec.5.2) and the Bayesian approach [START_REF] Chuang-Stein | Sample Size and the Probability of a Successful Trial[END_REF].

The introduction, according to [START_REF] Wang | Adapting the sample size planning of a phase III trial based on phase II data[END_REF], of a launch threshold in the setting of CSSE (i.e. phase III is launched, and its sample size is estimated, only when phase II shows effect size over a predefined threshold of a certain clinical importance) represents a strong link between theory and practice: on one hand the launching threshold rationalizes a step often adopted in practice, and on the other Some papers report comparisons among different CSSE strategies. [START_REF] Wang | Adapting the sample size planning of a phase III trial based on phase II data[END_REF] evaluated three frequentist CSSE strategies, mainly on the basis of launch probability and of the Average Power of phase III (i.e. the average of the random power given by the randomness of sample size estimators, provided that the phase III trial was launched). These performance indicators were computed under four different Scenarios: in Scenario 1 the effect size of phase II was equal to that of phase III; in Scenarios 2-4 phase III effect size was lower than that of phase II. The authors concluded by suggesting the adoption of the conservative strategy consisting in the use of the effect size observed in phase II minus one time its standard error, in order to then compute the sample size for phase III. Note that in Scenarios 2-4 the authors studied the robustness of the three strategies.

Let us now introduce the Overall Power (OP) of the phase II and the phase III, that is the probability to launch times the Average Power. [START_REF] Fay | Accounting for Variability in Sample Size Estimation with Applications to Nonhaderence and Estimation of Variance and Effect Size[END_REF] compared, mainly on the basis of a modified for continuity version of the OP, the simple pointwise frequentist approach with two Bayesian techniques. Scenario 1 alone was considered and the authors suggested the adoption of a corrected Bayesian strategy. (In detail, to compute the OP [START_REF] Fay | Accounting for Variability in Sample Size Estimation with Applications to Nonhaderence and Estimation of Variance and Effect Size[END_REF] set the launch threshold at -∞ and when the effect size was estimated to be under the null hypothesis they estimated the sample size for the phase III, for continuity, to be ∞; then, in these cases, and when the alternative hypothesis was true, i.e. when estimating the sample size has a practical sense, the phase III power was considered to be 1. Further details and comments can be found in De [START_REF] De Martini | Adapting by calibration the sample size of a phase III trial on the basis of phase II data[END_REF] It is worth noting that, from the perspective of statistical methodology classification, [START_REF] Hung | Methodological issues with adaptation of clinical trial design[END_REF] argued that SSE, and CSSE too, can be viewed as a kind of adaptation by design.

Note also that the OP evaluates the performance of SSE strategies on phase II and phase III (i.e. independently on phase II results) and it is therefore indicated for comparing strategies globally, where the Average Power becomes of interest once the phase II succeeded, say at the beginning of phase III. amount of conservativeness -COS. To evaluate the results the OP of the different strategies, as well as the mean and the MSE of sample size estimators, were computed. The launch threshold was adopted. Bayesian strategies performed poorly since they showed a very high mean and/or MSE of sample size estimators. COS clearly performed better than the other frequentist conservative strategies, both in terms of OP and of sample size estimators behavior (viz. MSE). Costs and experimental times are, therefore, considerably reduced and standardized. These results were computed solely under Scenario 1.

Broader and more heterogeneous patient populations are often pursued in phase III clinical trials as compared to in phase II studies. This might induce larger variability and/or lower average differences between drugs in primary endpoints variables. Both phenomena imply that standardized phase III effect sizes are lower than phase II ones. [START_REF] Wang | Adapting the sample size planning of a phase III trial based on phase II data[END_REF] modeled these situations in Scenarios 2-4.

Our aim is now to study the robustness of some frequentist, Bayesian and optimized (viz. COS) CSSE strategies through the evaluation of their performances under these latter Scenarios. Then, some techniques for correcting the difference between phase III and phase II effect size, that are usually adopted under these Scenarios, are discussed and a new one is introduced. The robustness of CSSE strategies improved by this correction technique are, hence, examined.

The paper is as follows: Section 2 deals with the theoretical framework, and Section 3 with CSSE strategies which will be compared, under Scenarios 2-4, in Section 4. Section 5 is devoted to the techniques of bias correction and to the application of the new one to CSSE strategies. In Section 6, a comparison among the corrected versions of our CSSE strategies is presented; Section 7 contains the discussion, and Section 8 the conclusions.

Theoretical framework

In accordance with [START_REF] Wang | Adapting the sample size planning of a phase III trial based on phase II data[END_REF], we focus on the one-sided alternatives test in a comparison of the means of two normal distributions in phase III, as we did De Martini (2010). The common variance σ 2 is assumed to be known and equal to 1, in phase III as well as in phase II. The statistical hypotheses are, therefore, H 0 : µ 3,T = µ 3,C vs H 1 : µ 3,T > µ 3,C , and the standardized phase III effect size is

δ 3 = (µ 3,T -µ 3,C )/σ = µ 3,T -µ 3,C .
Being m 1 = m 2 = m the sizes of the samples drawn from each group in phase III, the test statistic is T m = m/2( Xm -Ȳm ). Given the type I error α, the true power of the phase III trial is

π δ3 (m) = P δ3 (T m > z 1-α ) = Φ(δ 3 m/2 -z 1-α ),
where

z 1-α = Φ -1 (1 -α),
and Φ is the distribution function of the standard normal.

The ideal sample size per group for phase III is, therefore,

M I = min{m | π δ3 (m) > 1 -β} = 2(z 1-α + z 1-β ) 2 /δ 2
The effect size of phase II is

δ 2 = µ 2,T -µ 2,C . Being n 1 = n 2 =
= 2(z 1-α + z 1-β ) 2 /(d γ 2,n ) 2 + 1.
Hence, the OP (i.e. the probability of rejecting during phase III, when M I is estimated on the basis of phase II data) is:

OP n (•) = Mmax m=2 P δ2 (M • n = m)π δ3 (m) (1)
The average of sample size estimators is then

E[M • n |d • 2,n > δ 0L ],
and their mean square error

(MSE) is E[(M • n -M I ) 2 |d • 2,n > δ 0L ].
Remark 1. Although we used δ 2 and δ 3 for defining the effect sizes of phase II and phase III, it should be noted that the effect the drug has is not a function of the development phase of the sponsor. Instead, there are different treatment effects for the population studied for particular phase II and phase III protocols. Nevertheless, when δ 2 = δ 3 a structural bias arises within SSE.

CSSE strategies

In this Section five different CSSE strategies are recalled and are to be compared in Section 4 in order to evaluate their robustness. For each strategy, motivation for inclusion in our study is explained.

Pointwise strategy (PWS). It is the fixed-γ conservative strategy with γ = 50%: in practice, the observed effect size d 2,n is adopted and the sample size estimator is

M 50% n = 2(z 1-α + z 1-β ) 2 /(d 2,n ) 2 + 1 = M n .
This strategy is considered because it is the simplest, and also because the results regarding the estimation of the sample size under Scenario 1 were quite good [START_REF] De Martini | Adapting by calibration the sample size of a phase III trial on the basis of phase II data[END_REF].

One standard error conservative strategy (1SES). In this strategy, 1 standard error is subtracted from d 2,n to obtain a conservative estimate of δ 3 . This corresponds to 84.1% conservative strategy, so that d [START_REF] Wang | Adapting the sample size planning of a phase III trial based on phase II data[END_REF], who also considered Scenarios 2-4, deem it should be adopted.

Third quartile conservative strategy (3QS). To adopt 75% conservative strategy appears a reasonable choice, because in this way the probability of planning an underpowered experiment is reduced to the fixed rate of 1/4. Hence, d • 2,n = d 75% 2,n and the sample size estimator is M 75% n . We consider this fixed-γ strategy too because the performances of 3QS in terms of OP under Scenario 1 were closer to COS than those of PWS and 1SES.

Calibrated optimal strategy (COS). This strategy is based on the fact that if a fixed-γ conservative strategy is adopted, then its OP (say OP n (γ)) turns out to be a concave function of γ. Consequently, we introduced an optimization for OP n (γ) (De Martini, 2010), with the constraint of not exceeding 1 -β. Being that optimum (say γ O,n ) unknown, it can be estimated through the plug-in principle and its estimate is g O,n . With COS d • 2,n = d 2,n and the sample size estimator is Bayesian truncated strategy (BAT). In a Bayesian framework, the posterior distribution of δ 2 given d 2,n is, with noninformative prior, N (d 2,n , 2/n), and it is used for making inference on δ 3 . Consequently the Bayesian estimate of the true power

M g O,n n = 2(z 1-α + z 1-β ) 2 /(d g O,n n ) 2 + 1. COS
is π Ban (m) = ∞ -∞ π z (m)φ d2,n,2/n (z) dz.
The simple Bayesian estimate of the sample size, suggested by [START_REF] Chuang-Stein | Sample Size and the Probability of a Successful Trial[END_REF] and [START_REF] Fay | Accounting for Variability in Sample Size Estimation with Applications to Nonhaderence and Estimation of Variance and Effect Size[END_REF], is, therefore, Remark 2. With γ conservative strategies the OP in (1) simplifies to

M BaS n = min{m | π Ban (m) > 1 -β}. Note that in some circumstances lim m→∞ π Ban (m) < 1 -β, so that M BaS
OP n (γ) = Mmax m=2 {Φ( n 2 ( z 1-α + z 1-β (m -1)/2 -δ 2 )+z γ )-Φ( n 2 ( z 1-α + z 1-β m/2 -δ 2 )+z γ )}Φ( m 2 δ 3 -z 1-α ) (2)
Without loss of generality, we assume δ 2 = δ 3 /k so that (2) becomes:

OP n (γ) = Mmax m=2 {Φ( n 2 ( z 1-α + z 1-β k (m -1)/2 -δ 2 )+z γ )-Φ( n 2 ( z 1-α + z 1-β k m/2 -δ 2 )+z γ )}Φ( m 2 kδ 2 -z 1-α ) (3)
Remark 3. It should be noted that [START_REF] Fay | Accounting for Variability in Sample Size Estimation with Applications to Nonhaderence and Estimation of Variance and Effect Size[END_REF] considered the variance to be unknown and that they used the fiducial distribution of the effect size, actually not a Bayesian posterior. Nevertheless, it is easy to show that in case the variance is known the fiducial distribution reduces to the Bayesian posterior with noninformative prior, i.e. N (d 2,n , 2/n).

Evaluating the robustness of CSSE strategies

Here the performances of the five strategies presented in the previous Section are considered under Scenarios 2-4 of [START_REF] Wang | Adapting the sample size planning of a phase III trial based on phase II data[END_REF]. In Scenario 2 the phase III effect size is quite a bit (20%) lower than in phase II, and the launch threshold is set equal to the phase III effect size. In Scenario 3 the ratio between phase II and phase III effects size is the same as in the latter case, and the launch threshold is set at a value somewhat lower than δ 3 . In Scenario 4 the phase III effect size is much lower (50%) than in phase II, and the launch threshold is set as in Scenario 3.

In order to evaluate the performances of the strategies their OP, which is an important evaluation tool, is computed. Then, focusing on the behavior of sample size estimators, their bias (i.e.

|M I -E[M • n |d • 2,n > δ 0L ]|)
and their variability must be evaluated. The MSE of sample size estimators is in particular taken into consideration -this concerns both bias and variability, since it is the sum of the square of the bias with the variance. We also report the mean of sample size estimators.

Design of the study

In a recent work we stated (De Martini, 2010) that phase II sample sizes n between M I /3 and 4M I /3 were the most important from a practical point of view, and that higher values of n were evaluated to look at the asymptotic behavior of SSE strategies only. Now, it should be noted that under Scenarios 2-4 sample size estimation is not

consistent, since M • n does not tend to M I (but to M I k 2 , with k < 1). Moreover, OP n (•) does not tend to 1 -β (but to π δ3 (M I k 2 ) < π δ3 (M I ) 1 -β).
Then, the behavior of SSE strategies is not relevant for high values of n either from a practical standpoint or from a theoretical one. As regards the level of the power to be achieved, Wang et al.(2006) argued that 1-β = 80% is a low power for CSSE, and this point is confirmed in De Martini (2010). Here, under Scenarios 2-4 where π δ3 (M I k 2 ) < 1 -β, a power choice of 80% appears even more penalizing.

Consequently, for evaluating robustness only three n settings are considered in all Scenarios (i.e. 2M I /3, M I and 4M I /3, avoiding n = M I /3 because of the poor performances of all strategies under Scenario 1), and the power is set at 1 -β = 90%.

As regards the effect size parameters, in Scenario 2 we set δ 3 = 0.2, 0.5, 0.8, k = 0.8 (that implies δ 2 = δ 3 /0.8) and δ 0L = δ 3 ; we, therefore, consider 9 settings (i.e. 3 ns × 3 δs). In Scenario 3 we set the launch threshold δ 0L at 0.1, with δ 3 = 0.2, 0.5, 0.8; then we set δ 0L = 0.25, with δ 3 = 0.5, 0.8; as in Scenario 2, we have δ 2 = δ 3 /0.8; we, therefore, consider 15 settings (i.e. 9 + 6). In Scenario 4 we set δ 3 = 0.2, 0.5, 0.8, k = 0.5 (i.e. δ 2 = δ 3 /0.5) and δ 0L = 0.1, so that 9 settings are considered.

Results

Under Scenarios 2 and 4 all strategies do not present robust behavior. Being Scenario 3 the closest to Scenario 1, the strategies show better performances.

In scenario 2 the best performer is BAT, which shows the maximum OP (62.7%) with δ 3 = 0.8 and n = 4M I /3; its minimum was 55.9%, with δ 3 = 0.2 and n = 2M I /3.

The OPs are so low that a discussion on the MSE and the average of sample size estimators is unnecessary. These poor performances are due to too high launch threshold settings with respect to true effect sizes, inducing low launch probabilities, with subsequent low OPs.

In Scenario 4 the best performer is 1SES, whose OP is around 50% with every δ 3 and n. A fortiori we do not discuss MSEs. Poor performances under this Scenario are caused by too large differences between the effect sizes of the two phases, i.e.

δ 3 << δ 2 , so that pilot data estimate a sample size (i.e. M I k 2 ) too much lower than the interesting one (i.e. M I k 2 << M I with k = 0.5). Consequently, all strategies provide small sample size estimates, inducing low OPs. Scenario 3 is the closest to Scenario 1, where COS was clearly the best performer [START_REF] De Martini | Adapting by calibration the sample size of a phase III trial on the basis of phase II data[END_REF]. We then expect the strategies performances not to be far from those under Scenario 1. Table 1 reports OPs, together with the MSE of SS estimators and their averages in the 9 settings with δ 0L = 0.1. All strategies are quite robust, performing better than under Scenarios 2 and 4: their OPs are higher than 70%.

Nevertheless, the OPs are a bit lower than those of Scenario 1, as are the MSEs, as a consequence of the differences between phase III and phase II ES.

As it concerns strategy comparison, the (expected) better performances of COS are not as clear as under Scenario 1. It can be easily noted that COS performs better than PWS (higher OP, lower MSE, average sample size closer to M I ). As regards BAT, although the average over all 15 settings of its OPs is 3.3% larger than that of COS (which seems in any case acceptable resulting 77.5%), we still prefer COS because its average sample size is closer to M I and it mainly presents a dramatically lower MSE: the absolute error (i.e. MSE 1/2 ) of BAT is more than 4 times higher than that of COS, on average over all 15 settings (i.e. the average of the rates of MSE 1/2 of BAT with respect to those of COS among the 15 settings is 4.16, in other words 316% higher). Among 3QS and 1SES, which perform similarly, we prefer the former: although the OP of 1SES is 2.0% larger, its MSE 1/2 is higher too (40%) with respect to that of 3QS, and the average sample size of 3QS is 2.2 times closer to M I . Now the point is which is the better performer between COS and 3QS. As regards OPs, 3QS provides, on average, 81.7%, i.e. 4.2% more than COS. COS sample size estimator however performs somewhat better: the average sample size of COS is 2.0 times closer to M I , and, mainly, the MSE 1/2 of 3QS is, on average, 2.2 times higher than COS. For these reasons we maintain that under Scenario 3 too COS is the best strategy.

Bias correction

In the previous Section we evaluated the robustness of CSSE strategies when a structural bias is present, resulting from differences between phase II and phase III effect sizes (see also Remark 1). This bias, as well as those present in other branches of statistical theory, can nonetheless be corrected or at least reduced. In this Section an intuitive technique of bias correction is presented, and the corrected CSSE strategies related to those recalled in Section 3 are derived.

In the context of CSSE the authors usually suggest to modify the type I and/or type II errors to reduce the bias. [START_REF] Fay | Accounting for Variability in Sample Size Estimation with Applications to Nonhaderence and Estimation of Variance and Effect Size[END_REF], although no structural bias was considered (i.e. δ 2 = δ 3 ), observed that the Bayesian CSSE strategy provided an OP higher than requested, and so applied a correction to the nominal power (e.g. a power of 76% was suggested for 1 -β = 80%). [START_REF] Wang | Adapting the sample size planning of a phase III trial based on phase II data[END_REF], in presence of a structural bias, suggested techniques based on the modification of the type I and/or type II errors. One of these techniques consists in applying β * < β. It is worth noting that the launch probability is not modified, whereas the correction influences sample size estimators merely. [START_REF] Wang | Adapting the sample size planning of a phase III trial based on phase II data[END_REF] argue that these strategies are quite suitable under Scenarios 2-4, when applied to 1SES. For example when δ 2 = 0.3 and δ 3 = 0.2, if α = 0.025 and 1 -β = 80%, then 1 -β * = 92% is suggested. Note that their conclusions are based on the computation of the average of sample size estimators only (see Table 4 in their work). Our proposal concerns a correction to be applied directly on the effect size. In practice, the corection should be applied to the effect size estimator d • 2,n . This idea stems from two observations: on one hand, CSSE aims at studying the performances of M • n , based on d • 2,n , which is, under Scenarios 2-4, a biased (and non consistent) estimator of δ 3 ; on the other hand, the structural bias can be modeled through

δ 2 = δ 3 /k.
c M max = 2(z 1-α + z 1-β ) 2 /(δ 0L * k c ) 2 + 1.
Finally, note that the launch probability is not modified by the correction suggested here, and this is in accordance with those proposed by [START_REF] Wang | Adapting the sample size planning of a phase III trial based on phase II data[END_REF].

Indeed, d • 2,n remains the same, M • n alone is modified.

Corrected CSSE strategies

Corrected fixed-γ strategies (viz. PWS, 1SES, 3QS). The γ conservative estimator of δ 3 is multiplied by the correction k c , obtaining c d γ 2,n = (d 2,n -z γ / n/2) * k c , and, consequently,

c M γ n = 2(z 1-α + z 1-β ) 2 /( c d γ 2,n ) 2 + 1.
Corrected COS. We first refer to the corrected version of the estimated OP, which, following from (3), is:

c ÔP n (γ) = c Mmax m=2 {Φ( n 2 ( z 1-α + z 1-β k c (m -1)/2 -d 2,n )+z γ )-Φ( n 2 ( z 1-α + z 1-β k c m/2 -d 2,n )+z γ )}Φ( m 2 k c d 2,n -z 1-α ) (4)
Then, being g O,n the argument of the constrained maximum of (4), we make use of

c d g O,n 2,n = (d 2,n -z g O,n / n/2) * k c to compute the sample size estimate c M g O,n
n .

Corrected BAS. k c modifies Bayesian strategies through the corrected power

c π Ban (m) = ∞ -∞ π z * kc (m)φ d2,n,2/n (z) dz, so that c M BaS n = min{m | c π Ban (m) > 1 -β}, and c M BaT n = min{ c M BaS n , c M max }.
Remark 4. From the mathematical perspective, the correction techniques of [START_REF] Wang | Adapting the sample size planning of a phase III trial based on phase II data[END_REF] and that we proposed here are connected. In fact, both approaches aim to correct 2(z 1-α + z 1-β ) 2 /(δ γ 2,n ) 2 + 1 in order to make it closer to M I . Corrections can be brought to the numerator [START_REF] Wang | Adapting the sample size planning of a phase III trial based on phase II data[END_REF] or to the denominator (as we suggest). For each correction on the numerator there exists an equivalent correction for the denominator: for example, for every β * < β there exists k c < 1 such that z 1-α + z 1-β * = (z 1-α + z 1-β )/k c . For this reason we will not include correction techniques based on the modification of the type I and/or type II errors in the ensuing comparison (Section 6).

Remark 5. Note that since k c < 1 the average and the standard deviation of c M γ n are increased by 1/k c times with respect to those of

M γ n . As a consequence, M SE[ c M γ n ] is often higher than M SE[M γ n ].
We expect analogous behavior of the corrected COS and BAT.

Remark 6. If the postulated correction k c is right (i.e. δ 2 = δ 3 /k c ), then all the considered strategies are consistent (i.e. c M • 2,n tends to M I ). We do not nevertheless fall under Scenario 1: although the mean of c d 2,n is actually δ 3 , its variance (i.e.

2k 2 c /n) is different from that of d 2,n under Scenario 1 (i.e. 2/n).

A study comparing corrected CSSE strategies

We here evaluate the performances of the five corrected strategies introduced in Section 5.

Design of the study

Scenario 3 alone of the previous Section 4 is examined, with the 9 settings where δ 0L = 0.1. We recall that k = 0.8. As regards k c , there are three postulated corrections: a smaller than necessary one, i.e. k c = 0.9, the right one, i.e. k c = 0.8, and a higher than necessary one, i.e. k c = 0.7. So, 27 settings are evaluated (3 ns × 3 δ 3 s × 3 k c s). Since our CSSE strategies are consistent when k c = 0.8, we also evaluate their asymptotic behavior, with n up to 4M I , for the three δ 3 s. Once again, we look at OPs of CSSE strategies, and at the MSEs and averages of their sample size estimators.

Results

Table 2 reports OP, MSE and average of sample size estimators obtained with δ 3 = 0.5 alone, i.e. under 9 settings out of 27.

When δ 3 < δ 2 , applying a correction k c < 1 improves the OPs of all strategies, but also increases their MSEs. The averages of sample size estimators increase too, and the bias augments in all circumstances but one (i.e. for PWS with k c = 0.9).

With k c = 0.9, the OPs improve, on average over the 9 settings, from 4% (1SES) to 6.8% (PWS), but they are still lower than 90% (from 80.1% of PWS to 88.5% of 1SES). The MSE 1/2 of corrected strategies increases, on average, with respect to the uncorrected ones, from 16% with PWS (i.e. MSE 1/2 of corrected PWS is 1.16 times higher than that of PWS without any correction) to 30% of 1SES. Hence, the correction, although smaller than necessary, works well because the little increase in MSEs is counter balanced by the good improvements in OPs for all strategies. In particular, the OP of COS increases 6.1%, on average, becoming 83.7% (3QS, 87.1%).

In practice, the corrected COS provides the second best improvement in OP and the second smallest increase in MSE 1/2 (19%). Moreover, the MSE 1/2 of COS is still the lowest: those of PWS, 3QS, 1SES and BAT are, on average, 1.28, 2.53, 3.53 and 5.04 times higher, respectively. Since COS was the best performer under Scenario 3, it remains the recommended strategy even with a smaller than necessary correction.

With a higher than necessary correction, i.e. k c = 0.7, the OPs of all strategies result higher than 90%, going, on average, from 92.3% (PWS) to 95.1% (BAT) (COS, 94.0%). Nevertheless, the MSE 1/2 increases, on average, from 2.14 times with PWS to 2.46 times with COS, with respect to those obtained without correction. Also, the averages of sample size estimators are much higher than M I . These performances are caused by too high sample size estimates provided by this excessive correction.

In the light of sample size estimator behaviors, it would be preferable to avoid higher than necessary corrections. As regards detailed results, the MSE 1/2 of COS is once again the lowest: the enlargements of the other strategies go from 12% of PWS to

331% of BAT.
With the right correction of k c = 0.8, all strategies converge, as n tends to ∞:

their OPs tend to 90% and the MSEs tend to zero. (see Figures 1 and2). It should also be noted that when 2M I ≤ n ≤ 4M I /3, all strategies provide OPs closer to 90% than those obtained with k c = 0.9. Furthermore, considering these pilot sample sizes of practical interest, COS converges faster. Indeed, the strategy with averaged OP (over the usual 9 settings) closest to 90% is COS (89.4%), where the highest differences are provided by 1SES (91.8%). The MSE 1/2 increases, as expected, from 51% of PWS to 74% of 1SES, with respect to that provided by the strategies without correction.

Also, the MSE of COS is the lowest: the enlargements in MSE 1/2 go from 21% of PWS to 4.75 times (i.e. 375%) of BAT. Once again COS is recommended, and this was expected, since postulating the right correction makes estimations quite close to those under Scenario 1.

Finally, we emphasize that the OPs provided by all corrected strategies are a little higher (and closer to 90%) than the respective ones computed in De Martini (2010) under Scenario 1, i.e. where correction was not needed (see Figure 1). Moreover the MSEs with k c = 0.8 are comparable to those under Scenario 1 (see Figure 2). These behaviors are due to higher launch probabilities with respect to Scenario 1 (due to δ 2 > δ 3 ), and to quite close sample size estimation.

Discussion

The statistical literature contains many works on SSE for planning a generical experiment, as well as, in particular, a randomized controlled trial. Over the last five years, the conservative approach to SSE, which takes into account the variability of phase II pilot data, has been proposed under some different approaches. [START_REF] Chuang-Stein | Sample Size and the Probability of a Successful Trial[END_REF] and [START_REF] Fay | Accounting for Variability in Sample Size Estimation with Applications to Nonhaderence and Estimation of Variance and Effect Size[END_REF] argued a Bayesian approach, which consists in averaging the estimated power on the basis of the posterior distribution of the effect size. In the frequentist framework, [START_REF] Wang | Adapting the sample size planning of a phase III trial based on phase II data[END_REF] suggested a simple strategy consisting in the estimation of the sample size on the basis of the 1 standard error conservative estimate of the effect size. However, all these authors did not put sufficient emphasis on the variability of sample size estimators, for example by computing their MSEs, and we believe that this point is mandatory.

Recently, we compared (De Martini, 2010) some Bayesian and some frequentist CSSE strategies with a new calibrated optimal γ conservative one (viz. COS).

The Overall Power (OP) of phases II and III, together with the average of sample size estimators and, mainly, at their MSE, were considered. The indispensable launch threshold criterion, introduced by [START_REF] Wang | Adapting the sample size planning of a phase III trial based on phase II data[END_REF], was adopted, and it was assumed that the effect size in phase II was the same as that of phase III (i.e.

the so-called Scenario 1). COS resulted the best strategy, with good OPs, and with MSEs considerably lower than those of the other strategies. Nevertheless, the size of the phase II sample used for CSSE should be around the ideal phase III sample size, the launch threshold should be lower than one half of the true phase III effect size, and the prefixed power should be set at 90%. These three assumptions were considered as operating CSSE conditions.

In practice, small deviations from Scenario 1 would not substantially change these results. Nevertheless, in some circumstances the difference between phase II and phase III effect sizes can be not negligible, for example in the presence of a more restrictive inclusion criteria in phase II, or even when a lack of knowledge of dose-response relationship occurs. In these cases, phase III effect size is often lower than phase II. Consequently, not only CSSE becomes inconsistent for large samples, but it might also provide bad results for finite phase II sample sizes of practical interest. [START_REF] Wang | Adapting the sample size planning of a phase III trial based on phase II data[END_REF] considered some of these situations (viz. Scenarios 2-4),

where CSSE was applied in the presence of bias.

In this work, we have compared the performances of some Bayesian and some frequentist CSSE strategies with COS under Scenarios 2-4, in order to evaluate whether COS would still have provided good results. The results were mainly based on OP and MSE. As regards Scenarios 2 and 4, COS, unlike under Scenario 1, did not improve the performances of fixed-γ or of Bayesian estimators, and we confirmed the results in [START_REF] Wang | Adapting the sample size planning of a phase III trial based on phase II data[END_REF], where all strategies performed poorly. In Scenario 2 there was the problem of a too high launch threshold, so we confirm that it would be better to set δ 0L < δ 3 /2. In Scenario 4 poor performances were due to the high difference between the effect sizes of phase II and phase III. COS was still the best performer (thanks to its low MSE) under Scenario 3, although its supremacy is not as clear as in Scenario 1. Assuming the continuity of the performances of different strategies when the ratio k between δ 2 and δ 3 changes, in the light of results under Scenarios 1 and 3 we find that for small differences between δ 2 and δ 3 (i.e. Now, it should be noted that there exist some techniques for correcting bias in CSSE. [START_REF] Wang | Adapting the sample size planning of a phase III trial based on phase II data[END_REF] suggested correcting α or/and β levels, in accordance with some other works in the field. Here, we proposed to apply the correction directly on the source of the bias, that is on the estimation of δ 3 . Indeed, in our opinion, to correct α and/or β is more complicated and counter-intuitive than directly correcting

d •
2,n , i.e the estimator of δ 3 . In particular, when the error levels are modified a check of the amplitude of the correction is missing. On the contrary, d • 2,n can simply be modified by expressing the postulated correction k c . Hence, if k c is right, then CSSE is consistent and the structural bias is, asymptotically, corrected; otherwise, if k c is at least close to k, then this bias can be reduced, specifically for finite pilot sample sizes of practical interest. In both situations, the framework of CSSE returns to being close to Scenario 1, so that the peculiarity of COS can be exploited.

We, then, evaluated the behavior of k c -corrected CSSE under Scenario 3, i.e. when the phase III effect size is 20% lower than phase II. When the postulated correction is the right one, COS clearly performs better than the other strategies, providing good OPs and the smallest MSEs. It is worth noting that in this case the OPs of COS are even higher than those observed under Scenario 1 in De Martini (2010): this is due to a higher launch probability, since δ 2 > δ 3 , and to a similar estimation performance. Hence, paradoxically, the bias can be exploited for improving CSSE, should the right correction be applied. When the postulated correction is smaller than necessary COS is still the best performer, and it works quite well in general.

A higher than necessary correction should be avoided.

Conclusions

We maintain, in the presence of structural bias too, the operating conditions for CSSE to be: the launch threshold should be set lower than one half the true phase III effect size; the pilot sample should be around the ideal one (i.e. n ≥ 2M I /3); a power of 90% should be adopted. As a consequence of phase III effect sizes lower than phase II, low OPs often occur. If the difference between δ 2 and δ 3 is quite small (i.e. 1 ≥ δ 3 /δ 2 ≥ 0.8), then CSSE can still provide acceptable results, mainly by adopting COS. Nevertheless, a correction to CSSE can be applied, and it affects only sample size estimates, not launch probabilities. If a good correction is applied, i.e. close to the right one, and preferably a bit smaller rather than a bit larger, all CSSE strategies provide improved results, and COS results the best. If the right correction is applied, COS works even better than in the absence of structural bias.

  an upper bound for sample size estimators, which is essential for practical purposes.

  Recently, we compared (De Martini, 2010) some frequentist and some Bayesian strategies with a new conservative strategy based on the calibration of the optimal

n

  does not exist, and in other circumstances M BaS n can be simply higher than M max . Consequently, we defined[START_REF] De Martini | Adapting by calibration the sample size of a phase III trial on the basis of phase II data[END_REF] the truncated Bayesian sample size estimator M BaT n = min{M BaS n , M max }. With BAT, d • 2,n = d 2,n . BAT strategy is included in this comparison because under Scenario 1 it was the best Bayesian performer, although its sample size estimator showed very high average and/or MSE (see De Martini, 2010).

  the correction for the effect size

Figure 1 .

 1 Figure 1. Overall Powers of PWS, 1SES and COS under Scenario 3 with the right correction k c = 0.8 and under Scenario 1, with α = 0.025, 1 -β = 0.9, δ = 0.5 and δ 0L = 0.1.

Figure 2 .

 2 Figure 2. MSEs of the sample size estimators of PWS, 1SES and COS under Scenario 3 with the right correction k c = 0.8 and under Scenario 1, with α = 0.025, 1 -β = 0.9, δ = 0.5 and δ 0L = 0.1.

  n the sample size for each group in phase II, let d • 2,n be a generic estimator of δ 3 based on phase II data, where "•" represents a generic CSSE strategy. We apply the launching criterion introduced by Wang et al.(2006): δ 0L represents the launch threshold and phase III is launched on condition that d • 2,n > δ 0L . In this case, M I is estimated by M • n , i.e. the sample size estimator of •, usually based on d • 2,n . Note that M •

	n
	is a discrete random variable whose distribution depends on δ 2 . Finally, assume
	that there exists a maximum for M • n and that this maximum corresponds to M max =
	2(z 1-α + z 1-β ) 2 /δ 2 0L + 1. For example, being • the fixed-γ conservative strategy, we
	have that d • 2,n = d γ 2,n = d 2,n -z γ / n/2, where d 2,n = Xn -Ȳn is the pointwise estimator
	of δ 3 , and M γ n

Page 6 of 23 URL: http://mc.manuscriptcentral.com/lssp E-mail: comstat@univmail.cis.mcmaster.ca Communications in Statistics -Simulation and Computation

	F o
	r
	P
	e e r
	R e v i e w
	O n
	l y

  performed very well under Scenario 1 (De Martini, 2010), but it has not yet been studied under different Scenarios.

  Hence, to improve CSSE performances we find it natural and intuitive to speculate first about k. Being k c the postulated correction, M • n can be directly modified by using d • 2,n × k c , instead of d • 2,n only, in its formula. We then obtain c M • 2,n . Consequently, the assumption on M max changes, and the new maximum is

  δ 3 /δ 2 ≥ 0.8) consequently a wrong sample size is being estimated. Nevertheless, if differences between effect sizes are small, i.e. lower than 20%, COS still performs well, provided that pilot sample sizes are not large (i.e. n ≤ 4M I /3) in order to avoid the wrong asymptotic convergence.

	F o
	r
	P
	e e r
	R e v i e w
	O n
	l y
	COS remains the best strategy.
	In practice, COS is a very good strategy when CSSE is consistent, that is under
	Scenario 1 when δ 2 = δ 3 : in this case COS converges quickly, and with small variabil-
	ity. Consequently COS suffers when a structural bias is present (i.e. δ 2 = δ 3 ), and

Table 1 .

 1 Performances of different strategies under Scenario 3, with δ 0L = 0.1.

	Table 1		OP, MSE and average of sample size estimators
	Overall Power	n	PWS	3QS	1SES	COS	BAT
		2M I /3	70.67% 75.04% 72.85% 75.31%	80.19%
	δ 3 = 0.2	M I	72.68% 79.95% 80.52% 77.42%	80.17%
		4M I /3	73.37% 81.63% 83.79% 77.55%	79.53%
		2M I /3	73.08% 83.03% 85.48% 79.49%	82.50%
	δ 3 = 0.5	M I	73.69% 83.75% 87.47% 78.13%	81.09%
		4M I /3	73.88% 83.44% 87.35% 77.20%	79.97%
		2M I /3	73.75% 84.05% 87.12% 78.96%	82.95%
	δ 3 = 0.8	M I	74.26% 84.30% 88.22% 77.68%	81.49%
		4M I /3	74.45% 83.80% 87.73% 76.92%	80.40%
	MSE of SS Est. n	PWS	3QS	1SES	COS	BAT
		2M I /3	105405 173763 232528	34649	423681
	δ 3 = 0.2	M I	78751 130186 181341	35620	227369
		4M I /3	62875	95904 137493	38027	131601
		2M I /3	12503	45324	78144	3971	229328
	δ 3 = 0.5	M I	3647	16275	32738	3349	58581
		4M I /3	1815	7154	15497	2543	14208
		2M I /3	3207	17894	34876	1055	164605
	δ 3 = 0.8	M I	544	4766	13690	719	31270
		4M I /3	272	1060	3849	474	4631
	Av. of SS Est. n	PWS	3QS	1SES	COS	BAT
		2M I /3	428.7	615.4	715.8	407.0	742.2
	δ 3 = 0.2	M I = 526	408.3	578.9	675.9	423.8	600.3
		4M I /3	392.3	543.2	633.6	423.7	519.4
		2M I /3	81.3	145.7	192.9	92.2	233.3
	δ 3 = 0.5	M I = 85	69.0	111.4	144.4	83.9	127.2
		4M I /3	63.9	94.8	118.3	75.7	91.0
		2M I /3	33.1	65.2	92.0	38.9	137.7
	δ 3 = 0.8	M I = 33	27.2	45.7	62.5	33.7	57.8
		4M I /3	25.2	37.2	47.5	29.6	36.7

Table 2 .

 2 Performances of corrected strategies under Scenario 3, with δ 0L = 0.1, and with δ 3 = 0.5.

+ 1, where 1 -β is the power to be achieved. In practice we are required to estimate M I , since δ 3 is unknown.

URL: http://mc.manuscriptcentral.com/lssp E-mail: comstat@univmail.cis.mcmaster.caCommunications in Statistics -Simulation and Computation


Luminita Damian
email: luminitadamian@microcal.eu.com

Claire Marty-Detraves
email: claire.detraves@free.fr

Mathias Winterhalter

Didier Fournier
email: didier.fournier@ipbs.fr

Laurent Paquereau
email: laurent.paquereau@ipbs.fr

Single-strand DNA translation initiation step analyzed

Keywords:

come

Introduction

Since the discovery of the genetic code [START_REF] Crick | General nature of the genetic code for proteins[END_REF] and of messenger RNA (for review see [START_REF] Jacob | Genetic regulatory mechanisms in the synthesis of proteins[END_REF]), the molecular biology dogma is an evident pathway for the expression from gene to protein.

Nevertheless, as early as 1964, DNA was used in vitro to study translation and the question still remains whether or not DNA could be directly translated into protein. The first published works tried to determine whether single-stranded DNA (ssDNA) might be able to directly program the synthesis of proteins [START_REF] Holland | Stimulation of Protein Synthesis in Vitro by Denatured DNA[END_REF][START_REF] Mccarthy | Denatured DNA as a direct template for in vitro protein synthesis[END_REF]. Single-stranded DNA and circular messenger DNA were then used in direct translation systems for different purposes: to check the fidelity of the direct translation [START_REF] Morgan | Studies on polynucleotides. LXXIV. Direct translation in vitro of single-stranded DNA-like polymers with repeating nucleotide sequences in the presence of neomycin B[END_REF], to demonstrate that ribosomes do not require a free end on a messenger to initiate protein synthesis [START_REF] Bretscher | Direct translation of a circular messenger DNA[END_REF], to show that binding of fmet-tRNA and ssDNA to the ribosome could occur without neomycin which is necessary for elongation with these matrix [START_REF] Bretscher | Direct translation of bacteriophage fd DNA in the absence of neomycin B[END_REF], to test a synthetic ssDNA as ribosome binding site [START_REF] Jay | Specific binding of a chemically synthesized prokaryotic ribosome recognition site. Prospect for molecular cloning and expression of eukaryotic genes[END_REF], and finally, to make comparisons between RNA and ssDNA in their ability to program ribosomes for initiation and termination of translation [START_REF] Ricker | Use of single-stranded DNA oligonucleotides in programming ribosomes for translation[END_REF]. In all papers cited above, all results converge towards a similar behavior of RNA and ssDNA in the initiation step, in spite of some discrepancy concerning the elongation step of translation.

The formation of the initiation complex involves many partners which are largely described in the literature but the results are sometimes contradictory. For instance, the absence of initiation factors, especially IF3, does not significantly affect the interaction of the 30S subunits of ribosome and mRNA [START_REF] Calogero | Selection of the mRNA translation initiation region by Escherichia coli ribosomes[END_REF], although its role in promoting mRNA binding on ribosomes was demonstrated elsewhere [START_REF] Grunberg-Manago | Initiation mechanisms of protein syntehesis[END_REF]. On the other hand, the Shine-Dalgarno (SD) sequence [START_REF] Shine | The 3'-terminal sequence of Escherichia coli 16S ribosomal RNA: complementarity to nonsense triplets and ribosome binding sites[END_REF], which is localized inside the Ribosome Binding Site (RBS), is clearly involved in the interaction between mRNA and the 30S ribosome subunit. Therefore, this sequence is an unambiguous signal to test the specificity of these interactions.

In this biological process, the secondary or tertiary structures of mRNA play a crucial role for the control of the translation initiation [START_REF] Hall | A role for mRNA secondary structure in the control of translation initiation[END_REF][START_REF] Looman | Secondary structure as primary determinant of the efficiency of ribosomal binding sites in Escherichia coli[END_REF][START_REF] Smit | Secondary structure of the ribosome binding site determines translational efficiency: a quantitative analysis[END_REF]. We choose to exclude this aspect because i) RNA and ssDNA do not shares the same secondary structures [START_REF] Jones | Protein-RNA interactions: a structural analysis[END_REF] ii) The structural organization of both RNA and ssDNA is difficult to predict and control in a biological context, even if fine structural results have pointed up the interactions between mRNA and 30S subunits [START_REF] Wimberly | Structure of the 30S ribosomal subunit[END_REF][START_REF] Williamson | Induced fit in RNA-protein recognition[END_REF].

In this context, we tried to establish the thermodynamic values of this interaction in a global way, in order to determine and to compare with a sensitive method the binding constants for these two templates (mRNA and ssDNA). Some studies have reported association constant in particularly for interactions between the Shine-Dalgarno sequence on mRNA and ribosomes, with a value near 2 x 10 7 M -1 [START_REF] Calogero | Selection of the mRNA translation initiation region by Escherichia coli ribosomes[END_REF]. This range of affinity corresponds to the sensibility of the Isothermal titration calorimetry method, which allows us to access not only the binding constants but also the thermodynamic values of these interactions.

Materials and methods

S30 extracts

E. coli S30 extracts were prepared accordingly to Zubay [START_REF] Zubay | In vitro synthesis of protein in microbial systems[END_REF], with some minor modifications introduced by Pratt [START_REF] Pratt | Coupled transcription-translation in prokaryotic cell-free systems, in Transcription and translation : A practical approach[END_REF]. The S30 extract was quantified by measuring the optical density at 260nm. DO 260nm of 1480 corresponds to 100mg/ml of ribosomic extract. The concentration of S30 extracts stored at -70°C was 1.3mg/ml.

Oligonucleotides

All oligonucleotides were purchased from Eurobio (Table 1).

In vitro translation and detection of peptide synthesis

E. coli S30 Extract System for Linear Templates purchased from Promega was used in translation assays. Oligonucleotides were tested between 10 to 500µg and for several incubation times (2h to overnight).

Mouse monoclonal antibody (purchase from Oncogene TM ) against c-myc epitope (EQKLISEEDL) was used in dot-blot experiments. C-myc synthetic peptide (MASMQKLISEEDL) was used as positive control. Primary antibody was detected and revealed with anti-mouse IgG peroxidase-linked whole antibody from sheep (Amersham Biosciences).

To enhance the sensitivity of detection, c-myc epitope was produced with a 6 Histidine tag allowing purification on Dynabeads® TALON TM (Invitrogen). Fusion peptide (6(His)/c-myc) was separated of translation mix by affinity with cobalt binds on Dynabeads. Mouse monoclonal primary antibody against c-myc epitope was detected with anti-mouse secondary antibody from rabbit. To amplify the signal a third anti-rabbit antibody FITC-linked from sheep was used (λexc=490nm; λem=520nm).

In vitro transcription

Synthetic RNA was obtained by T7 RNA Run off transcription of a double strand DNA containing T7 promoter [START_REF] Milligan | Oligoribonucleotide synthesis using T7 RNA polymerase and synthetic DNA templates[END_REF]. Oligonucleotides T7/SD/ATG and AntiT7/SD/ATG were hybridized in Ulh buffer (40mM TrisHCl pH 8, 0.01% Triton X100, 1mM Spermidine) and 100mM NaCl by heating 3min at 95°C and cooling slowly until room temperature. Run off transcription were achieved in Ulh buffer, with 1µM of double strand oligonucleotide, 5mM DTT, 5mM of each NTP, 38mM MgCl 2 and 2 units of T7 RNA polymerase (NEB) at 37°C ON. The reaction mixture was treated by 10 units of DNaseI for 1h at 37°C and phenol extracted at pH 4. RNA was precipitated in ethanol, resuspended in H 2 O and quantified at 260nm. All solutions except Spermidine and Tris-HCl were DEPC treated and autoclaved.

Isothermal Titration Calorimetry, ITC

Isothermal titration calorimetry was performed using a VP-ITC microcalorimeter from Microcal Inc. (Northampton, MA). Several experiments were performed to determine the binding constant values. Injections of 2 to 10 μl of oligonucleotide were added by computercontrolled 296 μl microsyringe at intervals of 200 to 300 seconds into the S30 extract (cell volume = 1.437 ml). The experiments were performed at 30°C (303°K), with a stirring speed of 300 rpm. The S30 extract concentration varied between 0.17-0.43 µM and the oligonucleotides concentrations between 5-10 µM except for the binding of anti-SD (25µM) with SD/ATG oligonucleotide (250µM). S30 buffer was used to resuspend and dilute oligonucleotides. For RNA, the S30 buffer was previously treated with DEPC. Interaction of SD and AntiSD oligonucleotides was performed in 50 mM Na 2 HPO 4 /NaH 2 PO 4 pH 8, 100 mM NaCl at 25°C. All solutions were degassed before use. Antibiotics were added to the S30 extract and to the oligonucleotide solution before measurements. Several blind titrations were performed to determine and correct for non-specific heat contributions (heat of dilution).

The experimental data were fitted to a theoretical titration curve, using software supplied by Microcal, with ΔH (enthalpy change in kcal/mole), K a (association constant in M -1 ) and n (number of binding sites), as adjustable parameters. The instrument was calibrated using the built-in mode of electric field heat pulses. Thermodynamic parameters were calculated from the relation:

a K RT S T H G ln - = Δ - Δ = Δ
where the ΔG, ΔH and ΔS represent the changes in free energy, enthalpy and entropy of binding; T is the absolute temperature and R=1.98 cal mol -1 K -1 .

Results

Design of oligonucleotides

Oligonucleotides were designed to avoid any secondary structures in accessing the binding constants between S30 extract and specific signal sequences. The Shine-Dalgarno sequence, for example, is clearly involved in the translational efficiency, whether or not it is localized in a structured region [START_REF] Smit | Translational initiation on structured messengers. Another role for the Shine-Dalgarno interaction[END_REF]. Since protein/nucleic acid interactions were mostly observed with purine [START_REF] Looman | Influence of the codon following the AUG initiation codon on the expression of a modified lacZ gene in Escherichia coli[END_REF], we chose a polyT as framework for all the oligonucleotides to reduce nonspecific interactions with S30 extracts. The two main sequence signals involved in the initiation of translation were incorporated into the oligonucleotides. First, the SD sequence, which is complementary to the 3' terminal sequence of E.coli 16S ribosomal RNA [START_REF] Shine | The 3'-terminal sequence of Escherichia coli 16S ribosomal RNA: complementarity to nonsense triplets and ribosome binding sites[END_REF][START_REF] Shine | Determinant of cistron specificity in bacterial ribosomes[END_REF], and second, the codon AUG, which is the most common initiator codon [START_REF] Ringquist | Translation initiation in Escherichia coli: sequences within the ribosome-binding site[END_REF]. When these two sequences were present simultaneously, a distance of 9 nucleotides was introduced between them [START_REF] Ringquist | Translation initiation in Escherichia coli: sequences within the ribosome-binding site[END_REF][START_REF] Chen | Determination of the optimal aligned spacing between the Shine-Dalgarno sequence and the translation initiation codon of Escherichia coli mRNAs[END_REF]. Oligonucleotides used in functional translation assays were designed with the same characteristics upstream the start codon but in order to use a fluorescent sensitive method to detect the corresponding peptide, SD/ATG/c-myc oligonucleotide as well as SD/ATG/His tag/c-myc, coding for c-myc epitope, were used to produce a covalent L tRNA/cmyc complex, which was more efficiently spotted on dot blot experiments. A SD/ATG/Histag/c-myc/Stop oligonucleotide was designed to verify that L tRNA did not interfere with anti c-myc primary antibodies.

Translation of single strand DNA

In attempting to realize a functional assay, we tested the translation of different singlestranded DNA oligonucleotides in E. coli S30 extract. The SD/ATG/c-myc oligonucleotide led us to detect the peptide with a specific antibody directed against the c-myc epitope coupled to a second fluorescent antibody. The c-myc peptide was not detected, even if a third antibody was used to amplify the signal. The same negative results were obtained with an oligonucleotide bearing a Histidin-tag which allows a strong purification of potentially translated peptides. These two oligonucleotides did not contain a stop codon. Therefore, a peptidyl-tRNA complex should be formed and the release of the translated peptide should be blocked. We then tested this hypothesis with an oligonucleotide (SD/ATG/His tag/cmyc/Stop) which bore a stop codon, but none of these methods allowed us to show that an efficient translation of the single-stranded DNA oligonucleotides had occurred. Thus, it is obvious that the elongation process required for protein synthesis is not efficient with these matrixes.

Binding of ribosome on single strand DNA

Basal binding on nonspecific sequences were tested with a polydT oligonucleotide (Table 2).

The dissociation constant (Kd= 1.16 10 -7 M) shows that the S30 extract shares an important affinity with this sequence, as it probably does with any random nucleotide sequence. The introduction of an ATG in the PolyT framework oligonucleotide (Oligo ATG) did not significantly change the thermodynamic values (Kd=1.34 10 -7 M) and the start codon alone did not seem to be significantly involved in this binding. On the other hand, the presence of SD (Oligo SD) alone on ssDNA gave a dissociation constant of 4.1 10 -8 M. (Table 2) which corresponds to a three-fold increase in affinity. The same three-fold increase in affinity was obtained when we compared ATG alone (Oligo ATG) or ATG in association with SD (Oligo SD/ATG)(Figure 1 and Table 2), showing dissociation constants of Kd=1.34 10 -7 M and Kd= 3.6 10 -8 M, respectively. The introduction of a known specific signal, such as SD, in a polyT framework leads to an increase of the affinity constant. Then we checked if this increase was mainly due to the formation of the initiation complex as expected, or directly to the nucleic acid interactions.

We tested the direct contribution of nucleic acids hybridization in this binding by incubating SD/ATG oligonucleotides with AntiSD oligonucleotides and we found a dissociation constant of 2.4±0. 3 10 -6 M. This corresponds to an affinity of two orders of magnitude lower than was observed with S30 extract (Figure 2 and Additional file 1). The involvement of ribonucleoprotein in this interaction was largely proven and these results are consistent with the theory of translation initiation.

Interaction with RNA versus DNA

RNA synthesized in vitro was used in control experiments. We used a full double strand DNA containing SD and ATG signals as template. After run off transcription, RNA was treated with DNAse I and extracted with acid phenol to eliminate any trace of DNA template. It was then quantified before the ITC experiments that were done in triplicate (Figure 3 and Additional file 2). The binding constants were compared to the values obtained from the corresponding ssDNA oligo SD/ATG (Table 3). Whatever the template, ssDNA or RNA, the affinity constant as well as the thermodynamic values were very similar and the ribosome binding site was recognized in the same way.

Antibiotic interference on binding

Several antibiotics act on the translation initiation step and were used to analyze the binding specificity of our system. In all experiments, antibiotics were added to both solutions, SD/ATG oligonucleotide and S30 extract, in order to avoid any dilution effect. Aurintricarboxylic acid (ATA) prevents the attachment of mRNA to the 30S ribosome subunit [START_REF] Grollman | Inhibition of the attachment of messenger ribonucleic acid to ribosomes[END_REF]. An inhibition of interaction was observed for 1µM of ATA and the binding disappeared at 10µM (Figure 4A). ΔH values measured with (-700 kcal/mole) or without ATA (-913 kcal/mole) were similar while the molar ratio did not differ. These results correspond to the competitive behaviour of ATA which bound to the same site as SD/ATG oligonucleotides.

Tetracycline, which blocked the binding of tRNA on the 30S subunit [START_REF] Chopra | Tetracycline antibiotics: mode of action, applications, molecular biology, and epidemiology of bacterial resistance[END_REF], shared the same global effect than ATA but at a higher concentration, since total inhibition was obtained for tetracycline concentrations between 100 and 250µM (Figure 4B). Kanamycine prevented the whole formation of the initiation complex by blocking the junction between the two ribosomal subunits [START_REF] Carter | Functional insights from the structure of the 30S ribosomal subunit and its interactions with antibiotics[END_REF]. The complete inhibition of this interaction was obtained with 1mM of kanamycin (Figure 4C).

These antibiotics acted with a different mechanism than ATA, since they bind different sites than oligonucleotides. Molar ratio values were also higher than with SD/ATG oligonucleotides alone, as well as ΔH, which were approximately -2200 kcal/mol and -2500 kcal/mol for tetracycline and kanamycine, respectively, as expected for a non-competitive inhibition.

Discussion

With these experiments, our purpose was not to test whether or not ssDNA could be used as a translational template in cells but rather to determine, with new approach, if general features of translation could be observed with ssDNA. All previous work had tested if ssDNA allowed direct translation, but results obtained were not really convincing and the possibility that only some steps could be restrictive was not checked. In order to gain access to global thermodynamics constants, we worked with a complex system of S30 extract, which has been largely used for in vitro translations as well as for binding experiments. Our results on RNA compared to ssDNA were very close to previous work which had been done on small oligonucleotides [START_REF] Ricker | Use of single-stranded DNA oligonucleotides in programming ribosomes for translation[END_REF]. In this study, the relative behaviour of each template, RNA or ssDNA, was checked for some translation steps, but binding constants were not accessible. Access to the thermodynamic values allowed us to tackle new aspects of ssDNA translation. All interactions were characterized by the enthalpy-entropy compensation phenomenon; so the system paid a high entropic penalty for the bonding network to be formed. In our experiments, ΔS was expressed in kcal/mole.K and not in cal/mole.K as usually encountered in biological binding interactions. These high entropic negative values, obtained for ssDNA as well as for RNA, indicate a strong organization in the system which probably corresponds to the initiation translational complex formation where many partners are involved. Moreover, the time interval between injections was longer that observed in more classical ITC experiments, since 200 to 300 seconds were required and slow thermal processes occurring upon injection were probably involved in reaching binding equilibrium.

Although the ribosome -polydT interaction was characterized by a high enthalpy, we presumed that this was due to non-specific hydrogen network formation. The ribosome does not have a defined binding site for the PolydT but a single strand nucleic molecule may present a large number of non-specific sites. By adding specific motifs, like ATG and SD sequences, to the ssDNA sequence, a decrease in the ΔH was observed. We assumed that hybridization occurred and that the binding to the specific sites was favored. Being confronted with non-specific binding and uncertain reaction stoechiometry, we had to prove that hybridization occurred precisely between the different single-strand DNA and the 30S ribosome subunit. For this reason, we chose to monitor the binding in presence of antibiotics which specifically inhibited the translation initiation step.

The inhibition mechanisms of antibiotics are precisely known and largely described. For each of them the binding was fully reactive, and therefore functional. Aurintricarboxylic acid inhibited the attachment of mRNA to the 30S ribosome subunit, and tetracycline prevented the binding of tRNA on the 30S subunit. Kanamycine blocked the junction between the two ribosomal subunits. In each case, the whole initiation complex was blocked but the earlier the antibiotics acted, the lower the concentrations were necessary for a full effect. The results obtained perfectly matched this behaviour and demonstrated the specificity of the interactions measured. Moreover, the formation of a translational initiation complex was strongly driven by the ribosome, and if nucleic acids were the main actors involved in the specificity of this interaction, they played a second role in the affinity of this binding (see [START_REF] Maguire | The ribosome in focus[END_REF] for an excellent minireview). We then tested the binding between nucleic acid bearing the anti-SD sequence with the complementary SD sequence, which was the most important nucleic acid signal test in our system. As expected, this interaction contributed poorly to the affinity constant with a two order of magnitude difference. These data strongly supported the binding constant validity obtained for ssDNA. From these results, it appeared that there was no difference between RNA and ssDNA in the binding step of initiation complex formation. Naturally, we tried to translate ssDNA, but as previously described elsewhere, direct DNA translation was not really efficient in absence of antibiotics such as neomycin [START_REF] Morgan | Studies on polynucleotides. LXXIV. Direct translation in vitro of single-stranded DNA-like polymers with repeating nucleotide sequences in the presence of neomycin B[END_REF][START_REF] Bretscher | Direct translation of a circular messenger DNA[END_REF]. It seemed that the elongation phase was the limiting step in the direct translation of single-stranded DNA. Whether direct participation of DNA in protein synthesis could play any role in vivo must be excluded, although such a role could be attractive for modelling the evolution of the earliest forms of life.

Figure 1 -

 1 Figure 1 -ITC profile and binding isotherm of S30 extract and SD/ATG oligonucleotide

Figure 2 -

 2 Figure 2 -Binding isotherm of anti-SD and SD/ATG oligonucleotides.

Figure 3 -

 3 Figure 3 -Binding isotherm of S30 extract on RNA.

Figure 4 Effects

 4 Figure 4


Vincent Balat
email: vincent.balat@univ-paris-diderot.fr

Pierre Chambart
email: pierre.chambart@crans.org

Grégoire Henry
email: gregoire.henry@pps.univ-paris-diderot.fr

Client-server Web applications with Ocsigen *

Keywords: D.3.3 [Software]: Programming languages-Language Constructs and Features; H.5.3 [Information Systems]: Information interfaces and presentation-Group and Organization Interfaces Web-based interaction Languages, Reliability Continuation-based Web programming, AJAX, Javascript

The Ocsigen framework offers a new way to develop sophisticated client-server Web applications. It makes it possible to write as a single program both the server and client sides of a Web application, thus simplifying a lot communications and data transfers, and avoiding code duplications. It also proposes a wide set of high level concepts to program traditional Web interactions in a very concise way while mixing them seamlessly with client side features. The use of a powerful type system improves a lot the reliability of programs, reducing debugging time, and making the code easier to maintain.

INTRODUCTION

Recent Web browsers have become powerful virtual machines in which more and more programs are executed. Desktop programs are currently being superseded very quickly by ever more sophisticated cloud applications.

Today, most Web developers are using distinct languages for programming traditional Web interaction (e.g. forms, links and bookmarks) and client side features. This creates an impedance mismatch that often requires explicit data conversions and code duplications, all of which being error prone. For instance, data validation should be performed typically both on the client for early feed-back and on the server for safety. This approach also makes it difficult to combine in a flexible and natural way client and server features. As a result, client features remain often restricted to small one page applications with no persistence when links are followed.

Even though there are good libraries that attempt to address these problems, we believe that rethinking programming tools could speed up the rise of applications taking full advantage of the new possibilities offered by modern browsers.

A few programming frameworks now propose to use the same language both client and server side [START_REF]Google web toolkit[END_REF][START_REF] Serrano | Hop, a language for programming the web 2.0[END_REF][START_REF] Dart | [END_REF]. On the client, they must rely on what browsers provide, that is, basically Javascript. The good news is that there now exists good and fast Javascript engines, available on servers as well [START_REF]Nodejs[END_REF].

The Ocsigen project [START_REF] Balat | Experience report: ocsigen, a web programming framework[END_REF] takes the alternative approach of using Javascript as the target language of a compiler. Indeed, we want to use advanced language features not available in Javascript. The goal is to abstract away technical details, proposing high level technical concepts corresponding closely to the very needs of Web developers. This makes it possible to express sophisticated behaviors in very few lines.

One of the most powerful features of Ocsigen is the way Web interaction, such as the effect of following links, is programmed through a unique concept of "services", and remains automatically compatible with client side features: the client side program persists while the user browses the Web site, and traditional features like bookmarks, forms, sessions or "back button" automatically remain available, in a transparent way.

Finally Ocsigen is making extensive use of the powerful type system of the underlying language, OCaml, to detect at compile time many programming errors. For example it makes it impossible to generate invalid HTML. All this reduces a lot debugging and testing time!

We start in section 2 by showing an example of application written with Ocsigen. Section 3 shows the basics for writing a client-server application as a single program, whereas the way to program traditional Web interaction is sketched in section 4. Then, section 5 shows how we benefits from static typing.

Figure 2 shows the full source code of a client-server drawing application (see also a screen-shot in figure1, and a working version at URL http://ocsigen.org/graffiti). This program allows the user to draw on the screen, choose the color of the brush, and see in real time what other users are drawing.

We won't go into the full details of the code, which is basically plain OCaml. For more explanations, there is a tutorial showing step by step how to write this program at URL http://ocsigen.org/tutorial.

The first thing to notice is that the code is short. And this is due not to the use of specialized libraries that would do everything for us (we are using a library only for the color picker) but rather to the expressiveness of concepts.

As you can see, the code is divided into several parts with special bracket to distinguish between client code, server code, and shared code. It mainly defines one service, corresponding to a URL and returning an HTML5 page, together with some code to be executed once the page is loaded.

CLIENT SERVER WEB APPLICATIONS

General principles

Writing client and server code together brings many advantages beside the use a single language. It allows a seamless communication of data between both parts of the program. Server side variables can be used from the client (see line 37); their values are sent along the page. This also ensures that client and server codes remain consistent. For instance, in this example, if the color of the brush were removed from the messages sent on the bus (on line 33), the compiler would complain if the remaining of the code were not updated accordingly, and it would point to the location of the error.

Interfaces are built in the usual declarative way, and sent together with the code corresponding to dynamic parts. Programs are written in way very similar to desktop applications. For example a very simple interface allows to use user variables server side as if there was a single user program.

The Js_of_ocaml compiler

The client side code is translated from OCaml to Javascript by the Js of ocaml compiler. A syntax extension makes it possible to directly invoke Javascript methods (line 23), set object properties (line 19) or create objects (line 22). This makes it easy to bind Javascript libraries (like Google Closure in our example [START_REF]Google closure library[END_REF]). Appropriate types can be used to ensure that objects are not misused. The compiler performs dead code elimination. Hence, the programmer does not have to restrict himself to minimal libraries.

Persistent client side programs

Ocsigen allows to combine the best of traditional and persistent applications. It is possible to write persistent applications in a classical declarative way. Only the reactive parts involves using DOM interfaces. This way, users with Javascript disabled and search engine crawlers can still browse the site and pages have real bookmarkable URLs.

When loading a page, links and forms are modified such that clicking links appears to work as usual, but does not in fact interrupt the Javascript program. This feature makes it possible to write very easily applications that need to keep a state during the navigation, such as an audio library with a music player, or persistent widgets (ex. a chat).

Having these features with other frameworks usually requires code duplication to implement a Javascript-less version of the site.

SERVICE BASED PROGRAMMING

Programming traditional Web interaction is done through the notion of service. A service is a kind of function invoked in reaction to an HTTP request, and that usually returns a Web page. Main services are identified by the URL they are attached to. Ocsigen proposes many other ways to identify a service: based on special parameters, on the HTTP method, on the names of parameters, on the current session, etc.

As an example, suppose you want to implement a login box on each page of your Web site. If a user logs in from a page, you want him to remain on this page after connection. With Ocsigen, you just need to create an extra service, not attached to a precise URL (that is: available from any URL), and that will perform the action of checking the password and registering session data. This only takes a few lines of code.

Another very convenient feature is the ability to dynamically create new services, for example services customized to one user and that depend on previous interactions with him. This simplifies a lot programming sequences of pages, each depending on the previous ones (like buying a plane ticket). This is usually known as continuation based Web programming [START_REF] Queinnec | Continuations and web servers[END_REF].

STATIC TYPING

Simply put, static typing is the principle of verifying at compile time that values are correctly used. It can provides some basics properties like ensuring that variables are declared and initialized. But, when the type system is powerful enough, it can also protect against more complex programming errors. With a powerful type inference mechanism, type safety can be combined with the lightweightness and flexibility of dynamic languages.

In Ocsigen we leverage the power of OCaml type system to enforce many properties at compile time. We can for instance ensure that Javascript APIs are correctly used, that database accesses are safe, that links go to existing services with the right parameters. With the whole application written as a single code base, we can moreover ensure that communications between client and server can't go wrong.

One of the most impressive use of static typing in Ocsigen is for checking at compile time the validity of function generating HTML. This makes it impossible for an Ocsigen program to generate a page that does not respect the W3C recommendations! let width , height = 700 , 400 Typing is useful to ensure code safety, but it also helps a lot in maintaining and refactoring code. When some parts of the code base is modified, the compiler can show the programmer where he forgot to propagate the changes.

CONCLUSION

The Ocsigen software is now mature and full featured. It has a growing community of users and is released under an open source licence (LPGL). Ocsigen provides the most advanced features of Web programming frameworks (like dynamic services or unified client-server programming) but also many unique features: service identification mechanism, static checking of HTML, persistence of the client program, etc. These unique features come to a large extent from the use of one of the most advanced programming language (OCaml), known to be very fast, expressive, concise and safe, with many robust libraries.

Many further features have not been presented in this paper: an advanced session mechanism, a very concise way to implement event handlers (see lines 41-44), or a flexible interface for server push events (see lines [START_REF] Queinnec | Continuations and web servers[END_REF]37,40). Ocsigen also guarantee the absence of code injection on the client, the server, nor in SQL queries, and provides mechanisms to protect angaisnt XSS and data leaks. In our experience, the safety provided by static typing helps a lot in reducing development time.

Figure 1 :

 1 Figure 1: The Graffiti program with two windows

3 4 }} 5 6Figure 2 :

 452 Figure 2: Full source code of the Graffiti program

Acknowledgements

Many acknowledgements are due to Jérôme Vouillon, Raphaël Proust, Benjamin Canou and Emmanuel Chailloux.

* Work partially supported by the French national research agency (ANR), PWD project, grant ANR-09-EMER-009-01, and performed at the IRILL center for Free Software Research and Innovation in Paris, France

Copyright is held by the authors/owner(s).

( fun () () -> (* The main service , at URL "/" , with no p a r am e t e r *)


Exact Comparison of Quadratic Irrationals

Keywords: . . . . . . . . . . . . . . . . . . . . . 4.3 Complexity . . . . . . . quadratic irrational, periodic continued fraction, exact comparison

quadratic irrational 3.3.1 Periodic continued fraction of a square root . . . . . . . . . . .

Introduction

During our recent study about discrete rotations based on hinge angles for 2D and 3D digital images [START_REF] Thibault | Rotation dans les espaces 2D et 3D[END_REF], we encounter the problem of comparing quadratic irrationals when we would like to observe the discretization of the rotation space induced by hinge angles. There exist various techniques for calculating quadratic irrationals; for instance Talor series transformation, Babyloian method, Exponential identity, Bakhshali approximation, etc. However these methods compare quadratic irrationals by using their approximative values, which means that the comparison is "true" with a given precision. For our purpose, we need to compare two quadratic irrationals with their "exact values". An exact approach, employing only integers, has been proposed in [START_REF] Thibault | Rotation dans les espaces 2D et 3D[END_REF] by using the square function. The disadvantage is that square function generates large integers, which are out of range of types int, long in C/C++ and thus, that is a limitation to implement. Our approach is based on another representation altogether for quadratic irrationals by using continued fraction expansion, which are represented by sets of integers. Indeed, quadratic irrationals can be represented exactly using periodic continued fractions, and this representation is unique. This provides an exact method that represent such continuing fractions, as well as compare them, which allows us avoid numerical errors. This paper is organized as follows: in section 2, we introduce some basic notions of quadratic irrationals. Section 3 is devoted to formulate the problem of comparing exactly two quadratic irrationals and to give a solution based on continued fraction representation. Then, in section 4 we compare our method with some others : Approximation method and square comparison method. Finally, we conclude the paper with discussion of future work and some applications in section 5.

Quadratic irrationals

In this section, we present several mathematical concepts that will be useful to understand the problem of comparison of quadratic irrationals. We first begin by the definition of irrational numbers. Definition 2.1 An irrational number is any real number which cannot be expressed as a fraction a b , where a and b are integers with b non-zero, and is therefore not a rational number.

Informally, this means that an irrational number cannot be represented as a simple fraction. In this paper, we are interested in a special kind of irrational numbers that are called quadratic irrational numbers. There are several ways in which quadratic irrationals can be represented. In this paper we refer the definition and results to mathematical analysis books given by K. Rosen [START_REF] Rosen | Elementary Number theory and its applications[END_REF].

Definition 2.2 A real number x ∈ R is a quadratic irrational 1 if there exist A, B, C ∈ Z and A = 0 such that Ax 2 + Bx + C = 0 and D = B 2 -4AC > 0 and D is not a perfect square 2 .
Consequently, the solutions of this equation are quadratic irrational numbers, that can be expressed as:

x = -B ± √ B 2 -4AC 2A .
According to Definition 2.2, quadratic irrationals have to satisfy two conditions: they must be the solution of a quadratic equation and be irrational.

We define now a special kind of quadratic irrationals, called reduced surd, because of their interesting properties associated with continued fraction that we will discuss in Section 3.

Definition 2.3 A quadratic irrational Q is said to be reduced if Q > 1 is
the root of a quadratic equation with integer coefficients whose conjugate root Q lies between -1 and 0.

We will present as follow a lemma that will be used in the algorithms of Section 3 in other to solve the problem of exact comparison.

Lemma 2.1 A quadratic irrational Q can be expressed in the form Q = p+ √ q
r , where q is not square free, r = 0 , p, q, r ∈ Z that satisfies r|q -p2 .

Proof : Suppose that Q = p + √ q r where p , q , r ∈ Z, r = 0 and r q -p 2 . By multiplying both numerator and denominator of Q with absolute value of denominator r , we obtain:

Q = |r |p + |r | √ q |r |r = |r |p + r 2 q |r |r .
Then, we have (|r |r )|(r 2 q -r 2 p 2 ).

Continued fraction expansion

It is known that a quadratic irrational can be represented by a continued fraction. The research of this relationship began from 17th century. Some useful properties are known [START_REF] Hardy | An Introduction to the Theory of Numbers[END_REF] [4]; For instance, Leonhard Euler proved that the infinite simple continued fraction of an irrational number is eventually periodic if this number is a quadratic irrational. Then in 1770, Joseph-Louis Lagrange found out the proof that the continued fraction expansion of any quadratic irrational is periodic.

Continued fraction

Before describing our method for the exact comparison, we first provide definitions of continued fractions and some important properties that will be used in the method.

Definition 3.1 A simple continued fraction has an expression of the form

a 0 + 1 a 1 + 1 a 2 + 1 a 3 + ...
where a 0 ∈ Z and a i ∈ Z + for i = 1, 2, 3.... They are called the partial quotients of the continued fraction.

Hereafter, we will deal with the cases when partial quotients are all positive.

The number of terms a i is limited for finite continued fractions, while it is unlimited for infinite continued fractions. In the case of finite continued fractions, we thus write

[a 0 ; a 1 , a 2 , a 3 , ...a n ] = a 0 + 1 a 1 + 1 a 2 + 1 a 3 + ... + 1 a n .
Similarly, we denote infinite continued fractions by [a 0 ; a 1 , a 2 , a 3 , ...] = a 0 + 1

a 1 + 1 a 2 + 1 a 3 + ... .
In this paper, when we say continued fractions, they mean infinite continued fractions; otherwise, we will specify the term for each case.

For the problem of comparison, there are two special forms of continued fractions that interest us: periodic continued fraction and purely periodic. The are defined as follow: Definition 3.2 An infinite continued fraction is called a periodic continued fraction, if its terms eventually repeat from some point until infinite. The minimal number of repeating terms is called the period of the continued fraction.

A periodic continued fraction thus has the form:

[a 0 ; a 1 , a 2 , a 3 , ..., a k-1 , a k , a k+1 , ..., a k+m , a k , a k+1 , ..., a k+m , ...],
for which we also write [a 0 ; a 1 , a 2 , a 3 , ..., a k-1 , a k , a k+1 , ..., a k+m ]. Definition 3.3 A continued fraction which is periodic from the first partial quotient is called purely periodic.

We denote as:

[a 0 ; a 1 , a 2 , a 3 , ..., a k-1 , a k , a 0 , a 2 , a 3 , ..., a k-1 , a k , ...] = [a 0 , a 1 , a 2 , a 3 , ..., a k-1 , a k ].
If the period starts with the second partial quotient, the continued fraction is called simply periodic which is represented by:

[a 0 ; a 1 , a 2 , a 3 , ..., a k-1 , a k , a 1 , a 2 , a 3 , ..., a k-1 , a k , ...] = [a 0 ; a 1 , a 2 , a 3 , ..., a k-1 , a k ].
Then, a purely periodic is also simplify periodic but the reverse is not true.

Continued fraction expansions of quadratic irrationals

So far we have studied the basic notions of quadratic irrationals and continued fractions. We are now interested in their relationship to solve the exact comparison of two quadratic irrationals. For this, we will first present two fundamental theorems that indicate the bijection between a quadratic irrational and a continued fraction. We just formulate the result and refer to the literature [START_REF] Hardy | An Introduction to the Theory of Numbers[END_REF] [3] for proofs.

The first one was proved by J-L Lagrange.

Theorem 3.1 Quadratic irrationals are the real numbers that can be exactly represented by periodic continued fractions.

The next one was shown by A. Ya. Khinchin [Theorem 14].

Theorem 3.2 There is a one-to-one correspondence between a real number and a continued fraction, which is either finite or infinite.

Galois then found out then an interesting property between reduce surds and purely periodic continued fractions. Theorem 3.3 The continued fraction which represents a quadratic irrational Q is purely periodic if and only if Q is a reduced surd.

In fact, Galois showed more than this result. He proved that if Q is a reduced quadratic surd and Q is its conjugate, then the continued fractions for Q and for 1 Q are both purely periodic, and the repeating block in one of those continued fractions is the mirror image of the repeating block in the other. From Theorem 3.3, we can state the following lemma that is useful for the exact comparison algorithm.

Lemma 3.1 For any positive integer S that is not a perfect square, the continued fraction of √ S is simply periodic and more precisely it has the form √ S = [a 0 ; a 1 , a 2 , ..., a n , 2a 0 ].

Proof. Let a 0 = √ S . Since S is a positive integer, √ S + a 0 > 1. Because S is not a perfect square, we have 0 < √ S -a 0 < 1 and its conjugate lies between -1 and 0, which means -1 < -√ S + a 0 < 0. Therefore, from Definition 2.3, √ S + a 0 is a reduced quadratic irrational. Applying Theorem 3.3 we then obtain:

√ S + a 0 = [2a 0 ; a 1 , a 2 , ..., a n ],
which is equivalent to √ S + a 0 = [2a 0 ; a 1 , a 2 , ..., a n , 2a 0 ].

Consequently, we have √ S = [a 0 ; a 1 , a 2 , ..., a n , 2a 0 ].

Algorithm for finding corresponding continued fraction of a quadratic irrational

Let us consider now two problems: finding the corresponding fraction of a square root, and of a quadratic irrational.

Periodic continued fraction of a square root

To find the continued fraction of a square root, one of the most known methods was proposed by K. Rosen, in [START_REF] Rosen | Elementary Number theory and its applications[END_REF]. This method base on the study of irrational numbers as continued fractions obtained by J.L. Lagrange. Let consider that we would link to calculate the continued fraction of a square root √ S, where S is a positive integer and non perfect square. We obtain √ S = [a 0 ; a 1 , a 2 , a 3 , ...] as result. The iterative algorithm, Algorithm 1, uses two sequential integer parameters : m i and d i , to obtain a i . Lemma 3.1 says that the corresponding continued fraction of square roots of S is simply periodic. It means the terms of expansion will repeat from some points until infinite. Because a program cannot be executed infinitely, we can stop it at the first repetition of the extension. Thus, the stop condition is given by verify the doublet m i and d i goes back as the same one that is encountered before.

Algorithm 1: Calculate the periodic continued fraction of a square root Input: A positive integer S Output:

A periodic continued fraction of √ S, [a 0 ; a 1 , a 2 , a 3 , ...] 1 m 0 ←-0 2 d 0 ←-1 3 a 0 ←- √ S 4 i ←-0 5 repeat 6 m i+1 ←-d i a i -m i 7 d i+1 ←- S-m 2 i+1 d i 8 a i+1 ←- a 0 +m i+1 d i+1 9 i ←-i + 1 10 until (m i-1 , d i-1 ) = (m 1 , d 1 );
According to Lemma 3.1, algorithm can terminate if we have 2a 0 while calculate the a i with i > 0. For this, in Algorithm 1 we replace the condition at line 10 by a i = 2a 0 .

Periodic continued fraction of a quadratic irrational

K. Rosen proposed in [START_REF] Rosen | Elementary Number theory and its applications[END_REF] a method which allows us to get the corresponding continued fraction of a quadratic irrational. Let Q = p 0 + √ q 0 r 0

, where q 0 is not a perfect square, p 0 and r 0 are integers. Algorithm can be applied if p 0 , q 0 and r 0 are satisfying r 0 |q 0 -p 2 0 . For this condition, Lemma 2.1 says that any quadratic irrationals of non square free can be expressed in the form p+ √ q r that r|q -p 2 . We have Algorithm 2 as follows:

Algorithm 2: Calculate periodic continued fraction of a quadratic irrational Input: p 0 , q 0 and r 0 represent the quadratic irrational

Q = p 0 + √ q 0 r 0 Output: Corresponding periodic continued fraction : Q = [a 0 ; a 1 , a 2 , a 3 , ...] a 0 ←- p 0 + √ q 0 r 0 i ←-0 repeat p i+1 ←-a i r i -p i r i+1 ←- q 0 -p 2 i+1 r i a i+1 ←- p n+1 + √ q 0 r n+1 i ←-i + 1 until a i-1 = 2a 0 ;
Notice that p n , r n and a n are integers.

Comparison of quadratic irrationals

In this section, we first formalize the problem of comparing quadratic irrationals, then explain how to solve this problem with an exact approach using the continued fraction expansion.

Problem statement

Using Definition 2.2 in previous section, we now formulate the problem as follows. Suppose that we have two quadratic irrationals Q 1 and Q 2 that have forms:

Q i = p i + √ q i r i
where p i , q i , r i ∈ Z , q i > 0 and r i = 0 for i = 1, 2. Then, we would like to compare Q 1 and Q 2 , with an exact calculation by using only integers.

According to Theorems 3.1 and 3.2, for each quadratic irrational we can find out exactly one corresponding periodic continued fraction. Let us consider

Q 1 = p 1 + √ q 1 r 1 = [a 0 ; a 1 , a 2 , a 3 , ..., a k-1 , a k , a k+1 , ..., a k+m ], Q 2 = p 2 + √ q 2 r 2 = [b 0 ; b 1 , b 2 , b 3 , ..., b l-1 , b l , b l+1 , ..., b l+n ]
where a 0 , b 0 ∈ Z and a i , b j ∈ Z + for i = 1, 2, ..., k + m and j = 1, 2, ..., l + n. This representation allows us to describe the quadratic irrationals not by their numerical values, i.e. real values, but rather by their corresponding continued fractions which requires only integers. Thus, the comparison between quadratic irrationals becomes the problem of comparison their corresponding periodic continued fractions.

Algorithm for comparing two quadratic irrationals

We would like to compare two quadratic irrationals Q 1 and Q 2 , given by:

Q i = p i + √ q i r i
where p i , q i , r i ∈ Z , q i > 0 and r i = 0 for i = 1, 2. As discussing, our exact comparison method is based on calculating the corresponding continued fraction. However, for comparing we dont need to compute all terms of continued fraction expansions but the comparison happen for each coup of terms having the same level of continued fractions. Precisely, let us consider

Q 1 = p 1 + √ q 1 r 1 = [a 0 ; a 1 , a 2 , ...] and Q 2 = p 2 + √ q 2 r 2 = [b 0 ; b 1 , b 2 , ...].
Without loss of generality, we can assume k ∈ Z + is the smallest index for which a k is unequal to b k . In order to compare Q 1 and Q 2 , we should calculate expression E = (-1) k (a k -b k ). There three cases:

1. Q 1 = Q 2 if and only if p 1 = p 2 , q 1 = q 2 and r 1 = r 2 (Theorem 3.2 about unique correspondence).

2.

Q 1 < Q 2 if E < 0. 3. Q 1 > Q 2 otherwise.
As explaining before, the comparison will be done for each coup of terms a i and b i . For calculating these terms we will use two methods that are described in section 3.

To apply Algorithm 1 for calculating continued factions, we need to modify quadratic irrationals Q i into square root's form, as follows:

Q 1 = r 2 p 1 + r 2 √ q 1 r 2 r 1 = r 2 p 1 + r 2 2 q 1 r 2 r 1 , Q 2 = r 1 p 2 + r 1 √ q 2 r 2 r 1 = r 1 p 2 + r 1 2 q 2 r 2 r 1 Thus, comparing Q 1 and Q 2 becomes comparing the numerators of Q 1 and Q 2 . If we have r 2 2 q 1 = [a 0 ; a 1 , a 2 , ..., a n , 2a 0 ], r 1 2 q 2 = [b 0 ; b 1 , b 2 , ..., b n , 2b 0 ].
Then we obtain the two numerators as

r 2 p 1 + r 2 2 q 1 = [a 0 + r 2 p 1 ; a 1 , a 2 , ..., a n , 2a 0 ], r 1 p 2 + r 1 2 q 2 = [b 0 + r 1 p 2 ; b 1 , b 2 , ..., b n , 2b 0 ].
Now we can use Algorithm 1 in order to calculate the periodic continued fractions of two square roots r 2 2 q 1 and r 1 2 q 2 . Function 1: Calculate i-th term of continued fraction using Algorithm 1

Input: A quartet of integer: a 0 , a i , m i and d i Output: A triplet of integer: a i+1 , m i+1 and

d i+1 m i+1 ←-d i a i -m i d i+1 ←- S-m 2 i+1 d i a i+1 ←- a 0 +m i+1 d i+1
By using Algorithm 2, we can get directly the corresponding continued fraction of a quadratic irrational without modify quadratic form into square root's form.

Function 2: Calculate i-th term of continued fraction using Algorithm 2

Input: A quartet of integer: q 0 , a i , p i and r i Output: A triplet of integer: a i+1 , p i+1 and r i+1 p i+1 ←-a i r i -p i r i+1 ←-

q 0 -p 2 i+1 r i a i+1 ←- p n+1 + √ q 0 r n+1
Finally, for comparing we have the algorithm using Algorithm 4 as follows:

Algorithm 3: Compare two quadratic irrationals Input: Two triplets : (p 1 , q 1 , r 1 ) and (p 2 , q 2 , r 2 ) represent two quadratic irrational Q 1 and Q 2 Output: Value indicating result of the comparison:

0 if Q 1 = Q 2 ; 1 if Q 1 > Q 2 ;
otherwise -1. if p 1 = p 2 and q 1 = q 2 and r 1 = r 2 then return 0

else E ←-0 i ←-0 p 1i ←-p 1 q 1i ←-q 1 r 1i ←-r 1 p 2i ←-p 2 q 2i ←-q 2 r 2i ←-r 2 while E = 0 do (p 1(i+1) , q 1(i+1) , r 1(i+1) ) ←-Function 2 (q 1 , p 1i , q 1i , r 1i ) (p 2(i+1) , q 2(i+1) , r 2(i+1) ) ←-Function 2 (q 2 , p 2i , q 2i , r 2i ) E ←-(-1) i (a 1i -a 2i ) i ←-i + 1 end if E > 0 then return 1 else return -1 end

Complexity

The complexity of comparing two periodic continued fractions depends on the number of terms to examine. In the worst case, we have to compare all terms in the periodic expansion. Thus, the complexity belongs to length of the repeating block of quadratic surd Q, given by:

Q = p + √ q r = [a 0 ; a 1 , a 2 , ..., a n , 2a 0 ].
If the length of the partial quotients of Q and q are denoted by L(Q) and L(q); we have L(Q) = L(q) = n + 1. Lagrange proved that the largest partial denominator a i in the expansion is less than 2 √ q, and that L(q) < 2q. More recently Hickerson [START_REF] Dean | Length of period simple continued fraction expansion of √ d[END_REF] and Podsypanin [START_REF] Podsypanin | Length of the period of a quadratic irrational[END_REF], based on the divisor function 3 , have shown that L(q) is given by L(q) = O( (q) ln q).

Discussion

In this section, we would like to compare our approach with several other methods: Approximation method and Square comparison method to show why this approach is more efficient than the others.

Approximation method vs. Exact method

Square root of a positive integer number S which is not a perfect square is an irrational number. There exist many methods for evaluating this value. It can be classified into two categories : approximation methods and exact methods. We say approximation method if we represent √ S by an equivalence which is a infinitive real number4 . For instance Newton's method, Exponential identity, Rough estimation, Babylonian method, etc. However, there are two problems with these approaches : (1) Computers normally use the IEEE5 representation for real numbers; this standard represents a floating point number up to some precision as supported by the computing hardware. However, irrational numbers, such as √ 2, or even many rational numbers cannot be expressed exactly in this format. They must be approximated. (2) An algorithm must terminate at some point when a sufficient approximation is reached; as a result, only an approximation of quadratic irrational is provided. For these reasons, the comparison of two quadratic irrationals is true only within some precision. By using periodic continued fraction to represent irrational numbers, our approach is an exact method because it employs only integers to represent and to compare them.

Square comparison vs. Continued Faction comparison

Square approach is an exact comparison by using square function to avoid the irrational numbers. For instance, we have two quadratic irrational Q 1 and Q 2

Q 1 = p 1 + √ q 1 r 1 and Q 2 = p 2 + √ q 2 r 2 . If Q 1 > Q 2 then Q 1 -Q 2 > 0 Q 1 -Q 2 = p 1 + √ q 1 r 1 - p 2 + √ q 2 r 2 = r 2 p 1 + r 2 √ q 1 r 2 r 1 - r 1 p 2 + r 1 √ q 2 r 2 r 1 > 0,
because r 1 r 2 > 0, thus r 2 p 1 + r 2 √ q 1 > r 1 p 2 + r 1 √ q 2 , take the square two times (r 2 1 q 2 + r 2 2 q 1 -(r 2 p 1 -r 1 p 2 ) 2 ) 2 > 4r 2 1 r 2 2 q 1 q 2 .

The below equation contains only integers. Therefore, we can verify Q 1 -Q 2 with integer calculation. However the fact of using the square function, computation perform on big values which are squares of integer values ! The comparison of quadratic irrationals with continued factions approach (Algorithm 1) permit to calculate with integers that is smaller than r 2 p 1 + r 2 √ q 1 and r 1 p 2 + r 1 √ q 2 .

Conclusion

In this paper we presented a discussion on relation between quadratic irrationals and continued fractions, from that we introduced an exact comparison of two quadratic irrationals that

It is also known as a quadratic irrationality or quadratic surd.

A perfect square is a number that can be written as the product of some integer with itself, i.e : √ x = a with a ∈ Z.

divisor function is an arithmetical function related to the divisor of an integer. It counts the number of divisors of an integer.

Irrational number can't be represented as terminating or repeating decimals.

Standard for Floating-Point Arithmetic is the most widely-used standard for floating-point computation.

is important in dividing parameter space of digital image transforms. We have determined the complexity of this comparison that permit to reduce the complexity of general discrete rigid transformation's algorithm.