Nicolas Brisebarre

Joldes ¸ Mioara

Érik Martin-Dorel

Jean-Michel Muller

Peter Kornerup
email: kornerup@imada.sdu.dk

Augmented precision square roots, 2-D norms, and discussion on correctly rounding x 2 + y 2

Keywords: Floating-point arithmetic, compensated algorithms, square-root, Correct rounding, 2D-norms, accurate computations

Define an "augmented precision" algorithm as an algorithm that returns, in precision-p floating-point arithmetic, its result as the unevaluated sum of two floating-point numbers, with a relative error of the order of 2 -2p . Assuming an FMA instruction is available, we perform a tight error analysis of an augmented precision algorithm for the square root, and introduce two slightly different augmented precision algorithms for the 2D-norm p x 2 + y 2 . Then we give tight lower bounds on the minimum distance (in ulps) between p x 2 + y 2 and a midpoint when p x 2 + y 2 is not itself a midpoint. This allows us to determine cases when our algorithms make it possible to return correctly-rounded 2D-norms.

I. INTRODUCTION

In some applications, just returning a floating-point approximation y h to the exact result y of a function or arithmetic operation may not suffice. It may be useful to also return an estimate y of the error (i.e., y ≈ y h + y ). For simple enough functions (e.g., addition or multiplication), it is even possible to have y = y h + y exactly. Having an estimate of the error makes it possible to re-use it later on in a numerical algorithm, in order to at least partially compensate for that error. Such compensated algorithms have been suggested in the literature for summation of many floating-point numbers [START_REF] Kahan | Pracniques: further remarks on reducing truncation errors[END_REF]- [START_REF] Rump | Accurate floating-point summation part I: Faithful rounding[END_REF], computation of dot products [START_REF] Ogita | Accurate sum and dot product[END_REF], and evaluation of polynomials [START_REF] Graillat | Algorithms for accurate, validated and fast computations with polynomials[END_REF].

We will call augmented-precision algorithm an algorithm that returns, in radix-β, precision-p floating-point arithmetic, an approximation y h + y (i.e., an unevaluated sum of two floating-point numbers) to an exact result y = f (x) (or f (x 1 , x 2 )) such that • y ≤ 1 2 ulp(y h ); • there exists a small constant C (say, C β p ) such that

|y -(y h + y )| < C • β -2p • y.
This work is partially supported by the CIBLE programme of Région Rhône-Alpes, France.

This work is partially supported by the TaMaDi project of the french Agence Nationale de la Recherche.

When y = y h +y exactly, the transformation that generates y h and y from the inputs of f is called an error-free transform in the literature.

The unevaluated sum y h +y is called an expansion. Several algorithms have been suggested for performing arithmetic on such expansions [START_REF] Priest | Algorithms for arbitrary precision floating point arithmetic[END_REF], [START_REF] Shewchuk | Adaptive precision floating-point arithmetic and fast robust geometric predicates[END_REF]- [START_REF] Briggs | The doubledouble library[END_REF].

In the first part of this paper, we briefly recall two wellknown error-free transforms used later on in the paper. Then, we analyze an augmented-precision algorithm for the squareroot. In the third part, we use that algorithm for designing augmented-precision algorithms for computing x 2 + y 2 , where x and y are floating-point numbers. Such a calculation appears in many domains of scientific computing. It is also an important step when computing complex square roots. The naive method-i.e., straightforward implementation of the formula x 2 + y 2 -may lead to spurious overflows or underflows. When there are no overflows/underflows, it is quite accurate (an elementary calculation shows that on a radix-2, precision-p floating-point system, the relative error is bounded by 2 -p+1 + 2 -2p ).

Friedland [START_REF] Friedland | Algorithm 312: Absolute value and square root of a complex number[END_REF] avoids spurious overflows by computing

x 2 + y 2 as |x| • 1 + (y/x) 2 if |x| ≥ |y|, and |y| • 1 + (x/y) 2 otherwise.
Kahan 1 , and Midy and Yakovlev [START_REF] Midy | Computing some elementary functions of a complex variable[END_REF] normalize the computation using a power of the radix of the computer system: in radix 2, if |x| ≥ |y|, let b x be the largest power of 2 less than or equal to x, what they actually compute is b

x • (x/b x ) 2 + (y/b x ) 2 .
Their solution is less portable (and possibly on some systems, less fast) than Friedland's solution, yet it will be in general slightly more accurate, since division and multiplication by b x is exact. Our augmented-precision algorithms will derive from this one. As noticed by Kahan, the IEEE 754 Standard for Floating-Point Arithmetic [START_REF]IEEE Standard for Floating-Point Arithmetic. IEEE Standard[END_REF] defines functions scaleB and logB that make this scaling of x and y easier to implement.

Hull et al. [START_REF] Hull | Implementing complex elementary functions using exception handling[END_REF] use the naive method along with the exception-handling possibilities specified by the IEEE 754-1985 Standard to recover a correct result when the naive method fails.

In the fourth part, we investigate the possibility of correctly rounding x 2 + y 2 (assuming round-to-nearest). This requires solving the table maker's dilemma for that function, i.e., finding a lower bound on the smallest possible nonzero distance (in ulps, or in terms of relative distance) between

x 2 + y 2 and a midpoint, where a midpoint is the exact middle of two consecutive floating-point numbers.

II. TWO WELL-KNOWN ERROR-FREE TRANSFORMS

A. The Fast2Sum algorithm

The Fast2Sum algorithm was first introduced by Dekker [START_REF] Dekker | A floating-point technique for extending the available precision[END_REF], but the three operations of this algorithm already appeared in 1965 as a part of a summation algorithm, called "Compensated sum method," due to Kahan [START_REF] Kahan | Pracniques: further remarks on reducing truncation errors[END_REF]. Under conditions spelled out by Theorem 1, it returns the floating term s nearest to a sum a + b and the error term t = (a + b) -s. Throughout the paper, RN(u) means "u rounded to the nearest even" (see [START_REF] Muller | Handbook of Floating-Point Arithmetic[END_REF]).

Algorithm 1. Fast2Sum s ← RN(a + b) z ← RN(s -a) t ← RN(b -z) return (s, t)
The following theorem is due to Dekker.

Theorem 1 (Fast2Sum algorithm [START_REF] Dekker | A floating-point technique for extending the available precision[END_REF]). Assume the floatingpoint system being used has radix β ≤ 3, subnormal numbers available, and provides correct rounding to nearest. Let a and b be floating-point numbers, and assume that the exponent of a is larger than or equal to that of b (this condition is satisfied if |a| ≥ |b|). Algorithm 1 computes two floating-point numbers s and t that satisfy the following:

• s + t = a + b exactly;
• s is the floating-point number that is closest to a + b.

B. The 2MultFMA algorithm

The FMA instruction makes it possible to evaluate ±ax±b, where a, x, and b are floating-point numbers, with one final rounding only. That instruction was introduced in 1990 on the IBM RS/6000. It allows for faster and, in general, more accurate dot products, matrix multiplications, and polynomial evaluations. It also makes it possible to design fast algorithms for correctly rounded division and square root [START_REF] Markstein | Computation of elementary functions on the IBM RISC System/6000 processor[END_REF].

The FMA instruction is included in the newly revised IEEE 754-2008 standard for floating-point arithmetic [START_REF]IEEE Standard for Floating-Point Arithmetic. IEEE Standard[END_REF].

If an FMA instruction is available, then, to compute the error of a floating-point multiplication x 1 • x 2 , one can design a very simple algorithm, which only requires two consecutive operations, and works for any radix and precision, provided the product does not overflow and e x1 + e x2 ≥ e min + p -1, where e x1 and e x2 are the exponents of x 1 and x 2 , and e min is the minimum exponent of the floating-point system.

Algorithm 2 (2MultFMA(x 1 , x 2 )).

r 1 ← RN(x 1 • x 2 ) r 2 ← RN(x 1 • x 2 -r 1 ) return (r 1 , r 2 )

III. AUGMENTED-PRECISION REAL SQUARE ROOT WITH

AN FMA Let us now present an augmented-precision real square root algorithm. That algorithm is straightforwardly derived from the following theorem, given in [START_REF] Boldo | Representable correcting terms for possibly underflowing floating point operations[END_REF] (see also [START_REF] Muller | Handbook of Floating-Point Arithmetic[END_REF]):

Theorem 2 (Computation of square root residuals using an FMA [START_REF] Boldo | Representable correcting terms for possibly underflowing floating point operations[END_REF]). Assume x is a precision-p, radix-β, positive floating-point number. If σ is √ x rounded to a nearest floatingpoint number then

x -σ 2 is exactly computed using one FMA instruction, with any rounding mode, provided that

2e σ ≥ e min + p -1, (1)
where e σ is the exponent of σ.

Algorithm 3 (Augmented computation of √ x). σ ← RN( √ x) t ← x -σ 2 (exact operation through an FMA) r ← RN(t/(2σ)) return (σ, r)
Notice that similar approximations are used in [START_REF] Markstein | IA-64 and Elementary Functions: Speed and Precision[END_REF] in a different context (to return a correctly-rounded square root from an accurate enough approximation), as well as in [START_REF] Hida | Algorithms for quad-double precision floating-point arithmetic[END_REF] for manipulating floating-point expansions. This is not surprising, since behind this approximation there is nothing but the Taylor expansion of the square-root. What we do claim here, is that we have been able to compute a very tight error bound for Algorithm 3 (indeed, an asymptotically optimal one, as we will see later on). That error bound is given by the following theorem, which shows that the number r returned by Algorithm 3 is a very sharp estimate of the error √ x -σ.

Theorem 3. In radix-2, precision-p arithmetic, if the exponent e x of the FP number x satisfies e x ≥ e min + p, then the output (σ, r) of Algorithm 3 satisfies σ = RN( √ x) and

(σ + r) - √ x < 2 -p-1 ulp(σ), and
(σ + r) - √ x < 2 -2p • σ. Proof. First, if e x ≥ e min + p then x ≥ 2 emin+p , so that √ x ≥ 2 e min +p 2 ≥ 2 e min +p 2
, which implies

σ = RN( √ x) ≥ 2 e min +p 2
, therefore,

e σ ≥ e min + p 2 ,
so that we have,

2e σ ≥ e min + p -1.
Therefore, Theorem 2 applies: t = x -σ 2 is a floatingpoint number, so that it is exactly computed using an FMA instruction. Now, since σ = RN( √ x) and σ is a normal number (a square root never underflows or overflows), and since the square root of a floating-point number is never equal to a midpoint [START_REF] Markstein | IA-64 and Elementary Functions: Speed and Precision[END_REF], [START_REF] Jeannerod | Midpoints and exact points of some algebraic functions in floating-point arithmetic[END_REF], we have

|σ - √ x| < 2 -p • 2 eσ , which gives |t| = |σ 2 -x| = |σ - √ x| • |σ + √ x| < 2 -p+eσ • (2σ + 2 eσ-p ).
Notice that 2σ is a floating-point number, and that ulp(2σ) = 2 eσ-p+2 . Therefore there is no floating-point number between 2σ and 2σ+2 eσ-p . Hence, since |t|/2 -p+eσ is a floating-point number less than 2σ + 2 eσ-p , we obtain

|t| ≤ 2 -p+eσ+1 • σ, implying t 2σ ≤ 2 -p+eσ .
Also, since 2 -p+eσ is a floating-point number, the monotonicity of the round-to-nearest function implies

RN t 2σ ≤ 2 -p+eσ .
From these two inequalities, we deduce

RN t 2σ - t 2σ ≤ 2 -2p-1+eσ . (2)
Notice (we will need that in Section IV) that

RN t 2σ ≤ 2 -p • σ. (3)
Now, define a variable as

√ x = σ + t 2σ + , where = √ x -σ - t 2σ = t √ x + σ - t 2σ = t • 2σ -( √ x -σ) ( √ x + σ) • 2σ = - (σ - √ x) 2 2σ
, from which we deduce

| | < 2 -2p+2eσ 2σ ≤ 2 -2p-1+eσ . ( 4
)
By combining ( 2) and ( 4), we finally get

σ + RN t 2σ - √ x < 2 -2p+eσ .
This gives an error in ulps as well as a relative error: since ulp(σ) = 2 eσ-p+1 we obtain

σ + RN t 2σ - √ x < 2 -p-1 ulp(σ),
and σ + RN t 2σ - √ x < 2 -2p • σ.
Notice that the bound given by Theorem 3 is quite tight. Consider as an example the case p = 24 (binary32 precision of the IEEE 754-2008 Standard). Assume x is the floatingpoint number

x = 8402801 • 2 -23 = 1.00169193744659423828125,
(σ + r) - √ x = 0.9970011 • • • × 2 -48 × σ,
to be compared to our bound 2 -48 × σ. Furthermore, the error bounds given by Theorem 3 are asymptotically optimal, as we can exhibit a family (for p multiple of 3) of input values parametrized by the precision p, such that for these input values, |(σ + r) -√ x| /σ is asymptotically equivalent to 2 -2p as p → ∞. Just consider, for p being a multiple of 6, the input number x = 2 p + 2 p/3+1 + 2, and for p odd multiple of 3, the input number

x = 2 p-1 + 2 p/3 + 1.
If p is multiple of 6 (the case where p is an odd multiple of 3 is very similar), tedious yet not difficult calculations show that

√ x = 2 p/2 + 2 -p/6 + 2 -p/2 -2 -1-5p/6 -2 -7p/6 +3 • 2 -1-11p/6 + • • • , σ = 2 p/2 + 2 -p/6 , t = 2 -2 -p/3 , t/(2σ) = 2 -p/2 • 1 -2 -p/3-1 -2 -2p/3 + 2 -p-1 +3 -4p/3 -• • • , r = 2 -p/2 • 1 -2 -p/3-1 -2 -2p/3 + 2 -p , σ + r = 2 p/2 + 2 -p/6 + 2 -p/2 -2 -1-5p/6 -2 -7p/6 +2 -3p/2 , so that √ x -(σ + r) ∼ p→∞ 2 -3p/2 • -1 + 3 • 2 -1-p/3 , from which we derive | √ x -(σ + r)| ∼ p→∞ 2 -p-1 ulp(σ), and √ x -(σ + r) σ ∼ p→∞ 2 -2p 1 -3 • 2 -1-p/3 ,
which shows the asymptotic optimality of the bounds given by Theorem 3.

IV. AUGMENTED-PRECISION 2D NORMS

We suggest two very slightly different algorithms. Algorithm 5 requires three more operations (a Fast2Sum) than Algorithm 4, but it has a slightly better error bound. Again, as for the square-root algorithm, these algorithms derive quite naturally from the Taylor series for the square root: the novelty we believe we bring here is that we provide proven and tight error bounds.

Algorithm 4 (Augmented computation of x 2 + y 2 ).

1: if |y| > |x| then 2: swap(x, y) 3: end if 4: b x ← largest power of 2 less than or equal to x 5: x ← x/b x {exact operation} 6: ŷ ← y/b x {exact operation} 7: (s x , ρ x ) ← 2MultFMA(x, x) 8: (s y , ρ y ) ← 2MultFMA(ŷ, ŷ) 9: (s h , ρ s ) ← Fast2Sum(s x , s y ) 10: s ← RN(ρ s + RN(ρ x + ρ y )) 11: r 1 ← RN( √ s h ) 12: t ← s h -r 2
1 {exact operation through an FMA} 13: r 2 ← RN(t/(2r 1 )) 14: c ← RN(s /(2s h ))

15: r 3 ← RN(r 2 + r 1 c) 16: r 1 ← r 1 • b x {exact operation} 17: r 3 ← r 3 • b x {exact operation} 18: (r h , r ) ← Fast2Sum(r 1 , r 3 ) 19: return (r h , r )
Algorithm 5 (Slightly more accurate augmented-precision computation of x 2 + y 2 ).

1: if |y| > |x| then 2: swap(x, y) 3: end if 4: b x ← largest power of 2 less than or equal to x 5: x ← x/b x {exact operation} 6: ŷ ← y/b x {exact operation} 7: (s x , ρ x ) ← 2MultFMA(x, x) 8: (s y , ρ y ) ← 2MultFMA(ŷ, ŷ) 9: (s h , ρ s ) ← Fast2Sum(s x , s y ) 10: s ← RN(ρ s + RN(ρ x + ρ y )) 11: (s h , s ) ← Fast2Sum(s h , s ) 12: r 1 ← RN( s h ) 13: t ← s h -r 2
1 {exact operation through an FMA} 14: r 2 ← RN(t/(2r 1 ))

15: c ← RN(s /(2s h )) 16: r 3 ← RN(r 2 + r 1 c) 17: r 1 ← r 1 • b x {exact operation} 18: r 3 ← r 3 • b x {exact operation} 19: (r h , r ) ← Fast2Sum(r 1 , r 3 ) 20: return (r h , r )
Notice that if one is just interested in getting a very accurate floating-point approximation to x 2 + y 2 (that is, if one does not want to compute the error term r ), then it suffices to replace the last Fast2Sum instruction by "r h ← RN(r 1 +r 3 )" in both algorithms. Also notice that if the functions scaleB and logB defined by the IEEE 754-2008 Standard are available and efficiently implemented, one can replace lines 4, 5 and 6 of both algorithms by e x ← logB(x)

x ← scaleB(x, -e x ) ŷ ← scaleB(y, -e x ) and lines 16 and 17 of Algorithm 4, or lines 17 and 18 of Algorithm 5 by r 1 ← scaleB(r 1 , e x ) r 3 ← scaleB(r 3 , e x ). We have the following result Theorem 4 (Accuracy of algorithms 4 and 5). We assume that a radix-2, precision-p (with p ≥ 8), floating-point arithmetic is used and that there are no underflows or overflows.

The result (r h , r ) returned by Algorithm 4 satisfies

r h + r = x 2 + y 2 + , with | | < 15 2 • 2 -2p + 30 • 2 -3p • r h , and r ≤ 1 2 ulp(r h ).
The result (r h , r ) returned by Algorithm 5 satisfies

r h + r = x 2 + y 2 + , with | | < 47 8 • 2 -2p + 26 • 2 -3p • r h , and r ≤ 1 2 ulp(r h ).
Since the proofs are very similar for both algorithms, we only give the proof for Algorithm 5.

Proof of the error bound for Algorithm 5.

The computations of b x , x, and ŷ are obviously errorless. We have

x2 + ŷ2 = s x + s y + ρ x + ρ y = s h + ρ s + ρ x + ρ y , with |ρ x | ≤ 2 -p s x , |ρ y | ≤ 2 -p s y , and |ρ s | ≤ 2 -p s h .
We easily find

|ρ x + ρ y | ≤ 2 -p (s x + s y ) = 2 -p (s h + ρ s )
and define u = RN(ρ x + ρ y ).

We have |u| ≤ RN (2 -p (s x + s y )), so that

|u| ≤ 2 -p s h and |u -(ρ x + ρ y )| ≤ 2 -2p • s h . (5)
We therefore get

|ρ s + u| ≤ 2 -p+1 • s h , so that s ≤ 2 -p+1 • s h when p ≥ 2. Also, |s -(ρ s + u)| ≤ 2 -2p+1 • s h .
This, combined with ( 5), gives

|s -(ρ s + ρ x + ρ y )| ≤ 3 • 2 -2p • s h , implying that x2 + ŷ2 = s h + s + 0 , with | 0 | ≤ 3 • 2 -2p • s h .
We also have,

s h + s = s h + s , |s | ≤ 2 -p • s h , and |s | ≤ 2 -p+1 • s h , so that s h ≤ 1 + 2 -p 1 -2 -p+1 • s h ≤ (1 + 3 • 2 -p + 7 • 2 -2p ) • s h , when p ≥ 4. Which gives x2 + ŷ2 = s h + s + 0 , with | 0 | ≤ (3 • 2 -2p + 10 • 2 -3p ) • s h , (6)
when p ≥ 5. Now,

x2 + ŷ2 = s h + s + 0 = s h • 1 + s + 0 2s h + 1 ,
with

| 1 | ≤ 1 8 (s + 0 ) 2 (s h ) 2 •   1 1 - s + 0 s h   .
From the bounds on s and 0 we get

s + 0 s h ≤ 2 -p + 3 • 2 -2p + 10 • 2 -3p ,
which gives

| 1 | ≤ 1 8 • 2 -p + 3 • 2 -2p + 10 • 2 -3p 2 1 -(2 -p + 3 • 2 -2p + 10 • 2 -3p ) < 2 -2p-3 + 2 -3p , when p ≥ 6. Hence, x2 + ŷ2 = s h • 1 + s 2s h + 2 (7)
with

| 2 | ≤ | 1 | + 0 2s h < 13 8 • 2 -2p + 6 • 2 -3p .
In Eq. ( 7), s h is approximated by r 1 + r 2 using Algorithm 3. Therefore, from Theorem 3, we have

s h = r 1 + r 2 + 3 , with | 3 | < 2 -2p • r 1 .
Since |s /(2s h )| ≤ 2 -p-1 , so that |c| ≤ 2 -p-1 too, and

c - s 2s h ≤ 2 -2p-2 , hence x2 + ŷ2 = (r 1 + r 2 + 3 ) • (1 + c + 4 ), with | 4 | ≤ c - s 2s h + | 2 | < 15 8 • 2 -2p + 6 • 2 -3p .
From the bound (3) obtained in the proof of Theorem 3, we have

|r 2 | ≤ 2 -p • |r 1 |. All this gives x2 + ŷ2 = r 1 + r 2 + r 1 c + 6 , with | 6 | ≤ | 3 | + |r 1 4 | + |r 2 c| + |r 2 4 | + | 3 | • |1 + c 4 | ≤ r 1 • 35 8 • 2 -2p + 63 8 • 2 -3p + 6 • 2 -4p + 15 16 • 2 -5p + 3 • 2 -6p ≤ r 1 • 35 8 • 2 -2p + 8 • 2 -3p ,
when p ≥ 6. From the previously obtained bounds on r 2 and c,

|r 2 + r 1 c| ≤ 3 2 • 2 -p • r 1 so that r 3 = r 2 + r 1 c + 7 , with | 7 | ≤ 3 2 • 2 -2p • r 1 ,
and

|r 3 | ≤ 3 2 • 2 -p • (1 + 2 -p ) • r 1 .
We therefore conclude that

r h + r = r 1 + r 3 = x 2 + y 2 + 8 , with | 8 | ≤ | 6 | + | 7 | ≤ 47 8 • 2 -2p + 8 • 2 -3p • r 1 . From r h + r = r 1 + r 3 , |r | ≤ 2 -p |r h |, and |r 3 | ≤ 3 2 • 2 -p • (1 + 2 -p ) • r 1 , we get |r 1 | < 1 + 2 -p 1 -3 2 • 2 -p • (1 + 2 -p ) • |r h | ≤ 1 + 5 2 • 2 -p + 11 2 • 2 -2p • |r h |,
when p ≥ 6. From this we finally deduce that when p ≥ 8,

| 8 | ≤ 47 8 • 2 -2p + 23 • 2 -3p • |r h |. V. CAN WE ROUND x 2 + y 2 CORRECTLY?
In the following we call a midpoint a value exactly halfway between consecutive floating-point numbers. In any radix, there are many floating-point values x and y such that x 2 + y 2 is a midpoint [START_REF] Jeannerod | Midpoints and exact points of some algebraic functions in floating-point arithmetic[END_REF]. A typical example, in the "toy" binary floating-point system of precision p = 8 is x = 253 10 = 11111101 2 , y = 204 10 = 11001100 2 , for which

x 2 + y 2 = 325 10 = 101000101 2 .
If the minimum nonzero distance (in terms of relative distance, or in terms of ulps) between

x 2 + y 2 and a midpoint is η, then correctly rounding x 2 + y 2 can be done as follows:

• approximate x 2 + y 2 by some value s with error less than η/2; • if s is within η/2 from a midpoint m, then necessarily

x 2 + y 2 is exactly equal to that midpoint: we should return RN(m);

• otherwise, we can safely return RN(s). Hence our purpose in this section is to find lower bounds on the distance between x 2 + y 2 and a midpoint. Notice that in the special case where x and y have the same exponent, Lang and Muller provide similar bounds in [START_REF] Lang | Bound on run of zeros and ones for algebraic functions[END_REF].

As previously, we assume a binary floating-point arithmetic of precision p. Let x and y be floating-point numbers. Without loss of generality, we assume 0 < y ≤ x. Let e x and e y be the exponents of x and y. Define δ = e x -e y , so δ ≥ 0. We will now consider two cases.

If δ is large

First, let us notice that if x is large enough compared to y, our problem becomes very simple. More precisely, we have

x 2 + y 2 = x • 1 + y 2 x 2 = x + y 2 2x + , with - 1 8
y 4 x 3 < < 0.
When y ≤ 2 -p/2 x, we have

0 < y 2 2x ≤ 2 -p-1 x < 1 2 ulp(x), so that x -x 2 + y 2 = y 2 2x + < 1 2 ulp(x).
Hence when y ≤ 2 -p/2 x, x 2 + y 2 is far from a midpoint. Furthermore, in such a case, correctly rounding

x 2 + y 2 is straightforward: it suffices to return x. Notice that δ ≥ p/2 + 1 implies y ≤ 2 -p/2 x. So, let us now focus on the case δ < p/2 + 1, i.e. δ ≤ (p + 1)/2 .

If δ is small

Since x and y are floating-point numbers, there exist integers M x , M y , e x , and e y such that

x = M x • 2 ex-p+1 y = M y • 2 ey-p+1 ,
with 0 < M x , M y ≤ 2 p -1. Assume x 2 + y 2 is within ulps from a midpoint of the form (M s + 1/2) • 2 es-p+1 (with | | nonzero and much less than 1/2). Notice that x ≤ s ≤ ∆(x √ 2), where ∆(u) means u rounded up, so that e s is e x or e x + 1. We have,

M 2 x • 2 2ex + M 2 y • 2 2ey = M s + 1 2 + • 2 es , which gives = 2 -es M 2 x • 2 2ex + M 2 y • 2 2ey -M s + 1 2 .
This implies

= 2 -es • 2 2ey M 2 x • 2 2δ + M 2 y -2 2es M 2 s + M s + 1 4 2 ey M 2 x • 2 2δ + M 2 y + 2 es M s + 1 2 = 2 -es N D . (8)
Now since M s ≤ 2 p -1, 2 es • M s + 1 2 is less than 2 p+es and | | < 1/2, then 2 ey M 2

x • 2 2δ + M 2 y is less than 2 p+es too, so that the term D in ( 8) is less than 2 p+es+1 .

Notice that if e s = e x + 1 we can improve on that bound. In that case,

M 2 x • 2 2ex + M 2 y • 2 2ey < √ 2 2p + 2 2p-2δ • 2 ex = 1 2 √ 1 + 2 -2δ • 2 p+es ,
so that, when e s = e x + 1,

D < 1 + 2 -2δ • 2 p + 1 • 2 es . (since | | < 1/2, M s + 1 2 is within 2 es-1 from M 2 x • 2 2ex + M 2 y • 2 2ey
). Let us now focus on the term N in [START_REF] Graillat | Algorithms for accurate, validated and fast computations with polynomials[END_REF]. It is equal to

2 2ex-2δ M 2 x • 2 2δ + M 2 y -2 2(es-ex)+2δ M 2 s + M s + 1 4 ,

therefore

• if e s = e x + 1 or δ > 0, then N is an integer multiple of 2 2es-2(es-ex)-2δ = 2 2es-2-2δ . Hence, if is nonzero, its absolute value is at least

2 -es • 2 2es-2-2δ √ 1 + 2 -2δ • 2 p + 1 • 2 es = 2 -2-2δ √ 1 + 2 -2δ • 2 p + 1 ;
• if e s = e x and δ > 0, then again N is an integer multiple of 2 2es-2(es-ex)-2δ . Hence, if is nonzero, its absolute value is at least

2 -es • 2 2es-2(es-ex)-2δ 2 p+es+1 ≥ 2 -p-1-2δ .
• if e s = e x and δ = 0 then

N = 2 2es M 2 x + M 2 y -M 2 s -M s - 1 4
is a multiple of 2 2es /4, so that if is nonzero, its absolute value is at least 2 -p-3 . To summarize what we have obtained so far in the case "delta is small", whenever = 0, its absolute value is lower-bounded by 2 -p-3

in the case δ = 0; and

min 2 -2-2δ √ 1 + 2 -2δ • 2 p + 1 ; 2 -p-1-2δ = 2 -2-2δ √ 1 + 2 -2δ • 2 p + 1 in the case δ > 0.
Now we can merge the various cases considered above and deduce Theorem 5. If x and y are radix-2, precision-p, floating-point numbers, then either x 2 + y 2 is a midpoint, or it is at a distance of at least

2 -2 (p+1)/2 -2 √ 2 • 2 p + 1 ulp x 2 + y 2 from a midpoint.
When x and y are close, we obtain a much sharper result. For instance, when they are within a factor of 2, δ is equal to 0 or 1, which gives Theorem 6. If x and y are radix-2, precision-p, floating-point numbers such that |x/2| ≤ |y| ≤ 2 • |x|, then either x 2 + y 2 is a midpoint, or it is at a distance at least

2 -p-3 √ 5 + 2 -p+1 ulp x 2 + y 2
from a midpoint.

Tables I and II compare the actual minimum distance to a midpoint (obtained through exhaustive computation) and the bounds we have obtained in this section, in the case of "toy" floating-point systems of precision p = 10 and 15 (an exhaustive search was not possible for significantly wider formats). One can see on these tables that in the cases δ = 0 or δ = 1, our bounds are close to the minimum distance (a consequence is that there is little hope of significantly improving the bound given in Theorem 6), and that for larger values of δ, our bounds remain of the same order of magnitude as the minimum distance.

CONCLUSION

We have given a very tight error bound for a simple augmented-precision algorithm for the square root. We have also introduced two slightly different augmented-precision algorithms for computing

x 2 + y 2 . Then, we have given bounds on the distance between x 2 + y 2 and a midpoint, where x and y are floating-point numbers and x 2 + y 2 is not a midpoint. These bounds can be used to provide correctlyrounded 2D-norms (either using one of our algorithms, or another one).

  then one easily gets σ = 8395702 • 2 -23 = 1.0008456707000732421875, and r = -16749427 • 2 -48 ≈ -5.950591841497 × 10 -8 , which gives

Table I COMPARISON

 I BETWEEN THE ACTUAL MINIMUM DISTANCE TO A MIDPOINT (OBTAINED THROUGH EXHAUSTIVE COMPUTATION) AND THE BOUNDS OBTAINED USING OUR METHOD, IN THE CASE OF A "TOY" FLOATING-POINT SYSTEM OF PRECISION p = 10. WHEN δ ≥ 6, p x 2 + y 2 IS NECESSARILY FAR FROM A MIDPOINT.Table II COMPARISON BETWEEN THE ACTUAL MINIMUM DISTANCE TO A MIDPOINT (OBTAINED THROUGH EXHAUSTIVE COMPUTATION) AND THE BOUNDS OBTAINED USING OUR METHOD, IN THE CASE OF A "TOY" FLOATING-POINT SYSTEM OF PRECISION p = 15. WHEN δ ≥ 9, p x 2 + y 2 IS NECESSARILY FAR FROM A MIDPOINT. VI. APPLICATION: CORRECT ROUNDING OF x 2 + y 2 Various properties can be deduced from the analyses performed in the paper. Examples are: • we can obtain x 2 + y 2 correctly rounded in the bi-nary32 format of the IEEE 754-2008 standard if Algorithm 4 or Algorithm 5 is run in the binary64 format (or a wider format); • we can obtain x 2 + y 2 correctly rounded in the bi-nary64 format of the IEEE 754-2008 standard if Algorithm 4 or Algorithm 5 is run in the binary128 format; • if |x/2| ≤ |y| ≤ |2x|, we can obtain x 2 + y 2 correctly rounded in the binary64 format of the IEEE 754-2008 standard if Algorithm 4 or Algorithm 5 is run in the INTEL binary80 format.

		2
	δ	actual minimum distance	our bound to
		to a midpoint	that distance
	0	1.24 × 10 -4 ulp	1.22 × 10 -4 ulp
	1	5.73 × 10 -5 ulp	5.45 × 10 -5 ulp
	2	9.49 × 10 -5 ulp	1.48 × 10 -5 ulp
	3	8.76 × 10 -6 ulp	3.78 × 10 -6 ulp
	4	2.01 × 10 -6 ulp	9.51 × 10 -7 ulp
	5	6.24 × 10 -7 ulp	2.38 × 10 -7 ulp
	δ	actual minimum distance	our bound to
		to a midpoint	that distance
	0	3.81 × 10 -6 ulp	3.81 × 10 -6 ulp
	1	1.71 × 10 -6 ulp	1.70 × 10 -6 ulp
	2	4.65 × 10 -7 ulp	4.62 × 10 -7 ulp
	3	2.38 × 10 -7 ulp	1.18 × 10 -7 ulp
	4	5.96 × 10 -8 ulp	2.97 × 10 -8 ulp
	5	1.49 × 10 -8 ulp	7.44 × 10 -9 ulp
	6	3.76 × 10 -9 ulp	1.86 × 10 -9 ulp
	7	9.86 × 10 -10 ulp	4.56 × 10 -10 ulp
	8	3.81 × 10 -5 ulp	1.16 × 10 -10 ulp

Note that in the case p = 15, there is a large difference between the actual minimum distance and our bound when δ = 8. This is due to the fact that, when δ = 8, a very large part of the pairs (x, y) fall in the category "y < 2 -p/2 x", for which we have seen that p x 2 + y 2 is always far from a midpoint.


Charles Pontonnier
email: charles.pontonnier@irisa.fr

Georges Dumont

USING A COCONTRACTION RATIO TO PREDICT ANTAGONISTIC BEHAVIOR DURING ELBOW MOTION

Inverse dynamics methods for muscle forces prediction are globally unable to predict antagonistic activity during a joint motion. This is due to a lack of physiological information describing how forces are shared between flexors and extensors. The aim of this study is the definition and the use of a new EMG-based cocontraction ratio in an inverse dynamics muscle forces prediction approach applied to the elbow flexion motion. Results show the relevance of the ratio.

INTRODUCTION

The mechanical impedance of each joint of the human body is controlled by the cocontraction of both agonistic and antagonistic muscles. Cocontraction can be generated when muscles are involved in multiple joint motions [START_REF] Herzog | Cocontraction of pairs of antagonistic muscles : analytical solution for planar static nonlinear optimization approaches[END_REF]. Mathematically, cocontraction in a single joint cannot be predicted with standard inverse dynamics approaches [START_REF] Ait-Haddou | Theoretical considerations on cocontraction of sets of agonistic and antagonistic muscles[END_REF]. It also seems necessary to add physiological information in the optimization problem used to predict muscle forces. Using experimental data has been proposed in order to predict cocontraction. EMG measurement can be used to frame the activation of antagonistic muscles [START_REF] Vigouroux | Estimation of finger muscle tendon tensions and pulley forces during specific sports-climbing grisp techniques[END_REF]. This solution leads to realistic contraction patterns, but it is subject to scaling problems and muscle model or experimental uncertainty (such as Maximum Voluntary Contraction MVC). We propose to use a ratio instead of a direct activation value to obtain antagonistic activation and to avoid uncertainties on MVC value. Our hypothesis is to consider that dynamics of the motions lead to specific cocontraction patterns that can be used to predict the muscle forces. These cocontraction patterns can only be obtained by using EMG measurement during the motion. In the elbow joint case, triceps is globally antagonistic during flexion, and also during extension for a neutral shoulder position. We will use this muscle in application of this method.

METHODS

Using a framework and a musculoskeletal model of the arm fully described in [START_REF] Pontonnier | Inverse dynamics method using optimization techniques for the estimation of muscle forces involved in the elbow motion[END_REF][START_REF] Pontonnier | From motion capture to muscle forces in the human elbow aimed at improving the ergonomics of workstations[END_REF], we process motion capture and EMG data to obtain muscle forces (Figure 1). We capture the trunk and right arm motion. For the elbow joint we use bipolar EMG measurement of Biceps and Triceps activity to build the cocontraction ratio, assuming that the behavior of flexors is close to the biceps one and behavior of extensors is close to the triceps one. EMG signals are processed in the same way as proposed in [START_REF] Buchanan | Neuromusculoskeletal modeling : estimation of muscle forces and joints moments and movements from measurement of neural command[END_REF] to obtain muscle activations. Five muscles are taken into account in our elbow joint model: Biceps, Triceps, Brachialis, Brachioradialis, Anconeus. Standard Hill-type models [START_REF] Rengifo | Distribution of forces between synergistic and antagonistic muscles using an optimization criterion depending on muscle contraction behavior[END_REF] are associated with these muscles and parameters are adjusted according to [START_REF] Pontonnier | Motion analysis of the arm based on functional anatomy[END_REF].

The optimization problem is defined at each frame as follows :

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) () ( ) ( ) (1 ) ( ) (1 ) () n k k k n kk k k k k biceps global global triceps Fi f Fi a i F i R i b F i F i F i Fi c k i k i Fi                  (3)
The constraints are (a) mechanical equilibrium of the joint (b) Physiological limits of the muscles (c) cocontraction constraint (with a confidence indice γ of 0.1).

The cocontraction ratio for the elbow flexion/extension joint () global ki is defined at each frame i as follows:

() () ()

EMG biceps global EMG triceps Fi ki Fi  (2)
Where ()

EMG biceps

Fi and ()

EMG triceps

Fi are computed from the muscle models and the EMG-based activations measured for the frame i.

Experimental data has been collected on 4 healthy right handed male subjects without any specific training. They performed each 40 elbow flexion/extension motions at 10 different speeds (Figure 2) and 4 different loads (barbells). In order to minimize the muscle fatigue effects, each subject realizes motions a randomized order, and 5 minutes pauses are managed every 5 motions, allowing the subject to rest, drink and eat if he needs to. Most restricting motions can be followed by a pause, on subjects request.

RESULTS AND DISCUSSION

The influence of the cocontraction constraint on computed forces is studied by comparing the EMG activations with computed activations, using a global scale factor to decrease calibration issues (Isometric maximum forces, MVC). Figures 3 and4 show the mean error for the whole subjects between measured and computed activation without and with co-contraction constraint for the 40 motions. Each bar represents the error made on a motion (defined by a speed and a load) for the whole 4 subject. Without any co-contraction information, the estimation is absolutely unrealistic. The addition of the co-contraction constraint leads to a decrease of 70% the error made on the estimation. Even if the cocontraction ratio seems to be relevant to predict the behavior of antagonistic muscles, uncertainties on muscle parameters and calibration issues leads to an important remaining error.

Figure 3:

Error made on triceps activation without cocontraction constraint.

CONCLUSION

The addition of a cocontraction constraint in the optimization problem of an inverse dynamics approach leads to a more realistic behavior of antagonistic activations. Global variations of the activity are well predicted, but it still remains a static error due to the numerous arbitrary parameters introduced in muscle models. In order to improve these first results, calibration of muscle models and MVC procedures has to be more precise.

Figure 1 :

 1 Figure 1: Inverse dynamics framework using both motion capture and EMG data to compute muscle forces.

Figure 2 :

 2 Figure 2: Experimentation situation. The subject follows a visual guide to perform motions.

Figure 4 :

 4 Figure 4: Error made on triceps activation with cocontraction constraint.


Thomas Schrader

Thomas Gersthagen

Carsten Schmuck

Artificial RGD Receptor Molecules

Keywords: RGD peptide, integrins, molecular recognition, artificial receptors, amino acids RGD peptide, integrins, molecular recognition, artificial receptors, amino acids Introduction

published or not. The documents may come

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Numerous academic and industrial groups have developed peptide-derived or non-peptidic RGD mimetics, most of which showed some biological activity. Most notably, an elegant approach was introduced by Kessler et al., who synthesized a series of RGD-containing cyclopeptides with 5and 6-membered rings containing single D-amino acid residues. In extended NMR-spectroscopic studies the structural principles governing the formation of preferred fixed conformations were unraveled and used for a conformational design of integrin-specific RGD mimetics, which showed promising in vitro biological effects. During the course of these studies, a whole set of cyclic hexapeptides and pentapeptides was synthesized and thermodynamic equilibria as well as dynamics of their three-dimensional structures in solution were analyzed by 2D-NMR techniques.

Finally, systematic tuning of conformational restrictions accompanied by MD simulations and subsequent bioassays culminated in the preparation of the strong lead compound cyclo-RGDf[NMe]V (Figure 2), a powerful integrin antagonist, which effectively inhibits tumorassociated angiogenesis. [9,10,11] This very small cyclopeptide, which is currently undergoing clinical trials, imitates the RGD-loop in Vitronectin. In 2002 Arnaout et al. gained a crystal structure of this cyclopeptide in its complex with integrin αVß3, the only crystal structure of an RGD-integrin complex which is available today. This structure reveals that the peptide is inserted in a cleft between the so called ß-propeller from the αV-unit and ßA domains from the ß3-unit on the integrin head. As expected, almost all peptide-protein contacts are established by the RGD sequence (Figure 1c). [12] Figure 2 displays some related RGD mimetics, some of which have already been approved as drugs. Eptifibatide, a somewhat more flexible cyclohexapeptide with an internal disulfide bond, inhibits integrin a IIb ß 3 . It was developed entirely by pharmaceutical industry, and is used against

Artificial RGD Receptor Molecules.

A completely different approach to block pathologic RGD-Integrin interactions is to construct integrin mimetics which recognize RGD-containing proteins. In principle this simplifies the task to developing a small artificial host molecule which can serve as a competitive "lock" for the RGD-sequence in native proteins. Due to the encapsulated RGD-"key" the whole extracellular protein is not able any more to bind to its respective target integrin (Figure 3a). However, since the RGD-sequence exists in a variety of different conformations, depending on the parent protein, it is mandatory to provide synthetic hosts which can distinguish between them (Figure 3 b). Some years ago, the Schrader and Schmuck groups embarked on a program to pursue this alternative approach towards conformation-specific RGD host molecules. The starting point was marked by the discovery that m-xylylene bisphosphonates 1 bind guanidinium cations in a chelate-type manner reminiscent to the natural diaspartate binding site found in the crystal structure of the cRGDf[NMe]V/αVß3 complex. In both cases a bidentate salt bridge with a network of ion-pair reinforced hydrogen bonds firmly embraces arginine's guanidinium cation and holds it in place (Figure 3 c,d). [13] An early screening experiment for RGD-binders was carried out by Ruoslahti et al. In 1999 a patent was filed which announced the discovery that cyclic peptides with the consensus sequence (W/P)DD(G/L)(W/L)(W/L/M) bind to fibronectin and related RGD-containing proteins. The best candidate was cyclo-CWDDGWLC 2, with an internal disulfide bond, which is depicted in Figure 4 a. It was isolated from a library of short cyclic peptides and peptidomimetics in the course of specific assays for RGD-binding activity. The peptides were presented in form of a phage library which was used to isolate minimal receptor sequences that bind to fibronectin in affinity panning. Thus, the cyclic octapeptide c-CWDDGWLC was identified as the most potent inhibitor for RGD dependent cell attachment to fibronectin and vitronectin. Fibronectin itself tightly bound to a c-CWDDGWLC affinity column and could subsequently be reeluted with an RGD-containing peptide. Interestingly, a specific region within the ß-subunit of RGD-binding integrins had previously been shown by site-directed mutagenesis to be involved in ligand binding [14], which includes a the short peptide stretch KDDLW -quite similar to the above- described sequence. Short synthetic peptides correspondening to this integrin region bind to the RGD containing protein, but their affinities were markedly reduced if both aspartate residues are mutated to alanines. In summary, the combination of phage display technology, rational design and mutagenesis has lead to structural integrin mimetics which can significantly reduce the cell attachment of an RGD-containing protein [15,16]. However, these are still sensitive to proteolytic degradation. ) in order to generate a new chromatography stationary phase for the separation of RGD-peptides from mixtures. They used the free RGD peptide as template molecule, and prepared molecularly imprinted polymers (MIPs) from only two functional monomers, i. e., methacrylic acid (MAA) and acrylamide (AA). Three different crosslinking monomers were tried, with trimethylpropane trimethacrylate producing the best results in terms of RGD-selectivity. MAA was later postulated to form interactions with arginine, while AA's amide is both a moderate hydrogen bond donor and acceptor. To establish RGD selectivity, four other peptides where used in rebinding experiments, namely KGD and three other completely different peptides. Selectivity factor values ranged from 1.27 up to 1.98, net rebinding values for RGD and KGD were similar (~ 8-10%), while those for other peptides remained very small (2-4 %). These data indicate low affinity and only moderate selectivity for the template molecule. In addition, the free RGD peptide was examined in methanol solution, where K d values were in the high micromolar range. [17] Surprisingly, to the best of our knowledge, apart from our own project, these are the only artificial RGD receptor molecules published to date. We tentatively explain this with a lack of powerful concepts for the generation of water-soluble host molecules for zwitterions in physiological solution. In the aqueous medium, hydrogen bonds must be reinforced with electrostatic interactions and complemented by hydrophobic forces to achieve high affinities. Zwitterions are prone to undergo intramolecular as well as intermolecular self-association, which

H 2 N NH 2 N N N H 2 N NH 2 N O O O O H H H H H H N N N N O O O O N N N H H H H H CO 2 O 2 C O O CO 2 O 2 C O O 2 C CO 2 O CO 2 O 2 C N H H 2 N NH 2 H 3 N O O
often renders the desired host-guest complexes insoluble in buffered aqueous solutionespecially in the presence of high salt loads.

For isolated arginine and aspartate as well as for model peptides containing only anionic or cationic residues, some powerful receptor molecules have been developed: Hamilton et al. created a rigid concave scaffold 3 with two preorganized guanidinium moieties (Figure 5 a) which specifically bind two aspartate carboxylates in the i+3-relation of an α-helix. Small aspartate-containing model peptides are bound with millimolar affinities, albeit only in a 10% water-methanol mixture. [18] Bell et al. synthesized crescent-shape molecule 4 with multiple annulated pyridine and dihydrobenzene rings, which they coined "the arginine cork" because of its high preference for arginine (Figure 5 b). It prefers diarginine as its guest, and displays K d values in the low micromolar range for molecular recognition in water. The architectural concept is based on a remarkably rigid host skeleton which leads to a large degree of preorganization of hydrogen bond acceptors, reinforced by two strategically placed carboxylates for additional ionic interactions between the dianionic host and the dicationic guest [19]. Problematic features are limited solubility and difficult synthetic access.

Later the Dougherty group constructed a much larger, highly charged cyclophane receptor (5), which is selective for arginine and lysine dipeptides (almost micromolar affinities in aqueous solution, Figure 5 c). The clue is a combination of cation-π interactions with electron-rich receptor π-faces and multiple salt bridges, complemented by total inclusion of one amino acid inside the receptor cavity, leading to solvent exclusion; the second basic residue remains solventexposed. [20] In 1997 Schrader discovered that the m-xylylene bisphosphonate dianion is a powerful guanidinium binder in dipolar aprotic solution (DMSO) and forms a similar arrangement as the well-known Tat-TAR complex between the AIDS virus's RNA fragment (two phosphate groups) and a regulatory protein (arginine, Figure 3 d). [14] This recognition motif prefers delocalized, soft guanidinium cations over hard ammonium ions and especially arginine over lysine residues. Systematic investigations were carried out to study the specific interplay of noncovalent attractive forces with modified model compounds. To this end, additional substituents with varying electronic and steric properties were introduced in the 5-position of the central benzene ring (see selection in Figure 6). It was assumed that three kinds of noncovalent interactions play a key role in guanidinium recognition: electrostatic and hydrogen bond interactions with both phosphonate moieties and cation-π interactions with the receptor's central aromatic ring, which was placed in direct vander-Waals contact by force-field calculations. Direct proof of cation-π interactions was provided with small electronic-withdrawing and donating groups in 5-position of this central benzene ring; these substituents could not form any additional hydrogen bonds to the guanidinium cation because they were geometrically locked in the aromatic plane of the receptor molecule. Intriguingly, all electronic-withdrawing groups lowered and all donating groups raised relative affinities towards methylguanidine hydrochloride, reflecting weakened or strengthened interactions between the guanidinium cation and the aromatic system. A close correlation was found between free binding enthalpies and the electrostatic surface potential of the respective central rings. Dissociation constants could also be lowered by increasing the π -face of the receptor, e. g., by exchanging the central benzene for a naphthalene ring (as in 8).

Even better than all the preceding modifications, however, was the introduction of a third phosphonate anion at a certain distance from the benzylic bisphosphonate (9). Monte Carlo simulations and NOESY experiments confirmed the postulated additional hydrogen bond contacts to all five guanidinium NH protons of the model guanidinium salt, which was now surrounded by all three phosphonate arms (Figure 7). Strong binding and a pronounced selectivity for arginine over lysine and histidine derivatives was now observed in methanol, with K d values in the low micromolar range (~ 30 µM). Monte Carlo simulations in water showed that the long triphosphonate allows the arginine side chain to form a thermodynamically favoured extended conformation while an electrostatic interaction with the αammonium group still occurs. Unfortunately, affinities in aqueous solution remained small. [21] From the new trisphosphonate unit we subsequently developed the first synthetic RGD receptor molecule (Figure 8). Alkylation of the deprotonated benzamide-NH group with phthaloylprotected m-aminobenzyl bromide smoothly yielded 10 after hydrazinolysis and phosphonate ester cleavage with LiBr. Now a simple aspartate binding site is present, while at the same time the rigid benzyl spacer prevents intramolecular dimerization, and the ammonium functionality is a bad guest for the trisphosphonate and should thus circumvent an unwanted intermolecular dimerization. NMR titrations of the new RGD-receptor molecule with the free RGD pepide as well as the integrin antagonist cRGDfV became now possible in water, and furnished K d values in the high micromolar regime. Distinct NMR signal shifts were observed for the aspartate CH 2 protons, testifying to its additional interaction with the RGD host; complex stoichiometries were determined at a precise 1:1 ratio. This showed that the concept of combining an arginine binder with an aspartate binder was valid. [22] Inspired by this work the Gilson group chose our simple ditopic RGD receptor molecules and their complexes with RGD peptides as a biologically relevant model case to study the accuracy of their new program for the calculation of absolute Gibbs energies. Their results were in good agreement with the experimental values that had been independently determined in the Schrader group. For the interaction between 10 and cRGDfV, e. g., the experimental data showed a ∆G value of -3.91 kcal/mol, which was reproduced in silico with -3.11 kcal/mol. A most remarkable feature of the new program is its ability to include solvent and entropic effects in the M2 modeling method. In the course of these calculations, the Gilson group noticed, that in cRGDfVcomplexation by 10, electrostatic interactions play a much smaller role than for the linear RGD peptide, while van-der-Waals and nonpolar terms became dominating. This is in line with intuition, since cRGDfV has two charged groups less than free RGD, but carries two additional nonpolar groups. The M2 calculations also suggested that much of the receptor did not interact directly with the RGD sequence. Finally new receptor molecules were designed in silico and predicted to produce Gibbs energies of up to 5 kcal/mol higher than 10. One of the calculated structures reached predicted nanomolar affinities with the cyclic RGD model peptide, rivalling protein-ligand interactions. The underlying design concept either strived to extend the hydrophobic contact area between host and guest ligand or to reduce entropic or electrostatic desolvation penalties. The only problem with these optimized virtual RGD receptor molecules was that they were too difficult to synthesize [23].

In an independent investigation, the influence of the spacer unit was studied with simple model systems. The benzyl group in 10 was replaced by small peptides, capable of hydrogen bonding to the backbone amides of the RGD sequence. These carried on one end a bisphosphonate, and on the other end an anilinium unit. Interestingly, only short dipeptides turned out to constitute suitable spacers for RGD recognition. Obviously, the distance between arginine and aspartate recognition unit is critical for high affinity, and a better aspartate binder is needed. [24] In 2007 the next generation of artifical RGD receptor molecules was developed. To this end, the Schrader and Schmuck groups joined forces and connected the bisphosphonate motif with a guanidiniocarbonylpyrrol, developed earlier by Schmuck. This acylguanidinium species with lowered basicity (pK ~ 6-7) forms strong H-bond enforced ion pairs to carboxylate anions; these are further strengthened by additional hydrogen bonds from the pyrrole NH and its aromatic carboxamide. In toto, these combined hydrogen bond donors comprise the most potent aspartate and glutamate receptors for aqueous media known to date (Figure 9). Through the combination of ion pairing with multiple ionic hydrogen bonds, these receptor molecules effectively bind to single, isolated amino acid carboxylates with millimolar K d values in aqueous DMSO [25,26]. In order to investigate the influence of the spacer unit connecting the arginine and aspartate receptor units, four different hosts were synthesized with short peptidic linkers of varying length and rigidity (Figure 10). Compounds 11-13 were very well soluble in water (> 1mM), whereas the solubility of compound 14 with an aromatic spacer was still in the high micromolar range. Detailed binding studies were carried out with NMR titrations for hosts 11-13, and UV-vis as well as fluorescence titrations for 14. The pH was fixed at 6.1 (NaH 2 PO 4 /Na 2 HPO 4 buffer), and complexation was monitored and quantitatively evaluated by standard nonlinear regression methods for the free RGD peptide as well as for N/C-protected derivatives.

No binding occurs with receptor 13, weak complexation is found for hosts 11 and 12. As a potential explanation, force-field calculations produce stable structures for the isolated host molecules, with inter-or even intramolecular self-association between their bisphosphonate head and guanidiniocarbonylpyrrol tail. These were in part confirmed by dilution studies. Obviously, the peptidic linkers built into hosts 11-13 are too flexible. On the contrary, no intramolecular complexation is possible in receptor molecule 14 with its rigid m-aminobenzoyl linker. Consequently, UV-vis as well as fluorescence titrations pointed to a significant attraction of RGD peptides by host 14, with the N/C-protected Ac-RGD-NH 2 superior to the free RGD peptide. An attempt to probe the contribution of electrostatic interactions between bisphosphonate anion and guanidinium cation furnished a surprising result: Receptor 14 with methyl esters instead of phosphonate anions produced the tightest complex of all. This is counterintuitive, but may well reflect differences in the solvatation behaviour of both receptors. It is also in good agreement with observations from Gilson et al., who showed earlier that Coulomb interactions between the highly solvated phosphonate anions and arginine's guanidinium cation are of limited importance, since the structure and complex stability depend more on hydrophobic interactions (vide supra). [27] Conclusion and Outlook.

Research directed towards powerful RGD receptor molecules has been difficult, and is still in progress. Several steps forward have been taken, demonstrating the need for strong arginine and aspartate binders, as well as for suitable spacers which prevent self-association of the selfrecognizing complementary head groups and provide the optimal distance for complex formation of the RGD sidechains without introducing any strain. Mere Coulomb attraction seems to be outperformed by nonpolar interactions and hydrophobic effects. Affinities in buffered aqueous solution are still several orders of magnitude lower than those found for the RGD-protein/integrin pairs. In order to bridge this gap, the Schrader and Schmuck groups are currently synthesizing a third receptor generation in which the relatively weak binding and highly solvated bisphosphonate will be replaced by a much more powerful arginine binder which was discovered very recently. This is a concave belt like molecular tweezer with phosphonate tips, which binds N/C protected arginine derivatives with low micromolar affinity in water. [28] Interconnection of the tweezer with the guanidiniocarbonylpyrrole by an appropriate and optimized linker unit will hopefully not only lead to powerful RGD receptors, but at the same time provide some selectivity for different RGD conformations, another prerequisite for a potential application as therapeutic agents.

The best candidates will be tested in bioassays for their in vivo properties. For major therapeutic target integrins (αVβ3, αIIbβ3 and α5β1), Integrins represent the major class of biological receptors which mediate cell-cell and cell-matrix interactions. [1] They are heterodimeric glycoproteins, located inside the cell membrane and present the head portions of both subunits to the extracellular space. Cell recognition takes place when specific extracellular proteins dock onto the cell membrane and recognize a specific integrin type. This recognition process hinges on a small key peptide sequence, presented on a solvent-exposed loop of all approaching extracellular proteins. It contains 3 consecutive critical amino acids: arginine-glycine-aspartate, and is therefore generally coined RGD motif. This tripeptide sequence literally serves as a key which can be inserted into the integrin lock and open it.

Humphries
In 1984 it was discovered, that the RGD sequence mediates cell adhesion by Fibronectin; [2] subsequent findings quickly demonstrated that the RGD sequence is a general powerful recognition motif utilized by many other proteins like vitronectin, laminin or the von willebrand factor, which all play a pivotal role in cell adhesion biology. Fundamental biological processes such as cell differentiation, signaling, apoptosis and others depend on the interplay between RGD-containing proteins and their related integrin-receptors. [3,4,5] The integrin family is highly variable because its members differ in their composition through the combination of eight different αand β-subunits. A single integrin is able to discriminate between the RGD-sequences of different proteins by two main mechanisms; it recognizes not only the specific chemical nature of these three amino acids, but also the specific conformation of each of these individual RGDloops. The relative orientation of the alkylguanidinium and the alkylcarboxylate side chains can adopt all transitions from a linear, fully extended conformation up to a tightly folded cyclic loop.

In some cases, the integrin also interacts with the fourth amino acid residue, e. g., in the RGDS sequence typical for fibronectin. Dysfunction in this critical recognition system can cause pathogenic effects like tumor-induced angiogenesis, osteoporosis, myocardial infarction or thrombosis. [6,7,8] The resulting severe diseases reveal the medicinal and pharmaceutical importance of the RGD recognition system and render it an attractive drug target. A precise control over pathologic RGD recognition events by externally added small molecules would not only help to better understand this important natural signalling pathway, but also have immediate therapeutic relevance. In principle, inhibition of the RGD-integrin interaction can be reached in two different ways. One has been pursued intensively by pharmaceutical research; the other -more difficult one -has been totally neglected. A great deal of synthetic, computational and biological effort has been devoted to develop so-called RGD Mimetics which imitate polar groups and their relative orientations in a specific RGD-sequence, in order to compete with the native ligands of such integrins (Figure 1a and 1b). Numerous academic and industrial groups have developed peptide-derived or non-peptidic RGD mimetics, most of which showed some biological activity. Most notably, an elegant approach was introduced by Kessler et al., who synthesized a series of RGD-containing cyclopeptides with 5and 6-membered rings containing single D-amino acid residues. In extended NMR-spectroscopic studies the structural principles governing the formation of preferred fixed conformations were unraveled and used for a conformational design of integrin-specific RGD mimetics, which showed promising in vitro biological effects. During the course of these studies, a whole set of cyclic hexapeptides and pentapeptides was synthesized and thermodynamic equilibria as well as dynamics of their three-dimensional structures in solution were analyzed by 2D-NMR techniques.

Finally, systematic tuning of conformational restrictions accompanied by MD simulations and subsequent bioassays culminated in the preparation of the strong lead compound cyclo-RGDf[NMe]V (Figure 2), a powerful integrin antagonist, which effectively inhibits tumorassociated angiogenesis. [9,10,11] This very small cyclopeptide, which is currently undergoing clinical trials, imitates the RGD-loop in Vitronectin. In 2002 Arnaout et al. gained a crystal structure of this cyclopeptide in its complex with integrin αVß3, the only crystal structure of an RGD-integrin complex which is available today. This structure reveals that the peptide is inserted in a cleft between the so called ß-propeller from the αV-unit and ßA domains from the ß3-unit on the integrin head. As expected, almost all peptide-protein contacts are established by the RGD sequence (Figure 1c). Figure 2 displays some related RGD mimetics, some of which have already been approved as drugs. Eptifibatide, a somewhat more flexible cyclohexapeptide with an internal disulfide bond, inhibits integrin a IIb ß 3 . It was developed entirely by pharmaceutical industry, and is used against myocardial infarction or in heart surgery. Other potent drugs are drastically simplified and often rigidified RGD mimetics which contain only one stereogenic center and replace guanidinium ions by (hydroxy)amidines (Figure 2).

Artificial RGD Receptor Molecules.

A completely different approach to block pathologic RGD-Integrin interactions is to construct integrin mimetics which recognize RGD-containing proteins. In principle this simplifies the task to developing a small artificial host molecule which can serve as a competitive "lock" for the RGD-sequence in native proteins. Due to the encapsulated RGD-"key" the whole extracellular protein is not able any more to bind to its respective target integrin (Figure 3a). However, since the RGD-sequence exists in a variety of different conformations, depending on the parent protein, it is mandatory to provide synthetic hosts which can distinguish between them (Figure 3 b). Some years ago, the Schrader and Schmuck groups embarked on a program to pursue this alternative approach towards conformation-specific RGD host molecules. The starting point was marked by the discovery that m-xylylene bisphosphonates 1 bind guanidinium cations in a chelate-type manner reminiscent to the natural diaspartate binding site found in the crystal structure of the cRGDf[NMe]V/αVß3 complex. In both cases a bidentate salt bridge with a network of ion-pair reinforced hydrogen bonds firmly embraces arginine's guanidinium cation and holds it in place (Figure 3 c,d). [13] An early screening experiment for RGD-binders was carried out by Ruoslahti et al. In 1999 a patent was filed which announced the discovery that cyclic peptides with the consensus sequence (W/P)DD(G/L)(W/L)(W/L/M) bind to fibronectin and related RGD-containing proteins. The best candidate was cyclo-CWDDGWLC 2, with an internal disulfide bond, which is depicted in Figure 4 a. It was isolated from a library of short cyclic peptides and peptidomimetics in the course of specific assays for RGD-binding activity. The peptides were presented in form of a phage library which was used to isolate minimal receptor sequences that bind to fibronectin in affinity panning. Thus, the cyclic octapeptide c-CWDDGWLC was identified as the most potent inhibitor for RGD dependent cell attachment to fibronectin and vitronectin. Fibronectin itself tightly bound to a c-CWDDGWLC affinity column and could subsequently be reeluted with an RGD-containing peptide. Interestingly, a specific region within the ß-subunit of RGD-binding integrins had previously been shown by site-directed mutagenesis to be involved in ligand binding [14], which includes a the short peptide stretch KDDLW -quite similar to the abovedescribed sequence. Short synthetic peptides correspondening to this integrin region bind to the RGD containing protein, but their affinities were markedly reduced if both aspartate residues are mutated to alanines. In summary, the combination of phage display technology, rational design and mutagenesis has lead to structural integrin mimetics which can significantly reduce the cell attachment of an RGD-containing protein [15,16]. However, these are still sensitive to proteolytic degradation. In 2009, Liakopoulou-Kyriakides et al. tried to overcome this drawback by employing the Molecular Imprinting technology (Figure 4 b) in order to generate a new chromatography stationary phase for the separation of RGD-peptides from mixtures. They used the free RGD peptide as template molecule, and prepared molecularly imprinted polymers (MIPs) from only two functional monomers, i. e., methacrylic acid (MAA) and acrylamide (AA). Three different crosslinking monomers were tried, with trimethylpropane trimethacrylate producing the best results in terms of RGD-selectivity. MAA was later postulated to form interactions with arginine, while AA's amide is both a moderate hydrogen bond donor and acceptor. To establish RGD selectivity, four other peptides where used in rebinding experiments, namely KGD and three other completely different peptides. Selectivity factor values ranged from 1.27 up to 1.98, net rebinding values for RGD and KGD were similar (~ 8-10%), while those for other peptides remained very small (2-4 %). These data indicate low affinity and only moderate selectivity for the template molecule. In addition, the free RGD peptide was examined in methanol solution, where K d values were in the high micromolar range. [17] Surprisingly, to the best of our knowledge, apart from our own project, these are the only artificial RGD receptor molecules published to date. We tentatively explain this with a lack of powerful concepts for the generation of water-soluble host molecules for zwitterions in physiological solution. In the aqueous medium, hydrogen bonds must be reinforced with

F o r P e e r R e v i e w O n l y 7 H H H N H N NH 2 NH 2 H 2 N NH 2 N N N H 2 N NH 2 N O O O O H H H H H H N N N N O O O O N N N H H H H H CO 2 O 2 C O O CO 2 O 2 C O O 2 C CO 2 O CO 2 O 2 C N H H 2 N NH 2 H 3 N O O
electrostatic interactions and complemented by hydrophobic forces to achieve high affinities.

Zwitterions are prone to undergo intramolecular as well as intermolecular self-association, which often renders the desired host-guest complexes insoluble in buffered aqueous solutionespecially in the presence of high salt loads.

For isolated arginine and aspartate as well as for model peptides containing only anionic or cationic residues, some powerful receptor molecules have been developed: Hamilton et al. created a rigid concave scaffold 3 with two preorganized guanidinium moieties (Figure 5 a) which specifically bind two aspartate carboxylates in the i+3-relation of an α-helix. Small aspartate-containing model peptides are bound with millimolar affinities, albeit only in a 10% water-methanol mixture. [18] Bell et al. synthesized crescent-shape molecule 4 with multiple annulated pyridine and dihydrobenzene rings, which they coined "the arginine cork" because of its high preference for arginine (Figure 5 b). It prefers diarginine as its guest, and displays K d values in the low micromolar range for molecular recognition in water. The architectural concept is based on a remarkably rigid host skeleton which leads to a large degree of preorganization of hydrogen bond acceptors, reinforced by two strategically placed carboxylates for additional ionic interactions between the dianionic host and the dicationic guest [19]. Problematic features are limited solubility and difficult synthetic access.

Later the Dougherty group constructed a much larger, highly charged cyclophane receptor (5), which is selective for arginine and lysine dipeptides (almost micromolar affinities in aqueous solution, Figure 5 c). The clue is a combination of cation-π interactions with electron-rich receptor π-faces and multiple salt bridges, complemented by total inclusion of one amino acid inside the receptor cavity, leading to solvent exclusion; the second basic residue remains solventexposed. [20] In 1997 Schrader discovered that the m-xylylene bisphosphonate dianion is a powerful guanidinium binder in dipolar aprotic solution (DMSO) and forms a similar arrangement as the well-known Tat-TAR complex between the AIDS virus's RNA fragment (two phosphate groups) and a regulatory protein (arginine, Figure 3 d). [14] This recognition motif prefers delocalized, soft guanidinium cations over hard ammonium ions and especially arginine over lysine residues. Systematic investigations were carried out to study the specific interplay of noncovalent attractive forces with modified model compounds. To this end, additional substituents with varying electronic and steric properties were introduced in the 5-position of the central benzene ring (see selection in Figure 6). It was assumed that three kinds of noncovalent interactions play a key role in guanidinium recognition: electrostatic and hydrogen bond interactions with both phosphonate moieties and cation-π interactions with the receptor's central aromatic ring, which was placed in direct vander-Waals contact by force-field calculations. Direct proof of cation-π interactions was provided with small electronic-withdrawing and donating groups in 5-position of this central benzene ring; these substituents could not form any additional hydrogen bonds to the guanidinium cation because they were geometrically locked in the aromatic plane of the receptor molecule. Intriguingly, all electronic-withdrawing groups lowered and all donating groups raised relative affinities towards methylguanidine hydrochloride, reflecting weakened or strengthened interactions between the guanidinium cation and the aromatic system. A close correlation was found between free binding enthalpies and the electrostatic surface potential of the respective central rings. Dissociation constants could also be lowered by increasing the π -face of the receptor, e. g., by exchanging the central benzene for a naphthalene ring (as in 8).

Even better than all the preceding modifications, however, was the introduction of a third phosphonate anion at a certain distance from the benzylic bisphosphonate (9). Monte Carlo simulations and NOESY experiments confirmed the postulated additional hydrogen bond contacts to all five guanidinium NH protons of the model guanidinium salt, which was now surrounded by all three phosphonate arms (Figure 7). Strong binding and a pronounced selectivity for arginine over lysine and histidine derivatives was now observed in methanol, with K d values in the low micromolar range (~ 30 µM). Monte Carlo simulations in water showed that the long triphosphonate allows the arginine side chain to form a thermodynamically favoured extended conformation while an electrostatic interaction with the αammonium group still occurs. Unfortunately, affinities in aqueous solution remained small. [21] From the new trisphosphonate unit we subsequently developed the first synthetic RGD receptor molecule (Figure 8). Alkylation of the deprotonated benzamide-NH group with phthaloylprotected m-aminobenzyl bromide smoothly yielded 10 after hydrazinolysis and phosphonate ester cleavage with LiBr. Now a simple aspartate binding site is present, while at the same time the rigid benzyl spacer prevents intramolecular dimerization, and the ammonium functionality is a bad guest for the trisphosphonate and should thus circumvent an unwanted intermolecular dimerization. micromolar regime. Distinct NMR signal shifts were observed for the aspartate CH 2 protons, testifying to its additional interaction with the RGD host; complex stoichiometries were determined at a precise 1:1 ratio. This showed that the concept of combining an arginine binder with an aspartate binder was valid. [22] Inspired by this work the Gilson group chose our simple ditopic RGD receptor molecules and their complexes with RGD peptides as a biologically relevant model case to study the accuracy of their new program for the calculation of absolute Gibbs energies. Their results were in good agreement with the experimental values that had been independently determined in the Schrader group. For the interaction between 10 and cRGDfV, e. g., the experimental data showed a ∆G value of -3.91 kcal/mol, which was reproduced in silico with -3.11 kcal/mol. A most remarkable feature of the new program is its ability to include solvent and entropic effects in the M2 modeling method. In the course of these calculations, the Gilson group noticed, that in cRGDfVcomplexation by 10, electrostatic interactions play a much smaller role than for the linear RGD peptide, while van-der-Waals and nonpolar terms became dominating. This is in line with intuition, since cRGDfV has two charged groups less than free RGD, but carries two additional nonpolar groups. The M2 calculations also suggested that much of the receptor did not interact directly with the RGD sequence. Finally new receptor molecules were designed in silico and predicted to produce Gibbs energies of up to 5 kcal/mol higher than 10. One of the calculated structures reached predicted nanomolar affinities with the cyclic RGD model peptide, rivalling protein-ligand interactions. The underlying design concept either strived to extend the hydrophobic contact area between host and guest ligand or to reduce entropic or electrostatic desolvation penalties. The only problem with these optimized virtual RGD receptor molecules was that they were too difficult to synthesize [23].

In an independent investigation, the influence of the spacer unit was studied with simple model systems. The benzyl group in 10 was replaced by small peptides, capable of hydrogen bonding to the backbone amides of the RGD sequence. These carried on one end a bisphosphonate, and on the other end an anilinium unit. Interestingly, only short dipeptides turned out to constitute suitable spacers for RGD recognition. Obviously, the distance between arginine and aspartate recognition unit is critical for high affinity, and a better aspartate binder is needed. [24] In 2007 the next generation of artifical RGD receptor molecules was developed. To this end, the Schrader and Schmuck groups joined forces and connected the bisphosphonate motif with a guanidiniocarbonylpyrrol, developed earlier by Schmuck. This acylguanidinium species with lowered basicity (pK ~ 6-7) forms strong H-bond enforced ion pairs to carboxylate anions; these are further strengthened by additional hydrogen bonds from the pyrrole NH and its aromatic carboxamide. In toto, these combined hydrogen bond donors comprise the most potent aspartate and glutamate receptors for aqueous media known to date (Figure 9). Through the combination of ion pairing with multiple ionic hydrogen bonds, these receptor molecules effectively bind to single, isolated amino acid carboxylates with millimolar K d values in aqueous DMSO [25,26]. In order to investigate the influence of the spacer unit connecting the arginine and aspartate receptor units, four different hosts were synthesized with short peptidic linkers of varying length and rigidity (Figure 10). Compounds 11-13 were very well soluble in water (> 1mM), whereas the solubility of compound 14 with an aromatic spacer was still in the high micromolar range. Detailed binding studies were carried out with NMR titrations for hosts 11-13, and UV-vis as well as fluorescence titrations for 14. The pH was fixed at 6.1 (NaH 2 PO 4 /Na 2 HPO 4 buffer), and complexation was monitored and quantitatively evaluated by standard nonlinear regression methods for the free RGD peptide as well as for N/C-protected derivatives.

No binding occurs with receptor 13, weak complexation is found for hosts 11 and 12. As a potential explanation, force-field calculations produce stable structures for the isolated host molecules, with inter-or even intramolecular self-association between their bisphosphonate head and guanidiniocarbonylpyrrol tail. These were in part confirmed by dilution studies. Obviously, the peptidic linkers built into hosts 11-13 are too flexible. On the contrary, no intramolecular complexation is possible in receptor molecule 14 with its rigid m-aminobenzoyl linker. Consequently, UV-vis as well as fluorescence titrations pointed to a significant attraction of RGD peptides by host 14, with the N/C-protected Ac-RGD-NH 2 superior to the free RGD peptide. An attempt to probe the contribution of electrostatic interactions between bisphosphonate anion and guanidinium cation furnished a surprising result: Receptor 14 with methyl esters instead of phosphonate anions produced the tightest complex of all. This is counterintuitive, but may well reflect differences in the solvatation behaviour of both receptors. It is also in good agreement with observations from Gilson et al., who showed earlier that Coulomb interactions between the highly solvated phosphonate anions and arginine's guanidinium cation are of limited importance, since the structure and complex stability depend more on hydrophobic interactions (vide supra). [27] Conclusion and Outlook.

Research directed towards powerful RGD receptor molecules has been difficult, and is still in progress. Several steps forward have been taken, demonstrating the need for strong arginine and aspartate binders, as well as for suitable spacers which prevent self-association of the selfrecognizing complementary head groups and provide the optimal distance for complex formation of the RGD sidechains without introducing any strain. Mere Coulomb attraction seems to be outperformed by nonpolar interactions and hydrophobic effects. Affinities in buffered aqueous solution are still several orders of magnitude lower than those found for the RGD-protein/integrin pairs. In order to bridge this gap, the Schrader and Schmuck groups are currently synthesizing a third receptor generation in which the relatively weak binding and highly solvated bisphosphonate will be replaced by a much more powerful arginine binder which was discovered very recently. This is a concave belt like molecular tweezer with phosphonate tips, which binds N/C protected arginine derivatives with low micromolar affinity in water. [28] Interconnection of the tweezer with the guanidiniocarbonylpyrrole by an appropriate and optimized linker unit will hopefully not only lead to powerful RGD receptors, but at the same time provide some selectivity for different RGD conformations, another prerequisite for a potential application as therapeutic agents.

The best candidates will be tested in bioassays for their in vivo properties. For major therapeutic target integrins (αVβ3, αIIbβ3 and α5β1), we have now revised the paper according to the reviewers' comments and highlighted all respective changes in yellow.

Figure 1 .

 1 Figure1. a) Schematic of the integrin/RGD "lock and key" principle: An RGD-containing protein (key) docks onto the membrane-spanning α,β-integrin dimer (lock) and induces a biological effect. b) Conventional interference priciple: the ligand binding site on the Integrin is blocked by an RGD mimetic (grey); c) Crystal structure of the cRGDf[NMe]V peptide (grey) in its complex with the α,β-integrin dimer αVß3 (blue/red).

Figure 2 .

 2 Figure 2. Synthetic RGD-mimetics: cRGDf[NMe]V (Cilengitide, Merck KGaA, specific for αVß3-Integrin), Eptifibatide (Integrilin ® , Millennium Pharmaceuticals/GlaxoSmithKline/ Schering-Plough, specific for a IIb ß 3 -Integrin); Sibrafiban (Hoffmann-LaRoche, specific for a IIb ß 3 -Integrin); SB-265123 (GlaxoSmithKline, specific for αVß3-Integrin).

1 Figure 3 .

 13 Figure 3. a) Alternative concept for RGD/integrin interruption: RGD-capping by an artificial RGD receptor molecule (grey) prevents docking onto the membrane-bound integrin (blue/red); b) Folded and extended conformation found in the RGD-loops of Fibronectin and Vitronectin (from crystal structures); c) chelate-type binding motif from the crystal structure of the complex between cRGDf[NMe]V and αVß3; d) chelate-type binding motif in a m-xylylene bisphosphonate/arginine complex.

2 Figure 4 .

 24 Figure 4. a) Lead candidate cyclo-CWDDGWLC, identified from phage-display assays as a general RGD-protein binder b) Attempted Molecular Imprinting for the free RGD template peptide: proposed noncovalent interactions between functional monomer, crosslinker and template.

3 4 5 Figure 5 .

 55 Figure 5. a) Preorganized host molecules for aspartate and arginine: Hamilton's concave i+3-diaspartate binder 3; b) Bell's planar "arginine cork" 4; c) Dougherty's arginine-cavitand 5 with putative arginine inclusion.

6 7 8 Figure 6 .

 86 Figure 6. m-Xylylene bisphosphonate host molecules 6-8 with additional substituents in the 5-positionprobing the π-cation interaction. Below: Electrostatic Surface Potentials (ESP) of nitrobenzene, anisole and benzene (AM1-calculations).

9 Figure 7 .

 97 Figure 7. Summarized reciprocal NOE contacts measured between the new arginine host molecule 9 and methylguanidinium cation (500 MHz, 1:1 complex in CD 3 OD). Right: Monte Carlo simulation (MacroModel 7.0, water/GBA, Amber*, 1000 steps).

10

 10

Figure 8 .

 8 Figure 8. Primitive synthetic RGD-receptor molecule 10 with restricted rotational freedom. Left: Lewis structure; Right: molecular mechanics calculation of its 1:1 complex with cyclo(RGDfV) (MacroModel 7.0, water (GBA), Amber*, 1000 steps).

2 Figure 9 .

 29 Figure 9. General structure of the guanidiniocarbonylpyrrole recognition motif and the network of ionic hydrogen bonds formed on complexation of carboxylate ions. Right: CPK model showing the calculated complex structure of Ac-RGD-NH 2 peptide (pink) bound by superior receptor molecule 14 with a guanidiniocarbonylpyrrole skeleton (green).

  et al. developed (a) solid-phase proteinprotein assays and (b) cell-based adhesion assays, which allow determination of IC 50 values for cell attachment inhibition.[29,30] One day, such artificial receptors may serve as lead structures for the development of new therapeutic drugs for the treatment of diseases such as arthritis, heart attack and stroke, viral infections or cancer.Thomas Gersthagen, Carsten Schmuck, Thomas Schrader* TOC-Graphic.Abstract. Integrins play a pivotal role in cell-cell adhesion, signaling and apoptosis. Many extracellular proteins use the RGD sequence (arginine-glycine-aspartate) as a key to dock onto and unlock their respective binding partners at the cell membrane (α V β 3 -, a IIb ß 3 -and the a 5 ß 1integrin). Here, the RGD signal is transduced into the cytoplasm and triggers a variety of biological events like blood coagulation, cell matrix binding, cell differentiation and angiogenesis. A misfunction of this recognition system causes severe diseases, rendering the RGD recognition system an attractive drug target. Inhibition of RGD-integrin interactions can be reached in two different ways, by blocking integrins with RGD-mimetics or by capping RGDcontaining proteins by artificial RGD receptors. This review provides an overview over the very young history of artificial RGD receptor development, beginning with early research in arginine recognition, over the discovery of the first primitive RGD-receptor until the current state of research and future prospects.Key words: RGD peptide, integrins, molecular recognition, artificial receptors, amino acids Introduction.

Figure 1 .

 1 Figure 1. a) Schematic of the integrin/RGD "lock and key" principle: An RGD-containing protein (key) docks onto the membrane-spanning α,β-integrin dimer (lock) and induces a biological effect. b) Conventional interference priciple: the ligand binding site on the Integrin is blocked by an RGD mimetic; c) Crystal structure of the cRGDf[NMe]V peptide in its complex with the α,β-integrin dimer αVß3.

Figure 2 .

 2 Figure 2. Synthetic RGD-mimetics: cRGDf[NMe]V (Cilengitide, Merck KGaA, specific for αVß3-Integrin), Eptifibatide (Integrilin ® , Millennium Pharmaceuticals/GlaxoSmithKline/ Schering-Plough, specific for a IIb ß 3 -Integrin); Sibrafiban (Hoffmann-LaRoche, specific for a IIb ß 3 -Integrin); SB-265123 (GlaxoSmithKline, specific for αVß3-Integrin).

1 Figure 3 .

 13 Figure 3. a) Alternative concept for RGD/integrin interruption: RGD-capping by an artificial RGD receptor molecule (grey) prevents docking onto the membrane-bound integrin (blue/red); b) Folded and extended conformation found in the RGD-loops of Fibronectin and Vitronectin (from crystal structures); c) chelate-type binding motif from the crystal structure of the complex between cRGDf[NMe]V and αVß3; d) chelate-type binding motif in a m-xylylene bisphosphonate/arginine complex.

2 Figure 4 .

 24 Figure 4. a) Lead candidate cyclo-CWDDGWLC, identified from phage-display assays as a general RGD-protein binder b) Attempted Molecular Imprinting for the free RGD template peptide: proposed noncovalent interactions between functional monomer, crosslinker and template.

3 4 5 Figure 5 .

 55 Figure 5. a) Preorganized host molecules for aspartate and arginine: Hamilton's concave i+3-diaspartate binder 3; b) Bell's planar "arginine cork" 4; c) Dougherty's arginine-cavitand 5 with putative arginine inclusion.

6 7 8 Figure 6 .

 86 Figure 6. m-Xylylene bisphosphonate host molecules 6-8 with additional substituents in the 5-positionprobing the π-cation interaction. Below: Electrostatic Surface Potentials (ESP) of nitrobenzene, anisole and benzene (AM1-calculations).

9 Figure 7 .

 97 Figure 7. Summarized reciprocal NOE contacts measured between the new arginine host molecule 9 and methylguanidinium cation (500 MHz, 1:1 complex in CD 3 OD). Right: Monte Carlo simulation (MacroModel 7.0, water/GBA, Amber*, 1000 steps).

10 Figure 8 .

 108 Figure 8. Primitive synthetic RGD-receptor molecule 10 with restricted rotational freedom. Left: Lewis structure; Right: molecular mechanics calculation of its 1:1 complex with cyclo(RGDfV) (MacroModel 7.0, water (GBA), Amber*, 1000 steps).

2 Figure 9 .

 29 Figure 9. General structure of the guanidiniocarbonylpyrrole recognition motif and the network of ionic hydrogen bonds formed on complexation of carboxylate ions. Right: CPK model showing the calculated complex structure of Ac-RGD-NH 2 peptide (pink) bound by superior receptor molecule 14 with a guanidiniocarbonylpyrrole skeleton (green).

Figure 10 .

 10 Figure 10. New generation of RGD receptor molecules 11-14 with bisphosphonate and guanidiniocarbonylpyrrol moieties interconnected by various spacers.

Page 1

 1 figure 1a), b) colour

  Humphries et al. developed (a) solid-phase proteinprotein assays and (b) cell-based adhesion assays, which allow determination of IC 50 values for cell attachment inhibition.[29,30] One day, such artificial receptors may serve as lead structures for the development of new therapeutic drugs for the treatment of diseases such as arthritis, heart attack and stroke, viral infections or cancer.

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

URL: http:/mc.manuscriptcentral.com/tandf/gsch Email: suprachem@mail.cm.utexas.edu

URL: http:/mc.manuscriptcentral.com/tandf/gsch Email: suprachem@mail.cm.utexas.edu Supramolecular Chemistry

Acknowledgements. Funding of both the Schrader and Schmuck groups by the DFG (Deutsche Forschungsgemeinschaft) is gratefully acknowledged.

Acknowledgements. Funding of both the Schrader and Schmuck groups by the DFG (Deutsche Forschungsgemeinschaft) is gratefully acknowledged.