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Enzymes and Curvature.

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teraction of a ring boron atom with an olefinic C-C bond
was first presumed in 1964 on the basis of IR data."]
Crystals of 3[8h.c1
(triclinic, Pi) were obtained on allowing the liquid compound to stand for several days in a refrigerator. Data collections at 303 K for two crystals, one
of which had been annealed just below the melting point
(308 K) for two weeks, furnished comparable structural
data, as did also further measurements at 260 K, 180 K and
100 K. The vibration amplitudes of the anisotropically refined atomic parameters increase almost linearly from 100
to 303 K. Thus, in the case of 3, disorders can be ruled out
even just below the melting point. There were also no indications of a phase transition in the crystal in the polymorphic structure of
(monoclinic, P2,/n), cylindrical crystals of which were drawn from the melt between 301 K and
317 K in a Lindemann capillary (diameter 0.3 mm) using
the Bridgeman technique."'] Also several crystals of 2 prepared in this way gave comparable structural data. Data
collections and refinements at 298 K, 288 K and 248 K
gave, especially for the atoms CI1, C12, C13, C4 and C5,
anisotropic vibration amplitudes with highly disordered
structures at the measurement temperatures. The direction
of the disorders follows from the illustration of 2 in
Scheme 1. As to whether dynamic or static disorders are
involved cannot be decided, but the electronic patterns in
2 are consistent with dynamic effects. The disorders were
not resolvable, as also indicated, inter alia, by electron
density maps. Consequently the atomic positions could not
be determined accurately, and the R values and standard
deviations are correspondingly high. Determination of the
structure of 2 at T<248 K was not possible because of
broadening of the reflections.
MaK, radiation, R=0.048; maximum residual electron density
O.2OeA-'. Selected bond lengths [A] and angles ["I: N1-82 1.668(3), C5N1 1.282(3), C4-C5 1.482(3), C3-C4 1.535(4), B2-C3 1.642(4), C4-CI4
1.571(4), Cf4-Cf5 2.547(3), C15-B2 1.646(4), B2-C6 1.595(4), C3-CIl
1.528(3), C l l - C I 2 1.528(4), C12-Cl3 1.525(4), C13-Cl4 1.533(3); N1-B2C3 94.0(2), NI-B2-C15 98.6(2), C15-B2-C3 101.7(2), B2-Cl5-CI4
103.0(2), C4-Cl4-Cl5 103.3(2), C13-CI4-Cl5 I12.0(2), Nl-C5-C4
112.2(2), CS-C4-C3 101.9(2), C4-C3-B2 93.3(2), B2-Nl-CS 105.9(2).
c) Further details of the crystal structure investigation are available on
request from the Fachinformationszentrum Energie, Physik, Mathernatik
GmbH, D-7514 Eggenstein-Leopoldshafen 2 (FRG) on quoting the depository number CSD-52789, the names of the authors, and the journal
[9] 2 : cell data (288 K): a=6.493(3), b = 18.035(8), c=11.638(5)
p = 101.61(3)0,P2,/n, Z = 4 ; P250 independent reflections, 983 observed
(F0>4u(F)), w data collection, 28,4x=45:, Mo,, radiation, Rz0.135:
maximum residual electron density 0.58 e A - ' [Sc].
[lo] D. Brodalla, D. Mootz, R. Boese, W. Osswald, J. Appl. Crysrallogr. 18
(1985) 316.
Enzymes and Curvature
By Zoltan Blum,* Suen Lidin, and Sten Andersson
Numerous proteins and enzyme structures consist of two
building units-the a-helix and the 0-sheet. Many of these
structures seem to belong to two spatial arrangements of
the P-sheets-the barrel and the twisted sheet arrangement.
Recently Louie and Somorjai"] have shown that the barrel,
i.e., its inside, is catenoid-like and the twisted sheet is helicoid-like, and that such surfaces are adjoint, which means
one can be transformed into the other isometrically (Bonnet transformation, Fig. 1). Figure 2 schematically shows a
Received: October 20, 1987 [Z 2481 IE]
German version: Angew. Chem. 100 (1988) 956
Publication delayed at authors' request
CAS Registry numbers:
1, 114738-68.0: 3, 114738-69-1: Br(CH2),CH=CH2, 1119.51-3; 2,SdihydroI H-l,2-azaborolene (lithium salt), 84356-32-1.
[I] G. Schmid, D. Kampmann, U. Hohner, R. Boese, Chem. Eer. 117(1984)
121 G. Schmid, Comments Inorg. Chem. 4 (1985) 17.
131 G. Schmid, D. Zaika, R. Boese, Angew. G e m . 97 (1985) 581; Angew.
Chem. Inr. Ed. Engl. 24 (1985) 602.
[4] G. Schmid, 0. Boltsch, R. Boese, Urganomeral~ics6 (1987) 435.
[5] G. Schmid, D. Zaika, J. Lehr, N. Augart, R. Boese, Chem. Ber., in
161 1: Synthesis by dropwise treatment of a solution of 16.3 g (110.1 mmol)
of the lithium salt in 150mL of THF with 17.0g (110.1 mmol) of
Br(CH2)3CH=CH2at -78°C. After warming to room temperature and
removal of the solvent under vacuum all products volatile up to 200°C
torr were condensed in a trap which was cooled with liquid
nitrogen. On fractional distillation 13.4 g (65%) of 1 were collected as a
water-clear liquid at 126-127"C, (20 torr); correct elemental analysis;
'H-NMR ([D8]toluene): 6=2.38 (br, 1 H; H3), 5.43
MS: m / z 205 (M");
(dd, 'J=4, 4 J = 2 Hz, 1 H; H4), 6.53 (dd, '5=4, 4 J = 2 Hz, 1 H; H5), 0.72
(s, 3H: B-CHI), 1.19 (5, 9 H ; C(CH3),), 1.45, 1.78, 2.10 (each rn, each
2 H ; 3 C H 2 ) , 5 . 8 2 ( m , 1 H ; H 1 4 ) , 5 . 0 1 (dd,m,'J,,,=8.85,3J,,a,,=19.5Hz,
2 H ; H15): "B-NMR ([Dsltoluene): 6=50.2; "C('HJ-NMR ([D,]toluene,
373 K): 6=41.8 (br; C3), 114.1 (C4), 139.1 (C5), 53.6 (C(CH,),), 31.5
(C(CH,),), 1.53 (br: 8-CH,), 28.3, 29.1, 34.4 (3 CH2), 136.7 (C14), 115.8
[7] R. Koster, h o g . Boron Chem. I(1964) 314.
181 a) 3: 'H-NMR ([Ds]toluene): 6=2.38 (br, 1 H; H3), 2.00 (m, 1 H ; H4),
7.09(d,4J=7 Hz, 1 H; H5),0.55(~,3H;H6),0.96(~,9H;H8-H10), 1.46,
1.65, 1.88 (each m, each 2 H ; Hll-H13), 0.03 (dd, I H; H14), 1.98 (m,
2 H ; H15); "B-NMR ([Dsltoluene): 6=4.6; "C('H}-NMR ([Ds]toluene,
298 K): 6=46.7 (br; C3), 55.7 (C4), 171.2 (C5). 59.8 (C7). 28.9 (CS-ClO),
2.6 (br: C6), 18.5, 26.9, 32.4 (Cll-C13), 35.3 (C14), 22.6 (br; Cl5)
b) Cell data (100K): a=6.065(3), b=7.064(3), c = 14.960(10)A,
a=99.27(5), 8=94.82(5), y=97.01(4)', Pi, 2 = 2 ; 1644 independent reflections, 1266 observed (F0>4u(F)), w data collection, 28,,,=45°,
Angew. Chem. Int. Ed. Engl. 27 (1988) No. 7
Fig. 1. The Bonnet transformation
[*] Dr. Z. Blurn, S. Lidin, Prof. Dr. S. Andersson
Chemical Center, University of Lund
Inorganic Chemistry 2
Box 124, S-22100 Lund (Sweden)
0 VCH Verlagsgesellschafr mbH, 0-6940 Weinheim, 1988
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We have explained the adsorptive properties of zeolites
in terms of curvature [Eq. (l)]:
A H = const, K,, N
Fig. 2. a-Carbon representation of triose phosphate isomerase fitted to a
catenoid; a) side view; b) view down the central “hole.”
barrel protein with the catenoid fitted to the structure. The
complete topology of the molecule can be approximated
by a torus which has negative Gaussian curvature on the
inside and positive o n the outside. Louie and Somorjai also
showed that the Bonnet transformation can be used to explain leading features of the much debated protein folding
problem. They further propose that the Stieltjes integral,
[ FdA, describes the enzyme-substrate interaction. A is
given by the equation A = I + i ZZ, where I and I I are the
first and second fundamental forms of equations of surfaces describing the enzyme structure, and F is a function
describing the shape of the substrate. The Stieltjes integral
is complex ; therefore, projections onto the real plane are
needed and they claim this is a way to understand catalysis, recognition, etc.
We find the propositions of Louie and Somorjai of extreme interest. Our approach, however, is different (see below) and also of direct use. It is allied in some ways to the
DTE (dynamic transduction of energy) theory”] and, in
fact, it seems to unify the static geometrical theory of Louie
and Somorjai with the dynamic DTE theory.
In a number of articles we have demonstrated the use of
differential geometry and especially minimal surfaces in
chemistry.13-’] It has also been shown that solids have periodic equi- o r zero-potential surfaces, which are identical,
or nearly so, with periodic minimal surfaces.[81In two recent articles we have explained the adsorption properties
of two different zeolites in terms of their integral curvat~re.[~.’*I
The peculiar shape of the curves of differential
heat versus initial adsorbed amount was explained by assuming a quasi-liquid state of molecules moving in the
electrostatic field of the zeolite structure.”’] The initial heat
developed at very low charge is often extremely high
(Fig. 3).
ads amount
Fig. 3. Schematic representation of heat of adsorption versus loading on a
0 VCH VerlagsgeseNschafr mbH, 0-6940 Weinheim. 1988
K,, is the integral Gaussian curvature per surface area, A H
the heat of adsorption, and N an interaction parameter.
The first few molecules feel strong force fields focussed
(van der Waals lens) via saddle rings of high intrinsic curvature, and high intrinsic heat is developed. This is obviously one of the reasons why zeolites are catalysts; molecules transform or crack even at room temperature owing
to the intrinsic heat developed during their interaction
with the zeolite surface structure of high curvature.
A unifying description of proteintand enzyme structures,
giving them negative Gaussian curvature, provides a simple explanation of their role in biosystems. We have estimated their negative average integral curvature to be 0.010.03 AP2. Therefore, according to Equation (I), intrinsic
heats of adsorption high enough to start various reactions
are easily developed.
The following description of protein action thus
emerges ; the substrate, conveniently viewed as a freely
moving entity, reaches the vicinity of the protein. The latter can be either freely moving or bonded to some kind of
biological matrix or “contrivance,” e.g., a membrane. At a
certain distance the substrate is influenced by attractive
forces originating from the highly curved domain of the
protein. The a-helices distending the Ij-sheets can be envisaged as providing discrimination, allowing access for substrates of proper size and/or configuration only. The substrate, so accepted, then enters the space close to the surface set u p by the Ij-sheets. Each van der Waals active
structural unit of the Ij-sheets contributes to a force field,
which, owing to the curvature, is focussed, thereby enabling
relatively remote units to act upon the substrate. A barrel
protein can thus be conceived as a van der Waals lens.
As the substrate is intercepted by the protein, kinetic energy is transformed into potential energy; i.e., the deceleration furnishes thermal excitation. Thus, intramolecular
electron movement is substantially increased and bonding
integrity may be regarded as partially lost. Explicit manipulations on the substrate can then be undertaken.
The model of van der Waals focussing is also valid in
twisted sheet proteins because the surface set u p by the
twisted sheet, the helicoid, is isometric (curvature intact) to
the inside of a barrel, the catenoid. Thus, when the substrate is located close to the highly curved surface of the
protein it feels strong attractive forces from the entire protein.
Moreover, the relatively high flexibility of proteins permits a dynamic interaction with substrate molecules,
thereby tuning the curvature; i.e., the energy “release” is
partly governed by the entering substrate.
Other points of utmost importance in enzymatic reactions are concentration and extension. A serious shortcoming of the ordinary lock and key model is its failure to allow fully for the exceedingly low physiological concentrations of many enzyme substrates. Furthermore, a keyhole
can only be arrived at by means of highly precise docking
procedures; i.e., the reach of a keyhole into “space” is essentially nil. In view of the extraordinarily high turnover
numbers often encountered in enzymatic processes it
seems highly inexpedient to have a feeding mechanism
that requires a low-concentration substrate to interact with
the enzyme under conditions of extreme spatial ordering.
05750833/88/0707-0954 $ 02.50/0
Angew. Chem. Inl. Ed. Engl. 27 (1988) No. 7
A saddle surface, i.e., a catenoid o r a helicoid, on the other
hand, has none of the keyhole restrictions. It could be regarded as infinite in terms of interactive impact. Any point
on a saddle surface is the bull's-eye; the substrate would
be intercepted and guided to the region of maximum curvature even though the preliminary encounter was of a
random nature.
These ideas can be illustrated by a few examples: only
enzymes describable as barrels or twisted sheets have been
selected." 'I We have concentrated on enzymes displaying
nonsophisticated chemical transformations and where a
minimum of atom or group transfer is involved. Nevertheless, we believe that enzymes partaking in complex chemical transformations can be accounted for analogously; i.e.,
the major energy component is provided by the enzymesubstrate interaction.
Carbo.xypeptidase:a typical enzyme with the helicoid arrangement. It is a catabolic enzyme, degrading proteins by
detaching amino acids starting from the carboxylic end. If
we analyze the reaction in a strictly chemical sense, it is
obviously a hydrolysis of a n amide bond. Normally (i.e., in
vitro) we accomplish this by heating an amide in water,
adding a basic catalyst. The heat serves to energize the amide bond, i.e., to propagate the transient presence of highenergy entities and to confer additional thrust to the entering nucleophile (water). The catalyst simply transforms
water into a more reactive form (hydroxide ion). If we assume that, by means of efficiently focussed van der Waals
forces, carboxypeptidase imparts substantial thermal stress
to the entering peptide, the free carboxylic end of which
serves as an electron-rich target for nonbonding interactions, the transient existence of vibrationally excited entities coupled with heating of surrounding water molecules
should provide the impetus necessary to bring about hydrolysis.
Triose phosphate isomerase; an archetypal catenoid enzyme. The reaction catalyzed by this enzyme is the transformation of dihydroxyacetone phosphate to glyceraldehyde-3-phosphate. Chemically this could be viewed either
as a redox reaction, i.e., oxidation of a n alcohol to an aIdehyde and reduction of a ketone to an alcohol, or a re-enolization. Obviously there is no sophisticated chemistry involved and simple thermal excitation furnished by the van
der Waals lens (triose phosphate isomerase) could be envisaged to propel the reaction. By analogy to the carboxylate in the example above the phosphate group could serve
as an electron-rich target. In fact, one could seriously speculate that the presence of phosphate in biological systems, apart from buffering capacity, is explained by the
need for a nontoxic, reasonably electron-rich van der
Waals acceptor. Consequently, the role of phosphate esters
as biochemical energy-carriers could be due to van der
Waals targeting and the ease of hydrolysis of phosphate
Pyruvate kinase; a catenoid enzyme. This enzyme catalyzes the detachment of phosphate from phosphoenolpyruvate yielding pyruvate. The phosphate group is concomitantly transferred to ADP. Chemically, the reaction is once
again rather simple; hydrolysis of a phosphate ester followed by re-enolization. Again one can expect that thermal
excitation in the presence of water and ADP is the major
factor necessary for reaction.
Lactate dehydrogenase; a helicoid enzyme with the redox cofactor NADH. The reaction is a reduction of pyruvate to lactate. NADH is bound to the protein close to the
curved arrangement of the P-sheets. Thermal excitation assists the entering hydride supplied by NADH.
Angew. Chem Int. Ed. Engl. 2711988) No. 7
Albumin; a carrier protein that cannot be described as
either catenoid or helicoid. We have only been able to find
clear representatives of the structure of prealbumin. Assuming that the curved nature of prealbumin is essentially
preserved during its transformation to albumin, some interesting conclusions can be drawn. The structure can be
characterized as an arbitrary part of a minimal surface.
The curvature of the fi-sheet arrangement appears to be lower than in the aforementioned proteins. Considering the
function of albumin as a carrier of fatty acids, the relatively low curvature seems most reasonable. According to
Equation (I), the impact of a curved surface on an entering
molecule is dependent on the curvature of the surface. Obviously a carrier protein should have a curvature high
enough to retain the molecules that are carried but not so
high that damage is inflicted on the protein o r that release
of the molecules is too energy consuming. It is interesting
to note that longer-chain fatty acids bind more readily to
albumin than shorter-chain fatty acids. Furthermore, unsaturated fatty acids bind with higher affinity than saturated
ones. If van der Waals interception is at work, one would
indeed expect such behavior.
In this context it is tempting to propose that the extreme
toxicity of 2,3,7,8-tetrachlorodibenzo[l,4]dioxin (TCD)
might be a result of its interactions with highly curved proteins. If the interacting molecule is electron-rich and stable
(both characteristics of TCD), the heat developed owing to
the interaction could be sufficient to seriously damage the
protein. The resulting collapse of structural integrity would
release the TCD, undamaged and free to move to the next
protein-the function of the dioxin becomes that of a more
or less eternal projectile. Other chlorinated dioxins differ
in toxicity, with T C D being the most toxic; owing to the
arrangement of chlorine atoms, the slim, symmetrical T C D
molecule should provide the best general fit to an arbitrary
Extension of the latter proposal suggests a n unprecedented approach to antiviral agents. It has been recently
shown that the protein coat of polio viruses is compiled
of four discrete proteins. Three of these are barrels. It is
our opinion that these barrels are employed by the virus
when a host cell is attacked. After binding to the surface
of the cell, cell membrane components are attracted into
the barrel thereby creating a hole in the cell membrane.
Through this passage the virus can then inject the genetic
material, the action of which finally proves fatal for the
If this model of virus action is valid, it should be possible to exploit it advantageously in the treatment of viral
diseases. By careful molecular design, it may be possible to
prepare substances with the ability to interfere with the
protein coat of viruses. These substances might irreversibly
block the holes in the barrels, or, analogously to the proposed activity of dioxin, they might cause total destruction
of the discrete proteins in the coat.
We feel that the recognition of highly curved protein
structures represents a totally novel appreciation of protein function. Curvature-induced focussing of nonbonded
interactions could also help to account for the energetics
of many other biochemical events, such as transport across
membranes, antigen-antibody interactions, splicing of
mRNA precursors, action of tRNA in protein synthesis,
and oxygen interception by hemoglobin.
0 VCH Verlagsgesellschaji mbH, 0-6940 Weinheim, 1988
Received: October 5, 1987;
revised: March 21, 1988 [Z 2459 IE]
German version: Angew. Chem. 100 (1988) 995
0570-0833/88/0707-0955 $ 02.50/0
[ I ] A. H. Louie, R. L. Somorjai, J. Theor. Biol. 98 (1982) 189; J . Mol. Biol.
168 (1983) 143; Bull. Math. Biol. 46 (1984) 745.
[2] J. A. McCammon, P. G. Wolynes, M. Karplus, Biochemistry 18 (1979)
[3] S. Anderson, S . T. Hyde, H. G. von Schnering, Z . Kristallogr. 168
(1984) 1.
141 S . T. Hyde, S. Anderson, Z . Krisrallogr. 168 (1984) 221; 170 (1985) 225;
174 (1986) 225.
IS] S. T. Hyde, S. Anderson, B. Ericsson, K. Larsson, Z. Kristallogr. 168
(1984) 213.
[6] A.-C. Eliasson, K. Larsson, S. Anderson, S. T. Hyde, R. Nesper, H. G.
yon Schnering, Starch/Sturke 39 (1987) 147.
[7] S. T. Hyde, S. Anderson, K. Larsson, Z. Kristallogr. 174 (1986) 237.
[S] R. Nesper, H. G. von Schnering, Z . Kristallogr. 170 (1985) 138; Angew.
Chem. 98 (1986) 11 1 ; Angew. Chem. Int. Ed. Engl. 25 (1986) 110.
191 R. Thornasson, S. Lidin, S. Anderson, Angew. Chem. 99 (1987) 1056;
Angew Chem. Inr. Ed. Engl. 26 (1987) 1017.
[lo] Z. Blum, S. Lidin, R. Thomasson, J. Solid State Chem.. in press.
[ I 1 1 J. S . Richardson, Adu. Protein Chem. 34 (1981) 167.
rn CuCl
structure) dissociates from the complex [equilibrium reaction (c)].
a Novel
Cu' Cluster Containing Pentacoordinated Phosphorus
in p3-PRR' Bridges (R = iPr, R' = CHzPiPr)**
By Franjo Gol, Peter C. Kniippel, Othmar Stelzer,* and
William S . SheIdrick*
Dedicated to Professor UIrich Wannagat
on the occasion of his 65th birthday
+[Cus(iPrPCHzPiPr)~Cl,(py),]+ py
According to the results of the X-ray structure analystructure of 2 is based on a central
[Cu,(iPrPCH,PiPr),] unit, to which four CuCl(py), units
(x= 1,2) are joined through Cu-C1-Cu and Cu-P-Cu
bridges (Fig. 1). The Cu atoms C u l , Cu3, Cu5, and Cu7
and the P atoms P2, P4, P6, and P8 form an eight-membered ring having a distorted boat conformation. P2 and
P4 as well as P6 and P8 are linked in a strain-free fashion
by CH2 bridges (P2-C11-P4 105.1(4)", P6-C72-P8
105.3(4)", P2-Cl1 188.7(8), P4-Cl1 187.7(8), P6-C72
187.5(8), P8-C72 188.2(8) pm). With the exception of Cu6,
the arrangement of ligands around the copper atoms is distorted trigonal-planar (sum of the bond angles 358.3 to
359.8"). Of the two pyridine ligands on Cu6, one (N51) is
more loosely bound, as shown by a comparison of the
bond lengths Cu6-N51 (214.8(8) pm) and Cu6-N61
(205.2(8)). It is presumably this pyridine ligand which dissociates from 2 in solution [cf. equilibrium (c)].
S ~ S , ' ~ ] the
Methylenebisphosphanes HRPCHzPRH1']can be transformed by oxidative addition['] into difunctional phosphido-bridges RPCH,PR, bonding up to four transition-metal
atoms (A-C).
Attempts to synthesize tetranuclear complexes of
monovalent copper by reaction of copper chloride in
the presence of pyridine with dianionic PCP ligands
formed by cleavage of the P-Si
bonds in
(Me3Si)iPrPCH2PiPr(SiMe3)I3I[reaction (a)] resulted in the
formation of a product mixture. When excess CuCl (CuCI:
phosphane ligand 4 : I ) was used, however, a smooth reaction occurred which afforded [Cu8(iPrPCH2PiPr)2C14(py)5]
[reaction (b)].
The diamagnetic cu8 complex 2 is a nonelectrolyte in
CH2ClZsolution and exhibits a singlet at 6= -61.1 in the
31P('H)-NMR spectrum (CHZCl2, 25 "C), which is broad
(half-width ca. 50 Hz) owing to the quadrupole effect of
63Cu/65Cu.Molecular weight determinations (osmometric
in CH,Cl,) gave lower values (ca. 850) than that expected
for 2 (1369.95). These findings indicate that one of the five
pyridine ligands (see below the discussion of the X-ray
F. Gol, Dip1.-Chem. P. C. Kniippel
FB 9 (Anorganische Chemie) der UniversitstGesamthochschule
GauBstrasse 20, D-5600 Wuppertal I (FRG)
Prof. Dr. W. S. Sheldrick
Fachbereich Chemie der Universitat
Erwin-Schrodinger-Strasse,D-6750 Kaiserslautern (FRG)
[**I Linear Oligophosphaalkanes, Part 22. This work was supported by the
Deutsche Forschungsgemeinschaft and the Fonds der Chemischen Industrie.-Part 21: [ S ] .
[*] Prof. Dr. 0.Stelzer, Dip1:Chem.
0 VCH Verlagsgeselkchafi mbH, 0-6940 Weinheim, 1988
c 883
c: 84:
Fig. 1. Molecular structure of 2 [4]
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Ed. Engl. 27 (1988) No. 7
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