Cooperative effects in macromolecular self-ordering

some generalizations on grounds of theoretical and experimental results


Jaroslav Kříž

Institute of Macromolecular Chemistry

Academy of Sciences of the Czech Republic

Heyrovsky Sq. 2, 162 06 Prague 6, Czech Republic

E-mail: kriz (at)


Organization of macromolecules means transferring them into a state with lower probability, hence<0. We are interested in spontaneous processes, which usually correspond to a tendency of the system to adopt a state with higher probability, hence>0. This seeming paradox is solved in an equilibrium system by the well-known formula for the Gibbs’ free energy

Negative can be achieved either by trading for, i.e. strongly exothermal () process allows for<0, or by redistribution of entropy, i.e. absorbing the excessive entropy by waste. Both possibilities have to be considered in organization (or self-ordering) of macromolecules.

The enthalpy (energy) part must be provided for by some sort of interactions. All non-bonding interactions in molecular physics are forms of electromagnetic interaction and can be classified into Coulomb, H-bonding, dipolar and disperse interactions and their various combinations. All except the first are weak and short-range. To build up an at least moderately stable system, they have to cooperate.

Formally, we can partition the overall of e.g. coupling of two macromolecules into the contribution of (pairs of) units,doubles, triples etc.:

We can then classify the types of cooperation:

First order cooperation:

higher order cooperation:

First order cooperation:

consider a simple equilibrium

then the conversion of groups a will be given by



assuming we have and

simulation gives

i.e. we get substantial cooperation of groups if K > 1.1

(cooperation accumulates the already advantageous coupling)

Higher order cooperation:

simulation gives (examples will be given later on)

sigmoidal dependence of a on the number of cooperating units n is a clear sign of a genuine (higher order) cooperation

possible causes:

A: energetic (exothermic)

  • creation of a stable bonding system at a given n (e.g. key-lock arrangements)
  • additive long-range interactions (cumulative effect)

B: entropy producing

  • collective electrostatic phenomena (polyelectrolyte effect)
  • analogous phenomena in H-bonding
  • other collective phenomena (e.g. hydrophobic interaction)

C: others

  • in particular proximity effect (insular kinetics)

Quite often, a seemingly energetic case can be shown to be mostly entropic.

Example: cumulative long-range interactions between two polyions


i.e. a ammonium-type polycation with a polyphosphate polyanion:

the Coulomb attractions from more distant ions surely add to that of the pair in the immediate vicinity

as expected, we obtain a typical sigmoidal dependence of a on n


  • However, the stabilization energy per unit
    obtained by quantum-mechanical
    does not increase with n (quite to the contrary)

    The cause of cooperation can be shown to be entropic:

  • Whereas for n = 1 – 5 higher temperature exothermic process), for n > 10 (cases with leads to lower a (as it should for slightly growing cooperation) it leads to higher alpha.

    We thus must have D S > 0.. As for the polymers there must be D S < 0, (more ordered state), there must be other components absorbing entropy. According to our results, the following
    generalization is probably right:

    For most processes of macromolecular ordering, D H (energy stabilization) is usually the source of additive (first order) cooperation (except highly specific key-lock interactions); the source of non-linear (higher order) cooperation is mostly D S. As we need D S < 0 for the self-ordering part of the process, there must be some waste absorbing the excess entropy.

    In nature and culture, live cells, organisms, communities, cultures utilize nutrients or raw materials to gain energy (among other reasons), which they convert to negative entropy of their highly ordered state, the excess entropy being absorbed by waste.

    In non-living systems, there must be a rough analogy:

    we need some pre-ordered parts of the system, which are able, by their liberation into disordered state,
    absorb entropy

    Equilibrium effects:

    Sources of primary entropy-absorbing order of bound particles:

    active interactions:

    1. long-range Coulomb interactions: according to the charge density, lead to variously profiled density gradients (example: counterions around polyions)

    2. short-range interactions: lead to an equilibrium between an ordered (bound) and disordered (free) state (example: H-bond of a solvent)

    passive interactions:

    relative ordering enforced by the presence of the solute: lead to relative ordering of the solvent molecules with a steep gradient (example: hydrophobic hydration)

    1. long-range Coulomb interactions:

    we are looking for the radial distribution r (r) of the counterions. In simulations, we use the approximation of Poisson-Boltzmann:

    typical approximation: polyanion as a stiff cylinder with a charge distance d

    As most of our counterions posses magnetically active nuclei (with S=3/2 and an electric quadrupole) such as 7Li, 23Na, 39K, 35Cl, 79Br, we utilized NMR quadrupolar relaxation (developing the model by Halle, Wennerstrom and Picullel):

    ion density: Poison- Boltzmann

    diffusion: Smoluchowski

    relaxation is brought about by the diffusion of the counterion in the gradient of the electric field of the polyion

    for details, see papers at the end

    elaxation under progressive dilution offers
    From the best-fitted P(r) Poisson-Boltzmann simulation gives
    from it, we calculate:

    for densely charged polyions, there exists a relative ordering of counterions (already long assumed ion condensation or polyelectrolyte effect) at concentration < 10 mmol/L

    this relative ordering absorbs entropy in the coupling of oppositely charged polyions (liberated counterions adopt a disordered state)

    This conclusion has been verified by us for a number of polyion structures:

    (However, this effect diminishes rapidly with charge dilution on the polyion.)

    According to entropy simulation, conversion a should increase with increasing dilution of the polyions. This is verified by experiments:

    The effect of counterion condensation (and the resulting cooperativeness) depends also on the polyion length:

    longer sequences give larger P therefore larger a

    however, this also leads to weak preference of ordered structures:



    possible generalization:

    long-range interactions with high cooperation form organized structures readily but with low selectivity (they are scarcely used in nature)

    another flaw of high cooperation (reactivity): the parking problem



    we detect it via decreasing limiting alpha at higher n

    We have shown it to be in principle a kinetic effect: formation of pseudo-stationary states. Its nature can be shown by its gradual suppression when suppressing the reactivity by increased ion strength (addition of salt)


    simulation of a pseudo-limiting conversion of complementary pentamers at various salt concentrations and various degrees of cooperation

    (system of 32 differential equations, simulation proceeded up to 0.1% change)

    experiment: substantially larger effects for longer polyions

    In the case of low salt concentration, a increases after several months – thus a meta-stable state.


    2. short-range interactions: entropic effects, H-bond of the solvent

    a) ion pair hydration (H-bonds of water)

    contact separated

    according to our results (MP2/6-31G(d)), ion pairs in reactants and products are similarly hydrated, so that no important change in water ordering is expected

    b) hydrophilic hydration of peptides:

    important changes in water ordering: important part of the driving force of cooperation (though even more important is the hydrophobic hydration)

    c) H-bonds of another solvent:

    e.g. one of the polymers a H-donor: interesting case: groups B tand solvent s of analogous H-bond affinity:




    Striking example: pyridine cooperatively substituted by poly(4-vinylpyridine) – according to n , the polymer substitutes up to 100-fold molar excess of pyridine:

    no energy change, the sole driving force is entropy

    actual Delta S:
    distributive: bound to random position
    translational: access of faster translation modes
    rotational: change to faster and isotropic rotation

    by using pyridine-d5, rotational mobility can be measured by quadrupolar relaxation of 2H:

    larger T1 means faster rotation

    low value of T1 of the 4-2H in the bound pyridine demonstrates preferential rotation around the axis of N-H bond

    rotation of the free can be seen to be isotropic

    Passive interactions as a source of primary ordering:

    by passive we mean lack of direct interaction of the solvent with the solute

    hydrophobic hydration:

    enforced relative ordering of water molecules at the interface with hydrophobic micro-phase

    quantum-mechanical simulation of the build-up of a hydration sheaf of partly hydrophobic peptide unit


    IR blue shift of CH vibrations and NMR chemical shifts indicate that 5 molecules of H2O are stationary. NMR relaxation shows rapid exchange with the surrounding water

    Delta S connected with the change of hydrophobic hydration probably is the main part of cooperative hydrophobic interaction

    Examples of hydrophobic interaction as a cooperative phenomenon:


    1.Comparing electrostatic coupling with PDADMAC of:

    a) polymer PSS

    b) low-mol.-weight MBS

    c) low-mol.-weight DDBS with a hydrophobic chain

    (all have the same stabilization energy of coupling)

    in contrast to MBS, DDBS cooperates in coupling

    the model by Kuhn-Levin-Barbosa treating hydrophobic interaction in energy terms fails to describe cooperation fully

    2. Electrostatic coupling with ammonium salts with polyphosphates:

    whereas methylammonium chloride couples in a normal equilibrium,

    an analogous salt with hydrophobic groups shows cooperative behavior:

    the cause of this behavior is clearly in at higher molar ratio and higher T, the the increase in the mobility of water coupling appears to have D S > 0

    relaxation of D2O:

    increasing T1 of 2H2O at higher b clearly shows increased rotational mobility, hence increased entropy

    Hydrophobic interaction (change in hydrophobic hydration) is the main driving force of cooperative thermotropic behavior of some amphiphilic macromolecules such as poly(pentapeptides) and vinylic polymers (poly(vinylmethylether), poly(N-isopropyl-acrylamide) etc.) also studied in our laboratory (not reported here).

    Quasi-equilibrium effects:

    Apparent strongly cooperative behavior can be caused by effects, which are strictly speaking not entirely reversible, although equilibrium statistical thermodynamics can be used to some degree in their description

    1. Proximity assistance (insular kinetics)

    According to our findings, proximity assistance of neighboring bonding groups can explain some cases of cooperative coupling where neither D H nor D S give any reason for it. According to our theory, this behavior is connected with a perturbed statistics of collisions in an isle of the emerging complex (hence insular kinetics).

    Let us take a simple equilibrium:

    described in statistical thermodynamics by

    any of the partition functions can be rewritten with the translation part is (V is the volume). Then

    Assume now that pairs of molecules of p and q are restricted in their relative motion to some small volumes v. We have then

    with , , and with. It is easy to see that

    giving a huge effect if v is very small.

    Restriction of volume can be due to e.g. an already formed bond:






    v is the volume of a sphere with a radius = distance to the nearest group

    result: for [C]0 = 10-3 mol/L cooperation starts to operate at r<3.5 nm i.e. 15 – 17 monomer units

    experimental verification: electrostatic coupling of polyions with dilute charge:




    According to our NMR investigation, the ionic groups in both polymers behave like lone groups (no counterion condensation). According to our quantum-mechanical calculations, the charge distribution in the models is almost identical to that in the ionic groups of the respective polymer.


    model anion to polycation: model cation to polyanion:

    very weak bond very weak bond

    polycation to polyanion: 5 mmol/L

    our model correct in order of magnitude

    for independent ions expected about 40times less

    2. Irreversible effects of micro-phase separation

    preliminary results for PVPh-PVPy

    (polymeric polyvinylphenol, oligomers of


    measured by PFG NMR

    cooperation I: higher a at higher Pn

    cooperation II: igher a at higher b

    possible explanation: organized H-bonding leads to a stiff ladder polymer (we observe signs of aggregation, at higher a even flocculation and sedimentation).

    There is a possibility, that emerging microphase could shift the equilibrium


    1. Kříž, J.; Kurková, D.; Dybal, J.; Oupický, D. Cooperative interactions of unlike macromolecules: NMR study of ionic coupling of poly[2-(trimethylammonio)ethyl methacrylate chloride]-block-[poly(N-(2-hydroxypropyl)methacrylamide] polycation with polyphosphates in D2O, J. Phys. Chem. A 2000, 104, 10972-10985

    2. Kříž, J.; Dautzenberg, H. Cooperative interactions of unlike macromolecules: NMR and theoretical study of electrostatic binding of sodium poly(styrenesulfonate)s to copolymers with variably distributed cationic groups, J. Phys. Chem. A 2001, 105, 3846-3854

    3. Kříž, J.; Dybal, J.; Dautzenberg, H. Cooperative interactions of unlike macromolecules: NMR and theoretical study of electrostatic coupling of sodium polyphosphates with diallyl(dimethyl)ammonium chloride-acrylamide copolymers J. Phys. Chem. A 2001, 105, 7486-7493

    4. Kříž, J.; Dybal, J.; Kurková, D. Energy versus entropy in cooperative electrostatic interactions: Comparative study of binding of sodium poly(styrenesulfonate), dodecylbenzenesulfonate and methybenzenesulfonate to polycations J. Phys. Chem. B 2002, 106, 2175-2185

    5. Kříž, J.; Dybal, J.; Kurková, D. Cooperative counterion – polyion interactions in polyelectrolyte chain dynamics: NMR and quantum-chemical study of locally collapsed state in dilute PADMAC in NaCl/D2O solutions J. Phys. Chem. A 2002, 106, 7971-7981

    6. Kříž, J.; Dautzenberg, H.; Dybal, J.; Kurková, D. Competitive/cooperative electrostatic interactions in macromolecular complexes: Multinuclear NMR study of PDADMAC-PMANa complexes in the presence of Al3+ ions Langmuir 2002, 18, 9594-9599

    7. Dautzenberg, H.; Kříž, J. Response of polyelectrolyte complexes to subsequent addition of salts of different cations Langmuir 2003, 19, 5204-5211

    8. Kříž, J.; Dybal, J.; Kurková, D. Cooperativity in macromolecular interactions as a proximity effect: NMR and theoretical study of electrostatic coupling of weakly charged complementary polyions J. Phys. Chem. B 2003, 107, 12165-12174

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