Regression Analysis for NucleationâElongation Model of

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Regression Analysis for Nucleation-Elongation Model of Supramolecular Assembly: How to Determine Nucleus Size Shinnosuke Kawai, Mikako Kuni, and Kazunori Sugiyasu J. Phys. Chem. B, Just Accepted Manuscript • Publication Date (Web): 14 Sep 2018 Downloaded from http://pubs.acs.org on September 15, 2018

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The Journal of Physical Chemistry

Regression Analysis for Nucleation-Elongation Model of Supramolecular Assembly: How to Determine Nucleus Size Shinnosuke Kawai,*1 Mikako Kuni,1 Kazunori Sugiyasu2 1

Department of Chemistry, Faculty of Science, Shizuoka University, 836 Ohya, Suruga-ku, Shizuoka 422-8529, Japan

2

Molecular Design & Function Group, National Institute for Material Science, 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan

ACS Paragon Plus Environment

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ABSTRACT:

Nucleation-elongation is known to give satisfactory descriptions of many supramolecular polymerization systems in thermal equilibrium. Its key feature is the necessity to form a “nucleus” consisting of a certain number of monomer units before being able to grow into a longer polymer chain. The size of the nucleus has significant implications for the understanding of the supramolecular polymerization mechanism. Here we investigate how experiments can give information on the nucleus size by regression analysis of various types of measurements. The measurements of free monomer concentrations, diffusion coefficients, and calorimetric response as functions of concentration or temperature are considered. The nucleation-elongation model with a general value for the nucleus size is used to provide mathematical expressions for these experimental observables. Numerical experiments are performed where experimental errors are simulated by computer-generated random numbers, and it is investigated whether least-squares fitting analyses can give the correct values of the nucleus size in the presence of experimental errors. It is recommended that the calorimetric measurements such as differential scanning calorimetry (DSC) or isothermal titration calorimetry (ITC) be performed under various conditions to correctly determine the nucleus size experimentally.

INTRODUCTION Supramolecular polymers,1-3 defined as polymeric arrays of monomer molecules brought together through weak and reversible non-covalent interactions, have been attracting growing interest from the viewpoint of fundamental science as well as technological application. Polymer-like assemblies are spontaneously formed from the monomers by attractive interactions


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such as hydrogen bonds, electrostatic forces, hydrophobic interactions, and so forth. Due to the weakness of the intermolecular interactions, the formed polymers can undergo repeated dissociation and recombination, in contrast to covalently bonded polymers where the covalent bonds once formed do not dissociate in the relevant experimental time scale. Thus it is often possible to make the supramolecular system reach the equilibrium state where polymers of different lengths co-exist with the concentration ratios corresponding to the thermodynamically determined equilibrium constants. The equilibrium theory for supramolecular polymerization mechanism proceeds by first postulating equilibrium constants that determine the relative populations of polymer chains with different lengths. In principle, the equilibrium constant between the -mer and ( + 1)-mer can be different for all the different values of . However,

much simpler models are often found sufficient to analyze real molecular systems. One such model assumes the same value of the equilibrium constant for all the steps of chain growth. This is called the isodesmic model. Another model takes into account the necessity of forming a “nucleus” of a certain size in the process of chain growth. A small value of the equilibrium constant is assumed for the nucleation step, meaning that the formation of the nucleus is difficult. After the nucleation is complete, the chain growth proceeds readily with a larger equilibrium constant. This latter model is called the cooperative, or nucleation-elongation model. The nucleation-elongation model was first born in the study of protein4-6 and was recently introduced into the field of supramolecular science.1-3,7-10 Extensions of the model were also developed to explain chirality amplification in helical assemblies11 and self-assembly accompanied by adsorption onto 2-D surface.12,13 Theoretical considerations were provided7,8,14,15 for both kinetic and thermodynamic aspects of the nucleation-elongation model. Later, supramolecular selfassembly through the nucleation-elongation mechanism was experimentally proved9 by applying


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theoretical formula7,8 to the analysis of spectroscopic data. To study experimentally the supramolecular polymerization mechanisms, it is conventional to investigate the degree of polymerization as a function of temperature and/or concentration, and then compare the result with the predictions of the models. Thorough discussion of the effectiveness of experimental methods for distinguishing the isodesmic and nucleation-elongation mechanisms has been provided.8 It was concluded that the temperature dependence is more effective in identifying the mechanism than the concentration dependence, due to the limited number of data points that can be obtained with the latter, although concentration-dependent measurements can complement the characterization performed by temperature-dependent measurements. Recently, the study of supramolecular assemblies has extended its range also into kinetic aspects.10,16-18 In relation to the nucleation-elongation mechanism, Ref. 17 showed that existence of two pathways and their different (anti-)cooperativities are essential to explain the concentration dependence of the equilibration rate observed in experiments. Ref. 18 showed that the contribution of the coagulation and fragmentation processes to the time for the mean aggregate length to equilibrate is more pronounced in the cooperative aggregation than in the isodesmic aggregation. These findings elucidate the importance of the cooperativity also in the kinetic aspects of the supramolecular aggregation. The molecular mechanisms of cooperativity that give rise to the nucleation-elongation type of supramolecular polymer growth have a wide range of variety and are extensively reviewed in the literature.1-3,14 Just to pick up a few examples, there is firstly a type of supramolecular assembly through hydrogen bonding where redistribution of the electronic density along the polymer chain results in the strengthening of the hydrogen bonding upon polymer growth.1 Molecules that form helix-shaped supramolecular assemblies can also exhibit the nucleation-elongation type growth


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because, after the completion of one turn of the helix, the growth becomes more favored due to additional attractive interaction between monomeric units in the neighboring turns.1,14 Supramolecular assemblies formed through a hydrophobic effect can exhibit the nucleationelongation type growth because of the non-additivity in the de-wetting effect of the interface.1 Coexistence of hydrophobic and hydrophilic long chains in a monomer molecule can result in cooperative supramolecular growth because, after the assembly reaches a certain length, the hydrophobic part can be surrounded by the hydrophilic chains and shielded from the polar solvent.3 Whereas the π-stacked supramolecular assemblies often grow through isodesmic or

anti-cooperative mechanisms,2 it has been shown that combinations of π-π interaction and other interactions such as hydrogen bonding, metal-metal, and metal-ligand interactions can result in the cooperative type of supramolecular growth.3 The equilibrium theory of supramolecular polymerization is not only important for the understanding of thermodynamically controlled supramolecular systems, but also acts as an important basis for the understanding of supramolecular polymerizations operating under kinetic control. An example can be found in recent developments of living syntheses of supramolecular polymers resulting in a highly controlled size distribution.19-24 Briefly, the length control of the supramolecular polymers in these works were achieved by starting the polymerization intentionally by adding seeds19,21-23 or initiator molecule20 from outside or by photoexcitation.24 There it was correctly predicted that, before the initiation, monomer molecules would not effectively form supramolecular assemblies due to the large barrier height for monomer association, and that, once initiated, further elongation of the supramolecular polymer would proceed readily with lower barrier. These predictions were based on the nucleation-elongation


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model. Thus the insights provided by the nucleation-elongation model in the equilibrium theory gave important clues for the design of these living supramolecular syntheses.

One of the crucial parameters in the nucleation-elongation model is the polymer size below

and above which the equilibrium constants have different values. Thus the equilibrium constant

between the ( − 1)-mer and -mer for smaller than is assumed to be much smaller than that for larger than , reflecting the difficulty of the nucleus formation compared to the elongation.

This parameter will be called the nucleus size in the present paper. There have not been many studies on the experimental determination of the nucleus size. Prince et al.25 made spectroscopic

observation of the transition between folded and unfolded states of covalently bonded phenylene ethynylene (PE) oligomers upon changing the solvent. The free energy of transition as a function of the number of units implied that the single turn of the helix in the folded state consists of eight to ten PE units. Xue et al.26 measured kinetic curves of amyloid formation and analyzed them with a theoretical model that is formed by combining three modules each having several variant mechanisms. Coupled differential equations for the fractions of species up to 2400-mer were numerically solved and compared to 235 experimentally observed progress curves to identify the best-fit model. In the present paper, we perform numerical investigation of experimental measurements and nucleus size determinations with an emphasis on the sensitivity to experimental noise. We assume the nucleation-elongation model under thermal equilibrium in a single solvent. The measurements of concentrations, diffusion constants, and calorimetric properties are considered and tested on their abilities to distinguish different nucleus sizes. After reviewing the formulation of the nucleation-elongation model in the next section, we mimic experiments by numerical simulation dressed with computer-generated random numbers representing experimental errors.


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The possibility to identify the true nucleus size by regression analysis of the concentration- or temperature-dependence of the observed quantities is then systematically investigated. It is found that calorimetric measurements are promising tools for experimental determination of the nucleus size.


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THEORETICAL BASIS The Nucleation-Elongation Model for Supramolecular Polymer Growth The theory of the present study is based on the following scheme of polymerization equilibrium:

A + A ⇌ A ,

=

A + A ⇌ A ,

=

A + A ⇌ A ,

=

A + A ⇌ A ,

⋮

=

A , A

A , AA A , AA

(1)

A , AA

⋮ where A represents the monomer molecule, represents the degree of polymerization, the square

bracket denotes the concentration of each species, and is the equilibrium constant for each

polymerization step. In the case of the isodesmic model, all the equilibrium constants in Eq. (1)

are assumed to be equal. If the common value of the equilibrium constant is denoted by , the

isodesmic model is expressed as

= = ⋯ = = ⋯ = ,

(isodesmic).

(2)

On the other hand, the nucleation-elongation model is characterized by two different equilibrium constants. This model can be expressed as

= = ⋯ = ! = "#$ , !% = !% = ⋯ = &

(nuclation-elongation with nucleus size ),

(3)


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where the integer represents the size of the nucleus, and "#$ and & are the equilibrium constants for the nucleation and elongation steps, respectively. It is convenient to define the following parameter 1

"#$ (4) , & with 1 < 1, representing the ratio of the two equilibrium constants. According to Eqs. (1) and (3), 1≔

the concentration of the -mer is given by

A = (1& A) A A = 1 ! (& A) A

(for 1 ≤ ≤ ), (for ≥ ).

The total concentration 89 of monomer can be calculated as follows: ;

!

, (1 − 1& A) 1(1 − & A)

(6)

corresponding to Eq. (6) of Ref. 15. The degree of aggregation ?@AA is the fraction of the monomer molecules existing in the form of polymer. Under the nucleation-elongation model, it is calculated from Eq. (6) as

?@AA = 1 −

A 89

1 − ( + 1)(1& A)! + (1& A)!% =1−= (1 − 1& A) (1& A)! ( + 1 − & A) + > 1(1 − & A)

(7)

Equations (6) and (7) are general results that hold for the nucleation-elongation model given by Eq. (3) without any further assumptions or approximations. We next discuss some special cases obtained from Eq. (6). First, by substituting 1 = 1 into Eq. (6) we obtain


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A (8) , (for 1 = 1) (1 − & A) Note that 1 = 1 corresponds to the case of the isodesmic model (see Eqs. (2) and (3)). Equation 89 =

(8) is indeed the same as that obtained for the isodesmic model found in the literature.2,14 The

same result is obtained by formally setting = 1 in Eq. (6) for any value of 1. Note that setting

= 1 in Eq. (3) also reduces to the isodesmic model given by Eq. (2). Next, by inserting = 2 into Eq. (6) we obtain

1 D, (for = 2) (9) (1 − & A) which coincides with the result for the case with a single nucleation step in Refs. 2 and 14. 89 = A C(1 − 1) +

Behaviors in the limits of low concentration (89 → 0) and high concentration (89 → ∞) can

also be obtained from Eq. (6):

89 A ≈ I 1 &

(89 → 0),

(89 → ∞).

(10)

The first line is obtained simply by setting A → 0 in the curly bracket in Eq. (6). The second

line can be verified by noticing that setting A → 1/& makes the right hand side of Eq. (6)

diverge to infinity. Note that the two curves given by Eq. (10) cross at A = 89 = & , which is called the critical monomer concentration 8$ :

1 . &

(11)

A , (for → ∞ and A < 1/& ) (1 − 1& A)

(12)

8$ =

The limit of large nucleus size ( → ∞) for A < 1/& in Eq. (6) is given by

89 =

When 89 becomes large beyond 8$ , the free monomer concentration A approaches the critical

monomer concentration 8$ = 1/& . In this region, we may use the approximation A ≈ 8$ =


10

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1/& except in the denominator of the second term in Eq. (6) to obtain the following approximate expression

1 − ( + 1)1 ! + 1 !% 1 ! & 89 ≈ + , (for 89 > 1/& ) (1 − 1) (1 − & A)

(13)

When 89 becomes sufficiently large, the second term becomes dominant because (1 − & A) in the denominator becomes small. We can then obtain

& 89 ≈

1 ! L = , (for 89 ≫ 1/& ) (1 − & A) (1 − & A)

(14)

where L ≔ 1 ! is called the cumulative cooperativity.1,3 If we solve Eq. (14) for A, we obtain

?@AA

1 ! L & A ≈ 1 − N O =1−P Q ≈ 1, & 89 & 89

1 1 1 L = 1− − =1− − , / & 89 (& 89 ) & 89 (& 89 )/ (!)/

/

(for 89 ≫ 1/& )

(15)


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Figure 1. Free monomer concentration A as functions of the total concentration 89 for the

nucleation-elongation model with nucleus size = 1,2,3,4,6. Solid curves depict the asymptotic

behavior as → ∞.

Figure 1 shows the behavior of the free monomer concentration A as a function of 89 for

1 = 0.1 and nucleus size = 1,2,3,4,6. For low concentration, A increases with 89 as A ≈

89 (Eq. (10)). When 89 becomes larger and exceeds the critical concentration, the increase rate of A slows down and A approaches the limiting value 1/& . The limit of large nucleus size,

given by Eqs. (12) and (15), is shown by two solid curves in Fig. 1. It is seen that the value of

A indeed approaches these curves as increases. The plot for = 6 is already almost identical to the limiting curves.


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Temperature dependence of the aggregation is derived from that of the equilibrium constants. From elementary thermodynamics the dependence is given by

"#$ = "U exp X−

Δ" Z o 1 1 P − Q] , [ \ \U

(16) Δ& Z o 1 1 & = &U exp X− P − Q] , [ \ \U where \U is some reference temperature, "U and &U are the equilibrium constants for the

nucleation and elongation steps, respectively, at temperature \U , and [ is the gas constant. The heats of reaction, Δ" Z o and Δ& Z o for the nucleation and elongation steps, respectively, can be

different in general. They are usually assumed to be constant over the temperature range

considered. For fixed 89 and sufficiently low temperature, we have & ≫ 89 and A ≈ & by Eq. (15). Then the degree of aggregation is given by

?@AA = 1 −

A 1 Δ& Z o 1 1 ≈ 1− exp X P − Q], 89 &U 89 [ \ \U

(17)

which is essentially the same equation as that shown in Refs. 7-9 for the low-temperature regime, except for the way of expanding in \. Eq. (17) can be rearranged into a convenient form:

ln^1 − ?@AA _ ≈

Δ& Z o 1 1 P − Q − ln(&U 89 ), [ \ \U

(18)

which states that, when ln^1 − ?@AA _ is plotted against (1/\ − 1/\U ), the data points lie on a

straight line whose gradient and intercept are Δ& Z o /[ and − ln(&U 89 ) , respectively.

Nevertheless we should note that Eq. (18) is only applicable at low temperatures, and ?@AA in general must be calculated from Eqs. (6) and (16). To demonstrate the problem that we wish to discuss in the present paper, Fig. 2 shows some example results for the degree of aggregation calculated by the nucleation-elongation model (Eqs. (6), (7), and (16)). Figure 2(a) shows ?@AA plotted against the total concentration 89 for three ACS Paragon Plus Environment

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different combinations of (, 1): (2, 10` ), (3, 10 ), and (4, 10 ). It is seen there that these three parameter sets give almost identical results for the concentration dependence of ?@AA . In other words, it is impossible to distinguish these three models by experimental measurements of the concentration dependence of ?@AA unless extremely accurate measurements are performed. The coincidence of these three models in the plot of ?@AA can be understood from the asymptotic

formula given by Eq. (15). The expression of ?@AA depends on and 1 only through the factor

L = 1 ! . Thus the plots of ?@AA against & 89 become almost identical if they share a common

value of L. Note that all the three combinations of (, 1) plotted in Fig. 2 have L = 1 ! = 10` in common. The situation is not improved by observing the temperature dependence of the

same systems. Figure 2(b) shows the temperature dependence of ?@AA calculated for the same

three sets of (, 1) as in Fig. 2 (a). The temperature dependence is calculated by setting Δ" Z o =

Δ& Z o = −100 kJ mol . It is seen that the three models cannot be distinguished from the observation of the temperature dependence of ?@AA .


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(a)

Figure 2. The degree of aggregation ?@AA calculated for three different parameter sets. (a) ?@AA

as a function of the total concentration 89 . (b) ?@AA as a function of the temperature \.


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Equivalence with the Thermally Activated Self-assembly Model Cooperative supramolecular polymerization was formulated in Ref. 7 in a form that is slightly different from the one adopted in the present paper. The model there is called the thermally activated self-assembly model. The relation between the two formulations has been discussed in the supporting information for Ref. 8, where it is shown that the thermally activated selfassembly model results in approximately the same expression as the dimerization-elongation

model, that is, the nucleation-elongation model with the nucleus size = 2. The equivalence proved there is an approximate one for the condition of high concentration. Here we give a slightly different proof, with which the equivalence can be made exact for all conditions by giving a different interpretation for the “monomer concentration.” Following the notations in the supporting information for Ref. 8, the thermally activated self-

assembly model can be formulated as follows: The monomer A needs to undergo an activation

step forming A∗ , before assembling into a polymer A∗ . The equilibrium reactions are given by

A ⇌ A , ∗

A + A ⇌ ∗

A∗

∗

+ A ⇌ ∗

A∗ ,

A∗ ,

A∗ = , A ∗

A∗ = ∗ , A c

A∗ = ∗ ∗ , A A

(19)

c

⋮ where ∗ and c are the equilibrium constants for the activation and the elongation steps, respectively. The equilibrium concentrations of polymers are given by


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A∗ = ∗ A

A∗ = c A∗ ⋮

(20)

A∗ = c A∗ .

Here we define the “total monomer concentration” by

A ≔ A + A∗ ,

(21)

which is the total amount of the monomer molecules (per unit volume) with or without activation. The activated monomer concentration is then given by

Substituting this into Eq. (20) yields

A∗

∗ A . = ∗ +1

c∗ A∗ = N ∗ A O +1

Now we make the following identification

"#$

(22)

∗ A . ∗ + 1

(23)

∗ = P ∗ Q , +1

& =

c

c ∗ , ∗ + 1

(24)

"#$ ∗ 1= = ∗ . & +1

Then the equilibrium concentrations are

A∗ = 1 (& A ) A ,

The expression for the total concentration 89 is then

(for ≥ 2).

;

89 = A + : 1 (& A ) A = A C(1 − 1) + <

1 D (1 − & A )

(25)

(26)


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Equations (25) and (26) are exactly the same expressions as Eqs. (5) and (6) if we set = 2 (see

also Eq. (9)) and replace the “monomer concentration” A with A .

Numerical Simulations and Regression Analyses Real experimental results contain inaccuracies originating in the finiteness of data points and

measurement noise. Estimates for the parameters such as 1 , & , and are obtained from experimental data through least squares fitting of the theoretical equations to the scattered data points. The aim of the present paper is to examine how sensitive the estimate for the nucleus size

is to the experimental noise. For this purpose, we simulate the “experiment” by mimicking the experimental noise with computer-generated random numbers and adding them to the calculated values to obtain the “observed data.” For example, suppose that measurements of A are done at

89 = 89 , 89 , … , 89 , where f is the number of data points and 89 is the value of 89 at the ()

( )

(e)

(g)

hth data point. The experimental results for these points can be represented as follows:

A(g) = Ai89(g) ; & , 1k + lg ,

(27)

where A(g) denotes the measured value of the free monomer concentration at the hth data point,

and A(89 ; & , 1) is the theoretical value for the free monomer concentration given as a function of 89 , & , and 1 by solving Eq. (6) in terms of A. The second term lg in the right hand side of

Eq. (27) is random noise representing experimental errors. In the present analysis, we simulate the experiment by using computer-generated random numbers. The probability distributions of lg are assumed to be independent and identical normal distributions with zero mean 〈lg 〉 = 0

and designated standard deviation 〈lg 〉/ . Least-squares fitting minimizes the following residual error with respect to the model parameter 1 and & :


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e

o = : pA(g) − Ai89 ; & , 1kq . g
Δ& Z o as follows:

o () o ( o (1), A + A ⇌ A ,  Δ" Z o = Z − Z − 1) + Z o () o ( o (1), A + A ⇌ A ,  Δ& Z o = Z − Z − 1) + Z

Eq. (35) can therefore be rearranged into the following form:

Z(\; 89 )

(for > ).

(36)

!

o o (1)8 = Z 9 + :( − 1)Δ" Z A + Δ& Z , (1 − & A) (1 − & A) o A

(38)

where A is given by the solution of Eq. (6). Note that, in Eq. (38), dependence on \ and 89

enters into the right hand side through Eqs. (6) and (16). If the \-dependence of the molar enthalpies can be neglected, the first term in Eq. (38) cancels when it is inserted into Eq. (34). The DSC curve can therefore be calculated from just Δ" Z o , Δ& Z o , &U , and 1U . In reality, the \dependence of the molar enthalpies gives rise to a baseline with small but nonzero gradient in the


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DSC curve, and therefore appropriate subtraction of the baseline40 should be done before analyzing the experimental curve with the present theory. In the following numerical experiments, DSC measurements are simulated by calculating the

values of the heat capacity 8 () (Eq. (34)) at 300 equally spaced temperature points between

310 K and 370 K (δ\ = 0.2 K). Experimental noise of 〈l 〉/ = 0.1 kJ mol K is added to the calculated 8 (). This amount of noise corresponds to maximum S/N of 100 in the case of

rs#& = 10, although the value of S/N varies for different because of the difference of the peak

height. Figure 6 shows an example of the simulated DSC curve. The upper peak corresponds to heat absorption due to the dissociation of the supramolecular assembly as the temperature is

increased. The difference between the best fits with = 2 and = 3 is small but appreciable,

especially in the region 360 K < \ < 370 K. In the particular data set shown in Fig. 6, the

nucleus size is correctly assigned to 2. In Supporting Information (Sec. S2), the difference

between the fits with = 2 and = 3 is further analyzed in terms of the coefficient of determination for 100 runs of “experimental” data sets.


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Figure 6. Example of a simulated “experimental” DSC data set, that is, the plot of the values of

the heat capacity 8 against temperature, with rs#& = 2 and least-squares fitted curves.

Table 5. Number of runs (out of 100 for each rs#& ) where the nucleus size ̂ is estimated to each value by the least-squares fitting to the simulated experimental DSC data.

1 2 3 4 5 6 7 8 9 10

rs#&

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2

3

4

5

6

7

8

̂

9

10 11 12 13 14

0 100 0 0 0 0 0 0 0 0 0 0 0 100 0 0 0 0 0 0 0 0 0 0 0 100 0 0 0 0 0 0 0 0 0 0 0 100 0 0 0 0 0 0 0 0 0 0 0 100 0 0 0 0 0 0 0 0 0 0 0 100 0 0 0 0 0 0 0 0 0 0 0 100 0 0 0 0 0 0 0 0 0 0 1 92 7 0 0 0 0 0 0 0 0 0 6 66 26

0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

15 or higher 0 0 0 0 0 0 0 0 1


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Table 5 shows the results of assigning the nucleus size from simulated experiments of DSC. In contrast to the concentration and diffusion measurements considered above, excellent results are

found for rs#& ≤ 9. The result for rs#& = 10 is also fairly good since more than half of the runs

give correct assignment of the nucleus size. It is especially remarkable that, up to rs#& = 8, all the experiments result in correct assignment of the nucleus size. The success of the DSC experiment in determining the nucleus size may be understood from the viewpoint of “information” contained in the observable. The peak in the DSC curve indicates the occurrence of transition from aggregates to free monomers with increasing temperature. Roughly speaking, the horizontal position of the peak provides information on the temperature at which the transition occurs. This same information can be obtained from the temperaturedependent concentration measurement as shown in Fig. 4. However, the DSC curve can additionally provide information about the value of the heat exchange accompanied by the transition, which, roughly speaking, can be obtained from the vertical height of the peak. In this sense, the DSC curve contains more information than the temperature-dependent concentration

data. In the fitting, the parameter values, especially those of Δ" Z o and Δ& Z o , is required to explain both the horizontal position and the vertical height of the DSC peak. This requirement

poses strong constraint on the model, and only the model with the true can reproduce the observed data with high accuracy.


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Isothermal Titration Calorimetry In the experiment of Isothermal Titration Calorimetry (ITC),41 the exchanged heat is measured in the course of titrating a solution placed in a calorimetric cell with another solution placed in an automated syringe. Applications of ITC to supramolecular systems can be found in the literature.42-45 In the case of self-assembling mono-component systems, a concentrated solution of the compound is placed in the syringe, whereas the calorimetric cell is filled with pure solvent at the beginning.42 The concentrated solution in the syringe is then injected into the cell by steps.

Let Δ be the volume of solution injected per one step. After the -th step, the volume = Δ

of the solution has been injected into the cell. Let 8 be the concentration of the solution in the

syringe, and be the volume of the calorimetric cell. Because of the total-fill nature of the cell,41,42 the concentration in the cell after the -th injection is, as an approximation for ≪ , given by

(39) P1 − Q. 2 The heat exchanged during the -th injection is, after proper subtraction of the heat of dilution, 8 = 8

given by

(8 ) + (8 ), 2 where (8) is the “heat content” in the cell defined by Δ = (8 ) − (8 ) +

(40)

8 (41) . 8 Here Z(\; 8) is the enthalpy of the solution with the total monomer concentration 8, calculated (8) = Z(\; 8) − Z(\; 8 )

by substituting 8 and = into Eq. (35).

In the present paper, we propose to run five titrations with differing temperatures in one experiment. Figure 7 shows an example of ITC curves simulated with the same parameter set as


35


Page 36 of 48

previous sections, the syringe concentration 8 = 158$U , the volume injected per one step Δ = 0.005 , and experimental noise 〈l 〉/ = 1 kJ/mol of injectant , corresponding to

maximum S/N of 100. Heat is absorbed as the concentrated solution is injected into the cell due to the dissociation of the supramolecular assemblies. When a sufficient amount of solute has been injected, the heat absorption is reduced because the concentration difference between the syringe and the cell is smaller. For higher temperature, the heat absorption continues longer because of higher degree of dissociation.

Figure 7. Examples of simulated “experimental” ITC data sets of heat absorption against

injected volume with rs#& = 2 and least-squares fitted curves.


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Table 6 shows the result of assigning the nucleus size ̂ by fitting to simulated experimental

data sets of ITC. Excellent results are seen up to rs#& = 8, and a fairly good result for rs#& = 9. Naturally, with higher S/N, or equivalently with larger data sets, we obtain more accurate assignments for the nucleus size. Table 7 shows the results of the same experiments with maximum S/N of 200, where 95 % probability of correct assignment for rs#& = 9 is found. As in the case of the DSC experiment, the success of the ITC measurement in determining the nucleus size can be attributed to the fact that the information on the heat exchange is available in addition to just the occurrence of the transition. In conclusion, it is recommended to perform calorimetric measurements (ITC at different temperatures or DSC) in order to assign the nucleus size correctly. Since the numerical experiments in the present paper have been carried out only for one particular parameter set, it is nonetheless recommendable that, whenever an experiment is performed to determine the nucleus size of a specific supramolecular system, numerical simulations similar to the present paper be performed with data size and S/N being similar to that specific experiment in order to evaluate the uncertainty of the nucleus size assignment. It is also recommendable to perform more than one type of experiments (most preferably DSC and ITC) and check the consistency between them in order to increase the confidence of the assignment.


37


Table 6. Number of runs (out of 100 for each rs#& ) where the nucleus size ̂ is estimated to each value by the least-squares fitting to the simulated experimental ITC curve with maximum S/N of 100.

1 2 3 4 5 6 7 8 9 10

rs#&

2

3

4

5

6

7

8

̂

9

10 11 12 13 14

0 100 0 0 0 0 0 0 0 0 0 0 100 0 0 0 0 0 0 0 0 0 0 100 0 0 0 0 0 0 0 0 0 0 100 0 0 0 0 0 0 0 0 0 0 100 0 0 0 0 0 0 0 0 0 0 100 0 0 0 0 0 0 0 0 0 0 100 0 0 0 0 0 0 0 0 0 1 77 17 0 0 0 0 0 0 0 0 19 33

0 0 0 0 0 0 0 0 8

0 0 0 0 0 0 0 0 4

0 0 0 0 0 0 0 2 2

0 0 0 0 0 0 0 1 18

15 or higher 0 0 0 0 0 0 0 2 16

Table 7. Number of runs (out of 100 for each rs#& ) where the nucleus size ̂ is estimated to each value by the least-squares fitting to the simulated experimental ITC curve with maximum S/N of 200.

1 2 3 4 5 6 7 8 9 10

rs#&

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 38 of 48

2

3

4

5

6

7

8

̂

9

10 11 12 13 14

0 100 0 0 0 0 0 0 0 0 0 0 0 100 0 0 0 0 0 0 0 0 0 0 0 100 0 0 0 0 0 0 0 0 0 0 0 100 0 0 0 0 0 0 0 0 0 0 0 100 0 0 0 0 0 0 0 0 0 0 0 100 0 0 0 0 0 0 0 0 0 0 0 100 0 0 0 0 0 0 0 0 0 0 0 95 4 0 0 0 0 0 0 0 0 0 4 52 10

0 0 0 0 0 0 0 0 5

0 0 0 0 0 0 0 0 2

0 0 0 0 0 0 0 1 14

15 or higher 0 0 0 0 0 0 0 0 13


38

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CONCLUSION Systematic numerical simulations have been performed to investigate how the experimental determination of the nucleus size in the nucleation-elongation type supramolecular polymerization is affected by statistical errors. In the presence of experimental noise, analyzing only the data of concentration measurements can result in significantly wrong assignments of the nucleus size. However, it was found in the present investigation that the DSC and ITC measurements can remarkably improve the correctness of the nucleus size assignment. In order to obtain the correct nucleus size, therefore, it is suggested that measurements are performed of at least either of the DSC and ITC curves, or preferably both of them, to check the consistency. It is also suggested that, after the values of the parameters are obtained from the least-squares analysis of the experimental data, the numerical experiments as shown in the present paper should be carried out with the obtained parameter values to estimate the errors involved in the estimation. Since the nucleus size in the nucleation-elongation type polymerization is closely related to the molecular mechanism of polymerization, it is expected that the insights obtained in the present paper will contribute to correct determinations of the nucleus size in future experiments and so contribute to increasing our understanding of molecular pictures of supramolecular polymerizations.


39


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ASSOCIATED CONTENT Supporting Information. The following information is available free of charge. Detailed analyses of the fitting of the translational and rotational diffusion constants and the fitting of DSC and ITC data (PDF)

AUTHOR INFORMATION Corresponding Author * [email protected], TEL:+81-54-238-3014

ACKNOWLEDGMENTS Professor Masamichi Yamanaka at Shizuoka University is deeply acknowledged for introducing SK into the field of supramolecular chemistry as well as for his continuous support for this study. We thank Professor Takanori Oyoshi and Mr. Ryota Yagi at Shizuoka University for helpful information about ITC experiments. We are grateful to Professor Satoru Nagatoishi at the University of Tokyo for his kind instruction on ITC. This research was partially supported by JSPS KAKENHI Grant Numbers 16K17852 and 16KT0050. Part of the computation was performed by the supercomputer of ACCMS, Kyoto University.


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TOC Graphic


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