PROTEIN FOLDING, MOLECULAR DOCKING, DRUG DESIGN

The Role of the Derivative “Drift” in Complex Systems Dynamics

Corrado Giannantoni

ENEA - Casaccia Research Center - Via Anguillarese km 1,300 - 00060 S.Maria di Galeria, Rome, Italy

Keywords: Protein folding, Complex systems dynamics, Traditional Differential Calculus (TDC), Incipient Differential

Calculus (IDC), Molecular docking, Drug design.

Abstract: The relevance of Protein Folding is widely recognized. It is also well-known, however, that it is one of the

dynamic problems in TDC considered as being intractable. In addition, even in the case of solutions

obtainable in reasonable computation time, these always present a “drift” between the foreseen behavior of

the biological system analyzed and the corresponding experimental results. A drift which is much more

marked as the order of the system increases.

Both the “intractability” of the problem and the above-mentioned “drifts”, as well as the insolubility of the

problem in explicit terms (or at least in a closed form), can be overcome by starting from a different

gnoseological approach. This suggests a new definition of derivative, the “incipient” derivative.

The solution to the “Three-body Problem” obtained by means of IDC, and its extension to any number of

bodies, allows us to assert that the folding of even a macroscopic protein, such as dystrophin for example,

made up of about 100,000 atoms, can be carried out in a few minutes, when the model is run on next

generation computers (1 Petaflop).

The same methodology can also be applied to both Molecular Docking and computer-aided Drug Design.

1 INTRODUCTION

Mathematical models of Complex Systems

sometimes result as being intrinsically insoluble,

such as the famous “Three-body Problem”

(Poincaré, 1889).

In other cases they may result as being

insolvable “in practice” or, as usually referred to, as

being “intractable” (such as, for instance, Protein

Folding). This is because, although these problems

are thought of as being theoretically solvable in

principle, the time required in practice to be solved

may range from hundreds to some thousands of

years, even when run on the most updated

computers. Furthermore, even if they were solvable

in reasonable time, they would always present a

“drift” between the foreseen behavior of the system

analyzed and the corresponding experimental

results. A characteristic which is shared by all the

other mathematical models which result as being

solvable both in theory and in practice. A drift

which, in addition, is generally much more marked

as the order of the system increases.

This substantially depends on the fact that

mathematical models of Complex Systems are

generally formulated in terms of Traditional

Differential Calculus (TDC), that is by means of

linear and non-linear differential equations based on

the well-known concept of derivative. TDC,

however, often shows its limits, particularly when

describing biological Systems and, even more, social

Systems.

These in fact present such a richness of

characteristics that are, in the majority of cases,

much wider than the description capabilities of the

usual differential equations. In particular, because all

these Systems are habitually modeled as they were

mere “mechanisms”.

Such an aspect became particularly evident

during the research (Giannantoni, 2001b, 2002) for

an appropriate formulation of Odum’s Maximum

ergy-Power Principle (proposed by the same

Author as a possible Fourth Thermodynamic

Principle (Odum, 1994a,b,c)).

The original concept of Emergy, in fact,

introduces a profound novelty in Thermodynamics,

that is: there are processes which cannot be

193

Giannantoni C. (2010).

PROTEIN FOLDING, MOLECULAR DOCKING, DRUG DESIGN - The Role of the Derivative “Drift” in Complex Systems Dynamics.

In Proceedings of the First International Conference on Bioinformatics, pages 193-199

DOI: 10.5220/0002763401930199

 SciTePress

considered as being pure “mechanisms” (Odum,

1988). This is equivalent to say that they are not

describable in mere functional terms, because their

outputs show an unexpected “excess” with respect to

their inputs. “Excess” that can be termed as Quality

(with a capital Q) exactly because it is no longer

understood as a simple “property” or a

“characteristic” of a given phenomenon, but it is

recognized as being any emerging “property” (from

the considered process) never ever reducible to its

phenomenological premises or to our traditional

mental categories.

This evidently suggests a different gnoseological

approach with a correspondingly associated new

formal language, now represented by the definition

of a new concept of derivative, the “incipient”

derivative (see Appendix).

This enabled us to reformulate the Maximum

Em-Power Principle in a more general form, that is,

as the Maximum Ordinality Principle.

The successful application of such a Principle to

some decisively “critical points” of various

Disciplines, now enables us to assert that both the

“intractability” of the problem and related “drifts”,

as well as the insolvability of the problem in explicit

terms (or at least in a closed form), can be overcome

by starting from the above-mentioned new

gnoseological approach.

2 THE M. EM-P. PRINCIPLE

The Maximum Em-Power Principle asserts that:

“Every System reaches its Maximum Organization

when maximizing the flow of processed Emergy,

including that of its surrounding habitat”.

It thus suggests we focus our attention on those

processes which can be considered as more

specifically generative. Among them (as the same

Odum points out) there are three fundamental

processes (Co-production, Inter-action, Feed-back)

in which such an aspect is particularly evident.

These processes, in fact, when analyzed under

steady state conditions, can more appropriately be

described by means of a particular non-conservative

Algebra (Brown and Herendeen, 1996). This leads to

the introduction of the concept of Transformity,

which allows us to define Emergy as

Emergy = Energy Quality (Tr ) x Energy quantity (Ex )

(1)

that is as the product of a given quantity of available

Energy (represented by Exergy), by the product of

its corresponding Quality (expressed by

Transfomity).

The M. Em-P. Principle, through the introduction

of the new concept of derivative (given in

Appendix), can be reformulated in an extremely

more general form, by replacing the concepts of

Emergy and Transformity with the concept of

Ordinality (Giannantoni, 2008a).

The corresponding verbal enunciation then

becomes: “Every System tends to Maximize its own

Ordinality, including that of the surrounding

habitat”. In formal terms, this can be expressed as

0}{)/(

)/(

∼∼∼

rtdd

Maxnm →)/(

(2)

where:

)/( nm = the Ordinality of the System,

which represents the Structural Organization of the

System in terms of Co-productions, Inter-actions,

Feed-backs, while

}{

∼

r = the proper Space of the

System.

Such a more general formulation was thus

assumed as the preferential guide to recognize the

most profound physical nature of the basic processes

which particularly characterize self-organizing

Systems (such as Co-production, Inter-action, Feed-

back).

In such a perspective, we can now consider the

solutions to the “Two-body Problem”, to the “Three-

body Problem”, and to the more general “N-body

Problem”, respectively, in order to apply the latter

solution to the case of Protein Folding.

3 THE TWO-BODY PROBLEM

The new concept of derivative appeared as rather

surprising from the very beginning. In fact, although

originating from the description of self-organizing

systems, it seemed also valid when describing non-

living systems, such as those analyzed in Celestial

Mechanics. Let us think of, for example, Mercury’s

Precessions, the Three-body Problem, etc..

The initial idea of adopting IDC, to reconsider

such problems in a new light, originated from the

subsisting difference between the derivatives of the

exponential function

)(t

obtained on the basis of

the two distinct concepts of derivative (see Table 1).

In this respect it is worth noting that the assumption

of the exponential function as a reference function

does not represent a limitation, because any

function

)(tf can always be structured in the form

)()(ln

)(

ttf

eetf

(3)

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Table 1: Comparison between traditional and incipient derivates for the exponential function

)(t

, where the traditional

derivative of order n is expressed by Faà di Bruno’s formula (Oldham & Spanier, 1974, p. 37).

Such a choice, in addition, simplifies the

exposition of the basic concepts we are going to

present.

As Table 1 clearly shows, the traditional

derivatives present “additional” terms (from the

second order on) with respect to the incipient

derivatives. Such a specific “difference” suggested

the possibility of re-interpreting, by means of

incipient derivatives, the “failure” of Classical

Mechanics in foreseeing Mercury’s Precessions,

without modifying, in any form, the space-time

concepts, as vice versa happens in General

Relativity (Giannantoni, 2004b).

In fact the “Two-body Problem”, as traditionally

modeled in Classical Mechanics, is strictly

equivalent to solving a second order homogeneous

differential equation with variable coefficients

(Landau and Lifchitz, 1969, p. 46). At the same time

it is also well known that Classical

Mechanics underestimates the value of Mercury’s

Precessions, by foreseeing an angular anomaly of

“zero”, with respect to 42.6 ± 0.9 sec/cy, obtained by

astronomical measurements (ib.). It was precisely

this “discrepancy” which led us to think that such an

effect could be directly related to the “drift” of the

second order traditional derivative with respect to

the corresponding second order “incipient”

derivative. In fact, we obtained an estimation of the

angular anomaly of 42.45 sec/cy, which represents a

satisfactory agreement with the most recent

available data (Giannantoni, 2004b).

4 THE THREE-BODY PROLEM

The Three-body Problem was proved to be

intrinsically insoluble in Classical Mechanics

(Poincaré, 1899). In fact it is described by an 18th-

order system of ordinary differential equations

which, however, admits only 2 first order closed

form integrals (energy and areas). The concept of

“integral”, in this case, is not understood according

to the traditional sense of “solution”, but as a

“function of solutions” (ib., p. 8) structured in the

form

ttxtxtxF

cos)](),....(),([

(4)

where

),(

)(),....(

txtx

represent the generic

unknown variables of the considered problem. (ib. ,

vol. 1, p. 253).

Vice versa, when faced in terms of incipient

derivatives, the problem becomes perfectly solvable,

in the sense that: i) there exists at least one solution

in a closed form, as explicitly desired by Poincaré

(ib.); ii) such a solution, in addition, can be obtained

(always in a closed form) at there different

hierarchical levels of Ordinality, according to the

initial model adopted (Giannantoni 2007b, pp. 49-

60): a) as System made up of three distinct bodies;

b) as System made up of three “binary-duet ” sub-

systems, c) as one sole “ternary ” System made up of

three “binary-duet ” sub-systems (see also

Appendix).

The fact that the “Three-body Problem”, even in

its most general form, admits at least one solution in

a closed form when reformulated in IDC, is

substantially due to the intrinsic and specific

properties of the incipient derivatives (see

Appendix). In fact such a solution can be obtained

on the basis of the following: i) the Fundamental

Theorem of the Solving Kernel (Giannantoni, 1995),

which gives the general solution of any linear

differential equation with variable coefficients in

)(

⋅=



)(

⋅=

∼

)()(2

)(2

)()]([

etet

αα

⋅+⋅=



)(2

)]([

⋅=

∼

∑

⋅⋅=

!!...!

)(

kkk

⎟

⎠

⎞

⎜

⎝

⎛

⋅⋅

⎟

⎠

⎞

⎜

⎝

⎛

⎟

⎠

⎞

⎜

⎝

⎛

!!2!1

)(

ααα



)(

)]([

⋅=

∼

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195

terms of the sole Solving Kernel; ii) such a solution,

in particular, is already structured in a closed form

(according to Poincaré’s definition) and can directly

be transposed to the case of incipient derivatives

(Giannantoni 2007a, ch. 5); iii) in addition, since the

Solving Kernel is generally a function of function,

such a transposition can be directly obtained by

means of Faà di Bruno’s formula (ib., ch. 3); iv) this

in fact, being in turn structured in a closed form, can

directly be transposed to the derivatives of functions

of function when the latter are expressed in incipient

terms (the only difference is that, in such a case,

there are no longer “partitions” and, consequently,

related “sums”); v) finally, any traditional non-linear

differential equation in TDC can always be

transformed into a linear Ordinal differential

equation in IDC, with the same methodology as

already shown, for example, with reference to

Riccati’s Equation (ib. ch. 2).

On the other hand, such a general procedure,

already adopted in other papers and books (e.g.,

Giannantoni, 2004a,b, 2006a), is the same which

enabled us to sustain the general validity of a

Differential Calculus (namely IDC), which

contemporaneously operates in terms of Ordinality

and cardinality (Giannantoni 2007a, ch. 3).

These solutions, however, are still affected by

another form of “drift”, related to the supposed

independence of the space variables (x,y,z) from

each other. A hypothesis which, in reality, is merely

an aprioristic assumption about the geometrical

nature of the proper Space of the System.

If, vice versa, the proper Space of the System is

considered as being essentially “unum”, that is to

say the three coordinates (x,y,z) are so strictly

related to each other so as to form one sole thing of

Ordinal nature (ib., ch. 6), the problem admits an

extremely elegant solution in explicit terms and in

an Ordinal exponential form (ib.).

In such a case, the cardinal structure of the

System is nothing but the formal reflex of its Ordinal

nature, and it can be obtained through an adherent

reduction process (ib.).

5 THE N-BODY PROBLEM

The results obtained in the case of the Three-body

Problem can easily be generalized to the “N-body

Problem” (Giannantoni, 2008b, ch. 22), so as to get

an explicit solution in an Ordinal exponential form

(ib.). This solution is not affected by any form of

“drift”, either due to the “step by step” derivation or

to the supposed reciprocal independence of

coordinates, because the System analyzed is always

referred to its proper Space of generative nature.

On the other hand, such an extension can easily

be understood on the basis of what has been

previously said: any non-linear differential equation,

written in terms of incipient derivatives, can always

be transformed into a linear differential equation

(Giannantoni, 2007a, ch. 2).

This enables us to assert that the simulation of

Protein Folding, even in the case of a macroscopic

protein, such as dystrophin (made up of about

100,000 atoms), can be obtained in a few minutes,

when run on the next generation computers (1

Petaflop).

6 CONCLUSIONS

The solution to the “Three-body Problem”, thought

of as being a self-organizing system, and thus

modeled in the light of the Maximum Ordinality

Principle, after having been successfully extended to

the case of “N-body Problem”, enables us to assert

that any Protein Folding can be modeled as a

“tractable” problem. This can be solved by means of

appropriate computer code (in progress), to firstly

analyze biological systems made up of a limited

number of atoms (e.g, sugars and their mono-

chirality), before modeling the smallest proteins

(bout 2.000 atoms).

The computer time strictly required for its

solution (some minutes) evidently refers to the sole

availability of the three-dimensional space

configuration of the considered protein, given by an

appropriate set of data in the computer memory. The

subsequent analysis of the various parts of the folded

protein, however, could require much longer times.

In all cases, the analysis will always result as being

feasible and practicable.

What has been previously said about Protein

Folding can clearly be applied to Molecular

Docking, as well as to computer-aided Drug Design.

In this respect we have also analyzed the

possibility of adopting the same methodology when

analyzing the so-called “Phase 0”, recently defined

by US Food and Drug Administration (Shivaani K.

et al. 2009).

Clearly it is worth pointing out that the

mathematical procedure [code] here proposed

should not be considered as being reducible to a

mere mathematical “tool”, that is as simply being

able to solve the above-mentioned problems in a

more efficient way. This is because it represents a

radically new methodology, precisely because it is

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based on IDC. This new differential calculus, in fact,

“translates” into an adherent formal language a

gnoseological approach which is completely

different from the traditional one. This difference

resides on the three new basic presuppositions:

Generative Causality, Adherent Logic, Ordinal

Relationships (see also Appendix).

This an entirely different approach also enabled

us to recognize the reason for the mono-chirality of

proteins (Giannantoni, 2007a., ch. 18). That very

aspect which, really surprisingly, is ever present,

even in non-living Systems. For example, in the

motion of the planets in the Solar System. Albeit

mono-chirality is characterized in this case by some

“genetic” properties which, by keeping “memory” of

the generative process of the System, always reveal

the different nature of mono-chirality with respect to

biological systems (such as proteins, for instance)

(ib.).

REFERENCES

Brown M. T. and Herendeen R. A., 1996. Embodied

Energy Analysis and Emergy analysis: a comparative

view. Ecological Economics 19 (1996), 219-235.

Giannantoni C., 1995. Linear Differential Equations with

Variable Coefficients. Fundamental Theorem of the

Solving Kernel. ENEA, Rome, RT/ERG/95/07.

Giannantoni C., 2001a. The Problem of the Initial

Conditions and Their Physical Meaning in Linear

Differential Equations of Fractional Order. Applied

Mathematics and Computation 141 (2003) 87-102.

Giannantoni C., 2001b. Mathematical Formulation of the

Maximum Em-Power Principle. Second Biennial

International Emergy Conference. Gainesville

(Florida, USA), September 20-22, 2001, pp. 15-33.

Giannantoni C., 2002. The Maximum Em-Power Principle

as the basis for Thermodynamics of Quality. Ed.

S.G.E., Padua, ISBN 88-86281-76-5.

Giannantoni C., 2004a. Differential Bases of Emergy

Algebra. Third Emergy Evaluation and Research

Conference. Gainesville (Florida, USA), January 29-

31, 2004.

Giannantoni C., 2004b. Mathematics for Generative

Processes: Living and Non-Living Systems. 11

International Congress on Computational and Applied

Mathematics, Leuven, July 26-30, 2004. Applied

Mathematics and Computation 189 (2006) 324-340.

Giannantoni C., 2006. Emergy Analysis as the First

Ordinal Theory of Complex Systems. Proceedings of

the Fourth Emergy Conference 2006. Gainesville,

Florida, USA, January 17-22, pp. 15.1-15.14.

Giannantoni C., 2007a. Armonia delle Scienze (vol. I). La

Leggerezza della Qualità. Ed. Sigraf, Pescara (Italy),

ISBN 978-88-95566-00-9.

Giannantoni C., 2007b. Ordinal Benefits vs Economic

Benefits as a Reference Guide for Policy Decision

Making. The Case of Hydrogen Technologies.

Submitted to Energy (February 2008). In press.

Giannantoni C., 2008a. From Transformity to Ordinality,

or better: from Generative Transformity to Ordinal

Generativity. Proceedings of the Fifth Emergy

Conference. Gainesville, Florida, USA, January 31-

February 2, 2008.

Giannantoni C., 2008b. Armonia delle Scienze (vol. II).

L’Ascendenza della Qualità. Edizioni Sigraf, Pescara

(Italy), ISBN 978-88-95566-18-4.

Landau L. and Lifchitz E., 1969. Mécanique. Ed. MIR,

Moscow.

Odum H. T., 1988. Self-Organization, Transformity and

Information Science, v. 242, pp. 1132-1139,

November 25.

Odum H. T., 1994a. Ecological and General Systems. An

Introduction to Systems Ecology. Re. Edition.

University Press Colorado.

Odum H. T., 1994b. Environmental Accounting. Environ.

Engineering Sciences. University of Florida.

Odum H. T., 1994c. Self Organization and Maximum

Power. Environ. Engineering Sciences. University of

Florida.

Oldham K. B. and Spanier J., 1974. The Fractional

Calculus. Theory and Applications of Differentiation

and Integration to Arbitrary Order. Academic Press,

Inc., London.

Poincaré H., 1889. Les Méthodes Nouvelles de la

Mécanique Céleste. Ed. Librerie Scientifique et

Technique A. Blachard. Vol. I, II, III, Paris, 1987.

Shivaani K. et al. 2009. Phase 0 Clinical Trial of the Poly

(ADP-Ribose) Polymerase Inhibitor ABT-888 in

Patients with Advanced Malignancies. Journal of

Clinical Oncology. Published Ahead of Print on April

13, 2009 as 10.1200/JCO.2008.19.7681.

APPENDIX

The analysis of Generative Processes under dynamic

conditions suggests the introduction of a new

concept of “derivative”. This is because the same

adoption of the traditional derivative (d/dt) is

nothing but the formal reflex of three fundamental

pre-assumptions when describing physical-

biological-social systems: i) efficient causality; ii)

necessary logic; iii) functional relationships.

It is then evident that such an aprioristic

perspective excludes, from its basic foundation, the

possibility that any process output might ever show

anything “extra”, with respect to its corresponding

input, as a consequence of the intrinsic (supposedly)

necessary, efficient and functional dynamics of the

system analyzed.

Consequently, such a theoretical approach will

never see any “output excess”, exactly because it has

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197

already excluded from the very beginning (but only

aprioristically) that there might be “any”. In this

sense it is possible to say that such an approach

describes all the phenomena as they were mere

“mechanisms”.

Generative Processes, on the contrary, suggest

we think of a different form of “causality”, precisely

because their outputs always show something in

“excess” with respect to their inputs. This

“causality” may be termed as “generative” causality

or “spring” causality or whatsoever. In all cases the

basic concept is rather clear. In fact, any term

adopted is simply finalized at indicating that it is

worth supposing a form of “causality” which is

capable of giving rise to something “extra” with

respect to what it is usually foreseen (and expected)

by the traditional approach.

The same happens for Logic. In fact, a different

Logic is correspondently needed in order to

contemplate the possibility of coming to conclusions

much richer than their corresponding premises. This

new form of Logic, in turn, could correspondently

be termed as “adherent” Logic, because its

conclusions are always faithfully conform to the

premises. The conclusions, however, could even be

well-beyond what is strictly foreseen by the same

premises when interpreted in strictly necessary

terms.

As an adherent consequence of both previous

concepts, the relationships between phenomena

cannot be reduced to mere “functional” relationships

between the corresponding cardinal quantities. In

fact, they always “vehicle” something else, which

leads us to term those relationships as “Ordinal”

relationships. The term “Ordinal”, which might

appear as being simply adopted only to make a

difference with respect to its corresponding

“cardinal” concept, has in reality a much more

profound meaning (as shown later on).

At this stage we can clearly assert that the new

concept of derivative is nothing but the adherent

“translation”, in formal terms, of the three new

gnoseological concepts: Generative Causality,

Adherent Logic, Ordinal Relationships.

Such a new derivative was termed as “incipient”

(or prior derivative) because it describes the

processes in their generating activity or, preferably,

because it focuses on their pertinent outputs in their

specific act of being born. Its mathematical

definition is substantially based on the reverse

priority of the order of the three elements that

constitute the traditional definition:

()

→Δ 0

lim

(5)

that is: i) the concept of function (which is assumed

to be a primary concept); ii) the incremental ratio (of

the supposedly known function); iii) the operation of

limit (referred to the result of the previous two

steps). It is thus defined as follows (for further

details see also (Giannantoni, 2001a, 2002)):

() ()

Limtf

∼

→Δ

∼

⎟

⎠

⎞

⎜

⎝

⎛

−

∼

00:

(6)

where: i) the symbol

∼

Lim now represents a sort of

“window” or “threshold” (= Limen in Latin), from

which we observe and describe the considered

phenomenon, whereas

∼

→Δ 00:t indicates not

only the initial time of our registration, but also the

proper “origin” (in its etymological sense) of

something new which is being born; ii) the

“operator”

∼

registers the variation of the property

(

)

analyzed, not only in terms of quantity, but

also, and especially, in terms of Quality (as indicated

by the symbol “tilde” specifically adopted); iii) thus

the ratio (

∼

−1

) indicates not only a quantitative

variation in time, but both the variation in Quality

and quantity. That is, the Generativity of the

considered process or, in other terms, the output

“excess” (per unit time) characterized by both its

Ordinality and its related cardinality; iv) the

sequence of symbols in Eq. (6) is consequently

interpreted according to a direct priority (from left to

right); v) the sequence is also interpreted as a

generative inter-action (represented by the symbol

“

”) between the three considered concepts; vi) the

definition is valid for any fractional number

q . This

enables us to represent the three basic processes

(Co-production, Inter-action, Feed-back) in terms of

fractional derivatives of order 1/2, 2, and {2/2}

respectively. In such a case the order of derivation is

termed as Ordinality, because the corresponding

resulting “functions” (“binary”, “duet”, and “binary-

duet” functions, respectively) are structured in such

a way as to show an “excess” of Information, which

is never ever reducible to its sole phenomenological

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premises or to our traditional mental categories

(Giannantoni, 2004b).

On the basis of such a definition, by always

referring to a generic function structured in the

exponential form

)(t

(for the same reason

previously specified), the incipient derivative of

order

n is given by

)()(

)]([)(

tntn

ete

αα

⋅=

∼

(7)

where

)(t

represents the first order incipient

derivative of the function

)(t

. The symbol of a

little circle adopted to denote the incipient derivative

was evidently chosen in analogy to classical

Newton’s “dot” notation, which usually indicates a

first-order derivative.

The different symbology is here justified by the

fact that the former should now remind us the

conceptual difference between the incipient

derivative and the traditional one.

In fact, even if

)(t

and )(t



coincide from a

pure cardinal point of view, they are, on the

contrary, radically different from a Generative point

of view. The former in fact represents the specific

exit of a Generative Process, whereas the latter is

always understood as the result of a necessary

process (thought of as being a “mechanism” or a set

of “mechanisms”).

Such a quantitative coincidence, however, is

strictly valid only for

1=n . The right hand side of

Eq. (7), in fact, reveals an extremely important

property: a sort of “persistence of form”, which is

even more marked when the derivative is of

fractional order (

nm / ). This is precisely because it

represents an “adherent” consequence of a

Generative Process, characterized by specific

generation modalities. In other words, any

“generating process” (modeled by the left hand side

of Eq. (7)) gives origin to an output which

corresponds to a multiple structure functions

(multiple “binary” functions or multiple “duet”

functions (or both)), characterized by the Ordinality

)/( nm and described by the right hand side of Eq.

(7). (Giannantoni, 2006, 2007b). These functions are

similar to harmonic evolutions always in

“resonance” (as in a “musical chord”) with the

original function and at the same time with each

other, and they reach their maximum harmony in the

case of a perfect Ordinal Feed-back

}/{ nn . (ib.)

Such a more general modeling capacity of

incipient derivatives, associated with the afore-

mentioned property that any Ordinal dynamic model

always presents a solution (at least) in a closed form

(Giannantoni 2007a, ch. 5), confers to the Incipient

Differential Calculus much wider potentialities with

respect to the Traditional Differential Calculus, both

of integer and fractional order (ib.). This is also

confirmed by the fact that such a new mathematical

approach not only led us to the solution of the

famous “Three-body Problem” (ib.), but also paves

the way to the solution to the well-known “Three-

good Problem” which, on the other hand, remains

still unsolved in Neo-Classical Economics.

PROTEIN FOLDING, MOLECULAR DOCKING, DRUG DESIGN - The Role of the Derivative "Drift" in Complex

Systems Dynamics

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