Foresight Nanotech Institute Logo
Image of nano


Possible Principle of Controlling the
Self-Assembly of Biomolecular Materials

This is a draft paper for a talk at the
Fifth Foresight Conference on Molecular Nanotechnology.
The final version has been submitted
for publication in the special Conference issue of
Nanotechnology.

This page uses the HTML <sup> and <sub> conventions for superscripts and subscripts. If "103" looks the same as "103" then your browser does not support superscripts. If "xi" looks the same as "xi" then your browser does not support subscripts. Failure to support superscripts or subscripts can lead to confusion in the following text, particularly in interpreting exponents.


Abstract

One reports computational study aimed at possible principle of controlling the biomolecular self-assembly whose course may be specified by features characterizing changes in the patterns of expanding spatial regions in which molecular components exhibit appearance of required properties. Diversity in ways of the expansion influenced by stochastic factors is simulated by appropriate random process constituted of a number of independent (parallel) realizations of covering nodes of the hexagonal grid. For modelling the finite-size effects resulting from contribution of supramolecular structures to the expansion, the information about covering a node is transmitted: between members of a pair of neighbour nodes and due to conditional displacement of this pair (actually, information about organization of two nodes into a pair is displaced between grid nodes). The state at each stage is represented by the pattern constructed so that it is as close as possible, after situation in space and measure, to all the various sets being the process state realizations at this stage; this is mean measure set (MMS). The MMS evolution pattern enables us to pursue features of generating the collective effect resulting from the process realizations. We have parametrized conditions in which two subsequent rapid strong increments in the MMS area covered exhibit qualitative change in form, from sedimentation-like increment to percolation-like one (the events are separated by the resident time). We discuss conditions for using the process parameter to control appearance of the qualitative change and try to suggest actual methods which would be used to control the self-assembly corresponding to the simulations.

1. Introduction

Biomolecular materials and structures are those whose molecular-level properties are abstracted from biology and are structured or processed in a way that is characteristic of biological materials, but they are not necessarily of biological origin (NAS 1996). A key feature of biomolecular materials is their ability to undergo self-assembly that is coordinated action of independent objects or entities under influence of factors that are not centralized; this action leads to formation of structures with required properties. The self-assembly is one of strategies used in biology for the development of complex, functional structures and can incorporate them directly as components in the final systems which can be relatively defect-free and self-healing (Whitesides 1991,1995). One complex system of living Nature and the other belonging to unanimated world of mechanics have inspired us to suggest a class of biomolecular materials and structures whose self-assembly would be controlled by appropriate change in the ambient conditions of the process.

1.1 Two inspiring examples

A natural example appears to be provided by the ensemble composed of thousands of blood platelets experiencing independently transformations just after their adhesion to an artificial surface. The platelets are then situated on the surface within a single layer and spread in close proximity one to another. In those conditions, redistribution of certain transmembrane proteins on the flow side of the platelets' membranes results in collective effect that is facilitating the anchorage of the self-assembling fibrin net, forming the second platelet layer and thus further initiation of the thrombus. Displacements of the transmembrane proteins are associated with the platelet cytoskeleton remodeling (Mosher 1993, Smith et al 1992) and, indirectly, with membrane skeleton remodeling (Behnke and Bray 1988). Groups of the proteins tethered (even temporarily) by the cytoskeleton fragments (Sixma et al 1989) may contribute to the collective effect of the transmembrane proteins redistribution. The collective effect generation mechanism can be investigated using a model that requires considering the displacements of not tethered proteins or groups of tethered transmembrane proteins as point objects or finite-size entities, respectively. Displacements of the finite-size entities and the point objects associated with their origination contribute to the collective effect. Manifestations of this contribution represented in certain averaged description of the protein redistribution and referred to all platelets of the ensemble are identified as an example of finite-size effects. This representation at each stage of the platelet ensemble evolution may be realized by constructing a single pattern being as close as possible to all the patterns representing the transmembrane protein redistribution on membranes of all the platelets of the ensemble at the stage. The sequence of those single patterns corresponding to subsequent stages of the platelet ensemble evolution would be employed to pursue the collective effect generation.

The example from the world of mechanics is given by turbulent wall-bounded flow. Results of previous research on turbulent transport (Kozlowski 1996) have suggested that features of the collective effect generation may be controlled by appropriate change in ambient conditions of the process for systems exhibiting the finite-size effects. They have shown that appearance of the finite-size effects in a complex system admits regime in which the system may appear in either of the alternative states in ambient conditions unchanged. These conditions have been parametrized and it has appeared that the regime may occur only for certain intervals of the control parameter. Change in the conditions, specified by change in value of the control parameter such that it exits the interval mentioned implies jump in averaged values of the structural characteristic of the system state which indicates switch to the unambiguous regime. Values of the structural characteristics specifying the new state of the system differ from those specifying state of the system on the other side of the control parameter interval. Those structural characteristics represent average, collective effect of action of an ensemble of various complex finite-size organized fluid entities contributing to transport processes within the system. It has been found that neglecting the finite-size effects in the simulations of the system behavior implies absence of the characteristic features associated with change in value of the control parameter. This has suggested that features of collective effect generation may be controlled by appropriate change of ambient conditions for systems exhibiting the finite-size effects. On the other hand, direct influence on phenomena contributing to the finite-size effects would be a way of controlling the self-assembly of the biomolecular materials also.

1.2 A class of materials considered

Association of the revealed opportunity with the natural example has inspired us to distinguish a class of the biomolecular materials self-assembling from systems specified by the following features assumed:

  • They are composed in a form of the ensemble of many complex subsystems of the same kind (but they are not the same) that experience independently transformations under influence of stochastic factors in the same ambient conditions. Composition of the initial system (a molecular medium with suspended nanostructures or other ingredients) would permit for originating of a large number (incomparable, however, to the Avogadro number) of the subsystems that will further develop due to various chemical reactions.
  • The course of transformations of each complex subsystem can be specified by features characterizing changes in the pattern of expanding spatial regions in which certain components of the subsystem exhibit appearance of required properties.
  • Transformations of all the subsystems can result in a collective effect generated by whole the ensemble. Appearance of the collective effect can indicate finish or a discrete step in the self-assembling.
  • Expansion within each subsystem or within the vast majority of the subsystems can be considered of as resulting also from contribution of certain finite-size structures able to transmit information portion by themselves (between their ends) and due to effective finite displacement of those structures. Manifestations of this contribution which are represented in certain averaged description of the whole system transformation are identified as finite-size effects.

This class encompasses wide spectrum of biomolecular materials and structures including the future biomolecular membranes being much simpler than natural cell membranes and performing their selected functions (e.g. NAS 1996). The finite-size effects would result from synthesis and action of mechanically-interlocked molecules (like those reported by Amabilino and Stoddart (1995) which would contribute to the expansion process. On the other hand, extraordinary properties of carbon nanotubes pointed out by Globus (1997) make them attractive component of biomolecular materials or structures exhibiting finite-size effects in course of their self-assembly. It is worth considering also the opportunity for application of the self-replicating positional devices ,like assemblers proposed by Merkle (1996), for forcing occurrence of these effects under programmatic control. The self-assembly on surfaces seems to be here good candidate for performing the control in the way of changing the ambient conditions of the process: Actually there are known examples were self-assembly is function of temperature (MM 1996) ; there are also grounds for believe that encouraging the self-assembling material formation in particular regions of the surface may be possible in the way of the proper surface modifications (Goddard et al 1996).

The revealed opportunities would be used to control the self-assembly by varying certain parameters specifying ambient conditions of the process. Candidates for those parameters may be suggested basing on results of appropriately parametrized computer simulations of a model complex system possessing features of the real systems alluded to previously. In this paper we propose that simulation , parametrize it and perform variant computations revealing role of the particular parameters in the collective effect generation. This enables us to make certain suggestions concerning actual methods which would be used to control the self-assembly corresponding to the simulations.

2. Simulation: assumptions and method

The expansion of spatial regions in which molecular components exhibit appearance of certain required properties can be simulated of as resulting from an appropriately constructed process of information transmission in course of covering certain spatial domain assumed. Diversity in ways of the expansion realization influenced by stochastic factors is simulated by random expansion process (REP) whose states at subsequent stages of its development are finite random sets (FRS) being set multiplicity of realizations that have evolved independently (in parallel) within the same spatial domain. The state at each stage is represented by the pattern constructed so that it is as close as possible, after measure and situation in space , to all the various realizations at the stage. This is mean measure set (MMS) characterizing mean expected form of the REP and it is determined accordingly to method proposed by Vorobiev (1984). Results of the study on mechanisms contributing to appearance of changes in the MMS features will provide representative information on generation of the collective effect corresponding to final stage or an intermediate discrete step of the self-assembly. Let us note , herein, that sets constituting the multiplicity FRS are not necessarily to correspond , one to one, to the complex subsystems of the real system simulated ; a number of these sets is to be only as large as to provide opportunity for adequate representation of the diversity in expansion realizations for the complex subsystems. This number is to be established specifically for a particular process simulated. That manifestation of the finite-size effects by the complex system appears to provide opportunity for controlling the collective effect generation by changing ambient conditions has resulted from previous study (Kozlowski 1996) where application of discrete space for simulations enabled us to model the finite-size effects appropriately. In virtue of this result, the space with discrete topology (this is not Zariski space) has been chosen here for the simulations. Reference to the natural systems expanding on a surface implies specification of the discrete space so that its elements coincide with nodes of a regular grid on a plane.

We use the space S ={xi , i=1,2,...,Ns} whose finite number of Ns points are nodes of the regular hexagonal grid (here, we apply Ns=145x124 so that the grid covers the square 1x1). Each node is center of the regular hexagon that filled with certain color illustrates inclusion of the node to the subset of the S which is identified by this color. This grid has largest density of nodes among all the grids for which the same value of the smallest distance between the nodes is given (Hilbert and Cohn-Vossen 1932). For a discrete space (generally , for compact spaces) , a feature peculiar to each node which specifies situation of all its nearest neighbours is characteristic for whole the space as well (Janich 1984) . This implies superimposition of the form of a regular hexagon on the patterns resulting from processes of covering the nodes in course of simulating the expansion on the grid. It has appeared possible to avoid this effect in the way of specific randomization of distribution of realizations of the grid nodes' vicinities composed of such nodes being nearest neighbours of each node which are admitted for covering. Eventually, the local feature of the grid structural regularity has not been copied to local rules of covering the nodes ( rationale of this method will have been presented elsewhere). Normal covering is hereafter referred to the case for which all nearest neighbours of each node can be potentially covered due to signal transmission from the node surrounded. The term randomized covering indicates that choice which subset of a node nearest vicinity could be potentially covered due to signal transmission from the node surrounded has been done randomly for each node of the grid. Each of the two space covering specifications performed prior to the simulation could be referred to respective limit form of a prepatterned surface at which the self-assembly corresponding to the simulation might be realized. Therefore we report here results obtained with both the specifications.

We have incorporated the idea of discrete displacements employed to model the finite-size effects for certain example of complex systems (Kozlowski 1996) into the simplest variant of the Markov process proposed for modeling random displacements (Vorobiev 1984). In result we have obtained the modified version of the Markov process of independent realizations. State of this REP at a stage IT is given by a finite number NK of its realizations Ai accomplished independently one from another . This state is denoted as KIT ={Ai , i=1,2,...,NK}. With the purpose to sketch the way of constructing the KIT+1 from the KIT we note series of neighbourhoods Gz , Gzz with: node 'z' being element of the Ai at stage IT for each 'i' (i.e. the 'z' is also the element of KIT) ; node 'zz' being an element of the neighbourhood Gz ,and Hz being set-theoretical union of the neighbourhoods Gzz for all elements 'zz' of the Gz. This enables us to construct a vector HK such that each of the NK its elements indicated by index i=1,2,..., NK is set-theoretical union composed of sets Hz for all elements 'z' of the realization Ai at the stage IT. For each i=1,2,...,NK , realization (Ai)IT+1 of the KIT+1 is constructed as set-theoretical union of sets (Rx)i resulting from local expansions for all nodes 'x' belonging to respective element with index 'i' of the vector HK. Explanation of the (Rx)i requires presentation of a set {(Ai)IT.and.Gy} being common part of the set (Ai)IT and the neighbourhood Gy of the node 'y' being element of the neighbourhood Gx of the node 'x' in respect to which the (Rx)i is accomplished. For each element 'y' of the Gx, replacement of covering of each node 'xx' from the set {(Ai)IT.and.Gy} with covering of such node 'yy' from the neighbourhood Gxx of the 'xx' that the 'yy' is situated in respect to 'xx' as the 'y' is situated in respect to the node 'x' results in set of the nodes 'yy' which may constitute the (Rx)i. The transmission of information about organization of two nodes into a pair , (x,y).to.(xx,yy) identified as effective transposition of the pair (x,y) complements information portion about covering of a node which can be transmitted by the pair between its nodes , (x.to.y) or (xx.to.yy) . This is the way in which the finite size effects alluded to previously are modelled within frame of the REP. In virtue of the explanations given above for the Hz , the 'x' is an element of the Gzz and, generally, the situation y=x is valid also. The node 'yy' is covered , i.e. it is an element of the (Rx)i when certain probability Py(x) assigned for the simulation to a node 'y', being an element of the seven node vicinity ( SNV) of the node 'x', is not less than a random value f(y,i). The 'f' is generated , specifically for the 'y' and REP realization number 'i' , as an element of the random vector obtained with the help of available random number generator RANMAR (James 1990). The values Py(x) are elements of the vector of seven probabilities ( VSP) assigned to the hexagon of seven nodes with the 'x' in its center (i.e. to the SNV of the 'x' ). Here, it has been assumed that VSP is the same for all the nodes of the grid (a sum of the seven probabilities of the VSP is within the interval (0,1]).

Eventually, the REP is simulated as Markov process of covering nodes of a regular hexagonal grid and so that information about covering a node is transmitted : between members of a pair of neighbour nodes and due to conditional displacement of this pair. Displacement of information about organization of two nodes into a pair identifies the effective displacement of the pair of neighbour nodes. This may be considered as generalization of the discrete displacement method (DDM) developed previously (Kozlowski 1996) for modelling similar finite-size effects peculiar to turbulent transport and verified for example of the developed turbulent pipe flow.

Let us recall that computed realizations Ai , i=1,2,...,NK of the K are employed here to simulate specific diversity in the process realizations whose mean expected form is characterized by the mean measure sets at each stage IT. They are closest after their situation and measure to all the various realizations Ai at the stage considered . This multiplicity has unique , single set representation at the stage; this is also called mean measure set - MMS and has been determined using the method proposed by Vorobiev (1984). One searches also for the set of nodes covered in each realization Ai of the K ; this set denoted as ker(K) is called core of the K. One finds also a set of nodes covered at least in one realization of the K ; this set denoted as bas(K) is called base of the K. These characteristics can be found at each stage IT of the REP. Vanishing of the core for each IT>1 has appeared to be convenient criterion of establishing the sufficiently large value NK for the REP simulated here (we have found NK=100).

3. Results and discussion

We have searched for conditions of the expansion simulation which would result in the sequence of the MMS patterns with features revealing qualitative change in the MMS increment associated with jumps in value of the countable measure Imms of the MMS (just , number of nodes of the MMS). Tentative computations have suggested observing that situation after breaking symmetry of the expansion in the space. Therefore , for focusing attention just on the phenomenon searched for we have constructed the vector of seven probabilities (VSP) admitting expansion in one halfplane only. This has been done by specifying the VSP by a single parameter P25:=P2=P3 =P4 =P5 with P7= 0 and P1=P6 = 0 . The scheme

                               P2
                         P1=0        P3
                               P7=0
                         P6=0        P4
                               P5

of the VSP and its parametrization with the P25 , the same for all nodes of the S , fill in the requirement. Here, the value P7= 0 assumed excludes the possibility of coincidence of the node 'y' with 'x' and thus only pairs of neighbour nodes contribute to the expansion. Moreover, one admits only two possible instances of displacements of the pairs: In the first instance, no node of the pair displaced can occupy position of any node of the pair before the displacement (i.e. 'xx' can coincide neither with 'x' nor with 'y'). Lack of the displacement corresponds to the second instance (then 'xx' coincides with 'x' whereas 'yy' coincides with 'y'). Accordingly, it has been assumed that node 'zz' cannot coincide with the node 'z' ,and node 'x' cannot coincide with the node 'zz'. The process has been started from two straight separated chains of covered nodes to improve reliability of the results. Their vertical situation has been chosen in accordance with the VSP scheme. They are assigned to IT=0 so that bas(K0)=ker(K0)=MMS(K0) , although the physical start occurs at IT=1 from the K1 represented by the MMS(K1) whose pattern has a form of the two chains of small islands. We have constrained number of the time steps of the REP simulations with the purpose either to avoid superimposition of part of the bas(KIT) developing from one chain onto this expanding from the second one or to prevent the bas(KIT) from approaching the boundary of the area assigned to the S.

The research performed for randomized space covering specification has revealed an interval of the parameter, 0.1<P25<0.17 for which simulation of the REP results in a series of few jumps in increment of the MMS measure Imms preceded and followed by resident times with very weak specific variation of the Imms (see figure 1 [b/w] ). We have assigned numbers IA to subsequent stages at which the jumps occur so that IT(IA+1)>IT(IA)+1 and the resident time defined as number of stages that separates the subsequent jumps is expressed by equation, t(IA)=IT(IA)-IT(IA-1) with IA>0. Observation of two dimensional MMS patterns specified by the jumps in Imms has enabled us to find the first one {MMS(KIA)\MMS(KIA-1)} exhibiting the qualitative difference from the previous MMS increments (the backslash denotes the set-theoretical difference, e.g. for sets A, B : {A\B}={A.without.[elements of B]}). Then nodes constituting the increment {MMS(KIA)\MMS(KIA-1)} appear to be distributed throughout the MMS(KIA-1) , contrary to previous cases where area of the MMS increment resulting from the jump is just stuck to right of the MMS pattern preceding this jump. This qualitative change resembles switch from sedimentation-like form of the increment to percolation-like one. Appearance of this first specific jump in Imms can be observed for the second residence time t(IA=2) ; see figure 1 [b/w] and sequence of the patterns {MMS(KIA)\MMS(KIA-1)} depicted in series starting with figure 2 to figure 7 [b/w] that have been obtained for P25=0.16.

For analysing the organization process that may be recognized in the sequence of the MMS patterns we have constructed the entropy characteristic H. To this aim the notion of set theoretical ruler with length Sn has been proposed and the notion of gliding hexagon - GH has been used. The GH consists of seven nodes coinciding with the grid nodes so that a node indicating location of the GH is surrounded by its six nearest neighbours. The Sn is equal to the number of such locations of the GH within the smallest rectangle enveloping the bas(KIT) that for each of these locations there is a fixed number 'n' of nodes covered constituting a single chain within the GH (the 'n' is not less than 2 and not more than 7). Let the number of possible locations of the GH within that rectangle be denoted by MGH . One may consider increment in informative entropy of finding a sample node covered - SNC within the domain MGH which results from appearance of the SNC within single chains of 'n' covered nodes at the Sn locations of the GH. This value is denoted as Hn for the one number 'n' fixed whereas H identifies this value for all the 'n' being elements of the set {2,3,4,5,6,7}. That the SNC may appear within the chain just mentioned affords the opportunity for using the ruler Sn to find the SNC within the domain MGH with the probability Sn/MGH. The values Sn are determined elementary and the characteristics referred to the information entropy are expressed accordingly to general formulae known:
the H is expressed as a sum of Hn after n=2 to 7 with Hn= -( Sn/MGH)ln(Sn/MGH).
The characteristics just expressed are denoted as Hmms or (Hn)mms , respectively when they are determined for the MMS. Mathematical expectations Hav or (Hn)av of the H or Hn determined for all the realizations Ai , i=1,2,...,NK of the K have been presented here for comparison with these referred to the MMS. Alternations of the Imms within the 't' correspond to variation of the MMS pattern and its reference to bas(K) that can be seen in distribution of the Hmms against IT (see figure 8 [b/w] ). The Hmms diminishes for the residence time preceding the jump in the Imms. This jump is accompanied by the one in the Hmms to the Hmms local maximum from which this value diminishes for the residence time preceding the subsequent jump. Here, first act of the MMS increment, whose pattern shows the new features alluded to previously, is preceded by the residence time t(IA=2) extending just to left of the stage IT=IT(IA=2)=42 at which both the values Hmms and Hav are maximal. Additional simulations performed without regard to the finite size effects have not permitted for selecting conditions in which the patterns of the MMS increments show that feature.

Results for the normal space covering specification are depicted in figures 9 , 10 [b/w] (characteristics) and figures 11 , 12 [b/w] (MMS increments). They have been obtained for simplified version of the REP , for which the HK is replaced by the KIT i.e. node 'x' is taken as element of the KIT only (see section 2 for references). We have simulated this version of the REP also for the randomized space covering specification and found that the qualitative change in the MMS increment can be observed for the control parameter interval, 0.18<P25<0.23. On the other hand, applying the normal space covering specification results in reduction of that interval to the one extending less than 0.001 on both sides of the P25=0.104. Then, however, the switch in the form of the MMS increment is more distinct than for simulation with the randomized space covering specification. This seems to suggest that an intermediate space covering specification, between utterly randomized and the normal one, would afford opportunity for controlling features of the qualitative change in the MMS increment when supremum of the Hmms is achieved.

We were not able to achieve the qualitative change in the MMS increment for the symmetric expansion i.e. with the VSP such that PJ=P/6 with 0 < P < 1 for each J=1,2,...,6 ; those simulations were started either from chains of nodes or sets constituting filled rectangles situated in few ways in the space. This seems to suggest that asymmetry in local expansion may be the necessary condition here. The simulations performed with the VSP realizing this condition in several ways suggest that , for the aim pursued, the VSP should not prefer a single direction (i.e. only one of the six for this grid) to strongly but be close to the VSP used for simulations reported in this paper.

4.Conclusions and suggestions

The results analysed seem to reveal potential opportunity for controlling generation of the collective effect resulting from the biomolecular self-assembly for a class of the systems considered here (see Section 1.2). This would be done by changing ambient conditions of the process.

Realizing of the control would require imposition of a factor resulting in spatial asymmetry of the two-dimensional expansion , the same for all the expanding complex subsystems constituting the relevant real system in which self-assembly takes place. This might be done e.g. by an acoustic action, imposed shear, an electric field with an appropriate characteristic or concentration gradient of a chemical component in the initial environment. Application of those factors should be associated with properties peculiar to the self-assembling material or structure with the purpose to realize the asymmetry condition in the way corresponding to this used for the simulations reported (see section 3). Parameter specifying the asymmetry resulting from the particular value of that factor would correspond (qualitatively) to the parameter P25 . On the other hand , degree of ordering of the prepatterned surface on which the self-assembly is realized would correspond to an intermediate space covering specification in the simulation reported (see section 2 and 3). Then, selecting the value of the imposed factor together with the degree of ordering of the prepatterned surface would make possible controlling features of the collective effect resulting from various realizations of the expansion in all the complex subsystems of the ensemble constituting the system.

Comparison of the results obtained from simulation performed at normal and randomized space covering specification suggests that required selection of a value of the factor being imposed may appear to be hardly possible for the perfectly ordered prepatterned surface (then the acceptable value might be found within very short interval only). That qualitative change in form of the MMS increment is observed at supremum of the Hmms in course of the expansion simulated may suggest single occurrence of the corresponding event in the relevant real system.

The noted role of the finite-size effects makes worth considering the opportunity for forcing occurrence of these effects under programmatic control in particular regions of the surface.

Acknowledgments

Major part of the computations has been performed by using computational facilities in Institute of Biophysics and Biochemistry , PAS , Warsaw, Poland. A part of the results has been obtained by using resources of the Interdisciplinary Centre for Mathematical and Computational Modelling at Warsaw University, Poland.

References


to figures presented in black/white or gray-scale

Points corresponding to the MMS measure at
subsequent stages of the REP evolution assemble into graph having formof stairs
with several horizontal steps
Figure 1. Evolution of the MMS measure, P25=0.16 ; random space covering specification [b/w] (see section 3)

The form of the graph representing
entropiccharacteristic for the MMS resembles a saw with unequal teeth and
maximumat IT=42 (the largest tooth). At the IT=42 there is also the maximum
ofthe mathematical expectation of the ent

ropic characteristic determinedafter all realizations at the stage. The MMS
entropic characteristicjumps up rapidly at the stage IT together with the jump
up in the MMS measureand subsequently diminishes whereas the MMS measure
remains almost constanttill

 the next jump. More detailed discussion isgiven in section 3.\

Figure 8. Evolution of the entropic characteristics, P25=0.16 ; random space covering specification [b/w] (see section 3)

(to figures representing results with normal space covering specification [b/w]

The series starting with figure 2 to figure
7depicts the sequence of patterns representing increments in the area coveredby
the MMS between the subsequent jumps in the MMS measure and the MMS
entropiccharacteristic (one increm

ent in each figure). It is clearly seen qualitativedifference between pattern
of the increment from IA=1 to IA=2 and the previousincrements (IT(IA=2)=42 and
corresponds to maximum of the MMS entropiccharacteristic , IT(IA=1)=36 and
IT(IA=0)=30). This is t

he qualitative changedescribed in section 3

Figure 2. MMS increment (indicated by yellow color in Figures 2 to 7), P25=0.16 [b/w] (see section 3) ;

Figures 2 to 7 represent results obtained with random space covering specification ; scroll to compare

see description to figure 2

Figure 3. MMS increment, P25=0.16 [b/w]

see description to figure 2

Figure 4. MMS increment , P25=0.16 [b/w]

see description to figure 2

Figure 5. MMS increment from IA=0 to IA=1 , P25=0.16 [b/w] (see section3 for explanations)

see description to figure 2

Figure 6. MMS increment from IA=1 to IA=2 , P25=0.16 [b/w] (see section 3 for explanations)

see description to figure 2

Figure 7. MMS increment from IA=2 to IA=3 , P25=0.16 [b/w] (see section 3 for explanations)

(to Figure 8 [b/w] back to Figure 1) [b/w]

Points corresponding to the MMS measure
atsubsequent stages of the REP evolution assemble into graph having
formresembling stairs with horizontal steps (only few points of the secondstep
are shown). The first horizontal step e

xtends from IT=33 (it is IA=1)to IT=81 (it is IA=2). It is preceded by a
parabolic pattern of the MMS measurepermanent growth so that IT=12
corresponding to IA=0 has been found in graphsof the MMS entropic
characteristic and ratio of this value to the mat

hematicalexpectation of the entropic characteristic determined after all the
realizationsat a stage

Figure 9 Evolution of the MMS measure , P25=0.104 ; normal space covering specification [b/w]

Evolution of the MMS entropic characteristic
showsgrowth to saturation at IT=12 (it is IA=0) , next weak variation within
ahorizontal layer and strong jump up at IT=33 (it is IA=1). Further,
thischaracteristic smoothly dimini

shes to value only slightly higher than thisbefore previous jump and jumps up
as strong as previously achieving valueslightly higher than this at IT=33 (this
happens at IT=81 i.e. IA=2).The mathematical expectation of the entropic
characteristic determine

d after allthe realizations at a stage grows till IT=33and further varies not
remarkably

Figure 10. Evolution of the entropic characteristics , P25=0.104 ; normal space covering specification [b/w] (see section 3)

(to figures representing results obtained with random space covering specification )[b/w]

The figure 11 and figure 12depict the sequence of
patterns representing increments in the area coveredby the MMS between the
subsequent jumps in the MMS measure and the MMS entropiccharacteristic (one
increment in each figure

). It is clearly seen qualitativedifference between pattern of the increment
from IA=1 to IA=2 and the previousincrement ( IT(IA=2)=81 and corresponds to
maximum of the MMS entropiccharacteristic ; IT(IA=1)=33 and IT(IA=0)=12). This
is the qualitative cha

ngedescribed in section 3. This is more evident than for simulations with
therandom space covering specification.

Figure 11. MMS increment from IA=0 to IA=1 (indicated by yellow color in figures 11 and 12), P25=0.104 ; [b/w]

patterns depicted in figures 11 and 12 have been obtained with normal space covering specification

see description of figure11

Figure 12. MMS increment from IA=1 to IA=2 , P25=0.104 ; normal space covering specification [b/w] (see section 3)

(to figures representing results for random space covering specification )


to figures presented in colors

Points corresponding to the MMS measure at
subsequent stages of the REP evolution assemble into graph having formof stairs
with several horizontal steps
Figure 1. Evolution of the MMS measure, P25=0.16 ; random space covering specification [cr] (see section 3)

The form of the graph representing
entropiccharacteristic for the MMS resembles a saw with unequal teeth and
maximumat IT=42 (the largest tooth). At the IT=42 there is also the maximum
ofthe mathematical expectation of the entr

opic characteristic determinedafter all realizations at the stage. The MMS
entropic characteristicjumps up rapidly at the stage IT together with the jump
up in the MMS measureand subsequently diminishes whereas the MMS measure
remains almost constanttillt

he next jump. More detailed discussion isgiven in section 3.\

Figure 8. Evolution of the entropic characteristics, P25=0.16 ; random space covering specification [cr] (see section 3)

(to figures representing results with normal space covering specification [cr]

The series starting with figure 2 to figure
7depicts the sequence of patterns representing increments in the area coveredby
the MMS between the subsequent jumps in the MMS measure and the MMS
entropiccharacteristic (one increme

nt in each figure). It is clearly seen qualitativedifference between pattern of
the increment from IA=1 to IA=2 and the previousincrements (IT(IA=2)=42 and
corresponds to maximum of the MMS entropiccharacteristic , IT(IA=1)=36 and
IT(IA=0)=30). This is th

e qualitative changedescribed in section 3

Figure 2. MMS increment (indicated by yellow color in figures 2 to 7), P25=0.16 [cr] (see section 3) ;

Figures 2 to 7 represent results obtained with random space covering specification ; scroll to compare

see description to figure 2

Figure 3. MMS increment, P25=0.16 [cr]

see description to figure 2

Figure 4. MMS increment , P25=0.16 [cr]

see description to figure 2

Figure 5. MMS increment from IA=0 to IA=1 , P25=0.16 [cr] (see section3 for explanations)

see description to figure 2

Figure 6. MMS increment from IA=1 to IA=2 , P25=0.16 [cr] (see section 3 for explanations)

see description to figure 2

Figure 7. MMS increment from IA=2 to IA=3 , P25=0.16 [cr] (see section 3 for explanations)

(to Figure 8 [cr] back to Figure 1 [cr])

Points corresponding to the MMS measure
atsubsequent stages of the REP evolution assemble into graph having
formresembling stairs with horizontal steps (only few points of the secondstep
are shown). The first horizontal step ex

tends from IT=33 (it is IA=1)to IT=81 (it is IA=2). It is preceded by a
parabolic pattern of the MMS measurepermanent growth so that IT=12
corresponding to IA=0 has been found in graphsof the MMS entropic
characteristic and ratio of this value to the math

ematicalexpectation of the entropic characteristic determined after all the
realizationsat a stage

Figure 9 Evolution of the MMS measure , P25=0.104 ; normal space covering specification [cr]

Evolution of the MMS entropic characteristic
showsgrowth to saturation at IT=12 (it is IA=0) , next weak variation within
ahorizontal layer and strong jump up at IT=33 (it is IA=1). Further,
thischaracteristic smoothly diminis

hes to value only slightly higher than thisbefore previous jump and jumps up as
strong as previously achieving valueslightly higher than this at IT=33 (this
happens at IT=81 i.e. IA=2).The mathematical expectation of the entropic
characteristic determined

 after allthe realizations at a stage grows till IT=33and further varies not
remarkably

Figure 10. Evolution of the entropic characteristics , P25=0.104 ; normal space covering specification [cr] (see section 3)

(to figures representing results obtained with random space covering specification [cr] )

The figure 11 and figure 12depict the sequence of
patterns representing increments in the area coveredby the MMS between the
subsequent jumps in the MMS measure and the MMS entropiccharacteristic (one
increment in each figure)

. It is clearly seen qualitativedifference between pattern of the increment
from IA=1 to IA=2 and the previousincrement ( IT(IA=2)=81 and corresponds to
maximum of the MMS entropiccharacteristic ; IT(IA=1)=33 and IT(IA=0)=12). This
is the qualitative can

gedescribed in section 3. This is more evident than for simulations with
therandom space covering specification.

Figure 11. MMS increment from IA=0 to IA=1 (indicated in black in Figures 11 and 12), P25=0.104 ; [cr]

patterns depicted in Figures 11 and 12 have been obtained with normal space covering specification

see description of figure11

Figure 12. MMS increment from IA=1 to IA=2 , P25=0.104 ; normal space covering specification [cr] (see section 3)

(to figures representing results for random space covering specification ) [cr]



Donate Now

 

Foresight Programs

Join Now

 

Home About Foresight Blog News & Events Roadmap About Nanotechnology Resources Facebook Contact Privacy Policy

Foresight materials on the Web are ©1986–2014 Foresight Institute. All rights reserved. Legal Notices.

Web site development by Netconcepts. Email marketing by gravityMail. Maintained by James B. Lewis Enterprises.