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An information processing and body formation mechanism with its condition in development (Theory of Biomathematics and Its Applications X)

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(1)

An information

processing and body

formation

mechanism with

its

condition

in

development

MitsuoTakase

LINFOPSInc.

$6-21\cdot 1\cdot 503$OukurayamaKouhoku.ku Yokohama$222\cdot 0037$Japan

GZL03154@nity$\infty$m

Abstract.

Conditions and a mathematical mechanism for a body to be able to form according to the informationof genesin developmentwithoutmorphogenis$\infty$nsidered andproposed.In otherwords

the$\infty$nditions andthemethodgivethe methodto realize thebody usingthe$\infty$mpressedinformations.

usingthe memories ofgenes. One ofthe purposesof thisstudyis to make a mathematical modelon

PC for the formation simulation. The other is to find the wayhow for areal biological body to be formed.

1. Introduction.

Conditions and a mathematical mechanism for a body to be able to form aecording to the information ofgenesindevelopmentwithoutmorphogen is$\infty$nsidered andproposed. In otherwords

the$\infty$nditionsand the method give themethodtorealize thebodyusing the$\infty$mpressedinformations.

using the memories of genes. One ofthe purposesof this studyisto make amathematical model on PC for the formation simulation. The other is to find the way how fora realbiological body to be formed. Homeotic selector genesareknown to be used tomakethe bodyindevelopment(Ref.4

The$\infty$nditions for thebodyformation is that everypairofadjaoenttwo$\infty 11s$shouldhavea$s-lar$

expressivityofeachgene of all expressed genesin eachcell keeping slope$\infty$ntinuity.Theseareshown

insection3.

Cells with these conditionscanhavethefollowingabihties.

(1)If there is anexpressedgene in each cellbywhich the cell expresses the value ofapointofasine

curve

witha wave length, abig group of these$oe$]$k$canmake the sine curve alongthebody length

whichmeansthe grouphasastabilitywhentheyformthesine

curve

shape.Themultiplesinecurve

formations with differentwave lengths anddifferentphases canbe formedby havingthe different numberofpointsineachcycleofeach sinecurve.Thisisshowninsection4.

(2) Ifthree genes each of which gives an orthogonal sine cuuve are given, the $\infty$ndition with the

assumptionof theorthogonalcharacter of genes like sinecurvesinabodyshapeformation

can

have ability to make a body segmentation like what is done by gap gene, pairrule gene and

$segment-$pola$\dot{n}ty$gene(Ref.4).

The character of the mechanism shown hereseemsto be ableto haveability toreproduce alost part ofanorganlikeatail with any shapefromthe topoftheleftpartusing multiplesine and$\infty$sine

curveswithdifferentphases. This isalsoshownin section 6. All the mechanism shown here is based on the assumed character of orthogonahzation ofgenes whichis the main roleplayed prevalentlyin theinformationprocessingin the brain.

2. Thebackgroundtoshowwhya set ofcurves likesinecurves areused.

It isknown thattheactivationmechanism of genesisvery similarto that ofneural networks(Ref

4), moreover genes and neural networks have the mechanisms ofmutation and synapse plasticity respectively which change their memories. So it is thought that these two systems are in closely

(2)

In the formation of knowledge structure of neural networks, the basic character ofmemory compression works. Then in a hierarchical neural network, the formation of transformations, the productionofgeneral objectsand$\infty$ncreteobjectsfromlowerlayerstohigher layers

can

be done

ffif

1).A set ofsine

curves

canhavethecharactersof thetransformations like location transformation.

where the same objects with different locations are $re\infty$gnized as the same objects, and scale

transformation,where thesameobjects withdifferent scalesarerecognizedas thesameobjects

ffif.

2).Bythesetransformations,upperobjectscanbeexpressed byfewer informationseffectively. Ifasine

curve expresses some information $\infty$ncemed with many objects, it can be

said that the curve

contribute much to the memory$\infty$mpression. Orthogonalization in the existence ofmultiple

ceuves

like sinecurvesalso contribute tothememorycompression.

In thecaseof genes, peptidesorproteinsareexpressedeffectively by genes makinga hierarchical structure likepolypeptidechains,thepairof $\alpha$ helix and $\beta$sheet and $\beta\cdot\alpha^{-}\beta$ motif(Ref.4).

kingfrom these memorystructures,the existence ofthe character ofthe memorycompressionin theexpression bygenes

can

be inferred(Ref.3).

Fromthis background, amechanism where apartialsystemofabody like

an

organisexpressed byasetofcurvesisshown.

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3. Abouttheconditions and themechanism forgenesto express apartialsystem ofbodyusingcurves

likesinecurvesandto beabletodo$\infty$mputer simulation

When genes express a partial system of body using

curves

like sine curves, what kinds of

$\infty$nditions each gene or each set ofoells musthaveisshown here.

Fig. 1A shapeor acharacter distribution of

a

partial systemofabodyisexpressed by asetofgenes makingthe thepartialsystem$\infty$mectin$g$thepartsand their cells with the conditions shown in this

section.Atthebottomof thegraphsomeofthe partsofthecurveare shown. The meanings ofFig. 1areshownasfollows.

(1)It is assumed thatageneor aset of genes make thepartsofapartial system of thebody along thelongeraxisorthe shorter axis of thesystem.Eachpart$\infty$nsists ofoellswith$s-lar$expressed

charactersand$\infty$nditions shown below. Thelonger part$\infty$nsists ofmore oelk.

(2) So as to meet the followmg $\infty$nditions and align the parts with similar values keeping slope

$\infty$ntinuity, the set ofgenes put thenextpartat the topofthe lastpart making the system grow.

(3)Whenthe systemismade$ac\infty rd\dot{m}g$tothe following$\infty$nditions,thesystemisthoughtto havethe

biggestaffinityand thestabilitywhich makes the$\infty$mection among the oelk of thesystem strong.

Because all thepartsand the oells oftheparts

can

bejoinedmaking the totalaffinitythehighest.

[conditionsto$\infty$mectceks]

(1)Themostsimilartwopartswheretheexpressedcharacter value oftheoells ofonepartisneaoest to thatoftheother partareputadjaoently.

(2)The character values musthave slope$\infty$nhnuity.

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[Theexplanationof condition(2)]

Ifthere arethree sortsofcells

A

$B$ and$C$witha similar expressivity ofagene

where A hasthe highest,$C$hasthelowest and$B$hasthe average,thenthe partsarealignedin

the order$ofAB$and$C$

not being alignedlike in$A,$ $B$ and Abecause the expressivityof$B$

can

not be nearlythe average of

those ofthe adjacent two Acells. and this makes a little distortion. This means holding ofslope continuity

4.Formationofapartial systemofthe body

Under the $\infty$nditions and the mechanism shown in section

3, curves like sine curve can be expressedbyageneoraset of genes.

Thecasewhere two sine

curves are

expressed bytwogenesortwo setofgenesis shown in

Fig.2.1 Thetwosine

curves

make the

curve

shown inFig.2.2. Sogenes

can

express any

curve

bythe conditions and the mechanismusing multiplecurves(andmultiple setofgenes) making the affinity

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Center part ofthebody

Fig.2.1 Casewithtwo characters and two sets of genes for them

partofthe body

Fig.2.2 ${\rm Re}$shapeobtainedfromthe twocurvesin Fig2.1by algebraicsum

ThecurveofFig.2.2canexpress thedistributionofanyvalue.

So ifthe genes have the$\infty$nditionsand themechanism,genescannot onlyexpressanycurvebutalso

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[Examplesofthemeanings which thecurvein Fig.2.2canhave.]

(1)Widthdistributionalongthelongeraxisofapartialsystemofthe bodylikea tail (2)The distribution ofsomematerial inaninterstitialtissue

(3)Thedistribution of secretionabilityofanenzyme

5.Theeffect ofthe additionofcompetitionandthe necessityoftheslopecontinuity in theconditions The phenomena caused by positional value are shown in Fig 3. (Ref. 4) and can be partially explained by the $\infty$nditions and the mechanism in section 3, and by the addition of $\infty$mpetition

$\infty$nditioncanbeexplained.

The uppercase 1 of Fig. 3

can

be explained directly throughthe conditionsand the mechanism. But the lower case 2 of ofFig. 3 can not be explained directly, but by adding competition to the conditions thelowercase2ofofFig.3canbeexplained.

The competition is seen oftenin the dynamics of neural networks especially. The meaning of the competitionis thatastheresult of$\infty$mpetitionto connectanother parttotheterminalbeingbuiltnext

seeking thepartand its cellswith the biggest affinity, the partischosen.

Inthe lowercase2ofofFig. 3the upper partoftheleghassub-parts 1, 2, 3, 4,5,61inedin thisorder, and the lowerparthas $sub\cdot$parts4$.5.6.7$, 8, 9, 10linedin this order. $Sub\cdot$part7 ofthe upperpartof

thelegis connectedtosub-part4 ofthe lowerpartoftheleg.Sothesituation ofthe upperpartseeks8 oftheownpartsfortheelongation,butthis elongationcontradictstothe$sub\cdot$part4ofthe lower part

causingthecompetition.Soby addingthe conditionofcompetitiontothe conditions showninsection3, thiscasecanbe also explained.Butit is imagined that inusualmanycasestheconditionsshown in section 3 will beenough.

Evenifthe condition ofcompetitionisadded,the$\infty$ndition ofslopecontinuitycannot be removed.

Because cyclic patternsare seenoftenin the bodyasbeingable tobeexpressed bysinecurves, then therepeated

same

partscanbe connectedmutuallyanddirectly without slope continuity

6.Summaryofwhatthe$\infty$nditions andthe mechanismmean

Whatthe conditionsand themechanismmean aresummarizedasfollows.

(1) The conditions andthe mechanism canexpress not only any shape ofany partialsystem of the

body,butalso express any distribution of anyqualityin thesystem.

(2)By theconditionsandthemechanism,genescanhavethe possibihtytoexpress notonlythe partial

systembut alsothewholebody byfeweronescausingmemorycompression.Whenlikeasinecurve

some expression gives an information which is sharedby many cells, it $\infty$ntributes much to the

memorycompression.

(3) If theinformations memorizedby genesare made orthogonalandproduced throughanefficient

memorycompression,theconditionsand the mechanism giveamethodto realizeinto thebodyfrom theinformations.

(4) Even ifthetail ofabodyis cutoff, through the conditions andthe mechanism, the tailcanbe

reproduced$fi0om$theremained terminalthinking theconditionsandthemechanismlogically.

(5)By theaddition ofthe condition of$\infty$mpetition,behaviorslike those with positionalvaluescanbe

(7)

Fig.3Experimentsbylegsofa cockroach (FromMolecularbiologyofthe cell(Ref.4))

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Referenoes

1. Takase, M. (1998$\rangle$

Effect of feedback connections to memory compression and formation of knowledge structure based onHebb rule andinhibitorycells. Proc. Int. Conf NeuralInformation

Processing,981-986

2. Takase, M. (1994) Abihty to formtopologicaltransformations and data processing of

continuous pattern set in hierarchical neural network. Proc. Int. Conf Neural Information Processing,

1123.1128.

3. Takase, M (2005) Similarity in formation of knowledge structure between genes and neural networks and effect of memory $\infty$mpression. Proc. $8^{th}$ annual conference on

computational genomics.

Fig. 1 A shape or a character distribution of a partial system of a body is expressed by a set of genes making the the partial system $\infty$ mectin $g$ the parts and their cells with the conditions shown in this section
Fig. 2.2 ${\rm Re}$ shape obtained from the two curves in Fig 2.1 by algebraic sum

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