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An automatic control mechanism to ignite the immune system locally against a small cancer mass considering reliability (Theory of Biomathematics and Its Applications IX)

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(1)

An automatic controlmechanism to ignite theimmune systemlocally against a small

cancer mass

considering reliability

1.Introduction

Tumor-immunesysteminteractionis considered. Then it is known that$T$cellsplaya mainrole[2].

There should be two stepsforagroupof$T$cellsto attack. There isthedetection of tumorpeptides by

each$T$cell causing$T$cell activation, and there canbe theoretically the local ignition of theimmune

system which I already proposed [1]. While the former can be thought to have a threshold with probability oftheactivation,the latter has athresholdbytheeigenvalueequalto 1 which

causes

the local ignition. The realization of thesestepsisexpectedto lead to thecureof the tumor.

In this former situation for $T$ cell activation,

an

affimity between a tumor peptide and a $T$ cell

receptorcanbethoughttobeamechanical pattern matching, and thesituation is similar to neural networks[5]although separationbetweenhealthycells and the tumor cellsisimportantinthe tumor-immune interaction. But memory$T$cells

are

thoughttonecessarilymemorizetheantigen pattems.

[Problemandpurpose]

(1)$Here$,the attackagainstthe tumor cells isthoughtto be conductedthrough the step of thedetection

oftumor cells. But at the sametime, theprotectionofhealthycellsshould be considered. Especially the local ignition as a mass action isthought to give a serious damage to thehealthytissue ifthe attackis wrong. Onthe otherhand, it maynotbe reahstic to think that each$T$cell hasanenough

reliabilitytoprotect healthycells almostcompletelybecause thereliabilitymustbevery high..

Thethresholdsandtheprobabilities, especiallythe latterone,

are

expectedto besetto giveenough high reliabilitytoprotect healthycellsthinking naturalselection and the evolution.

On the other hand,ifwecanmake the thresholdscorrespondto the reliability, wecan grabthe behavior ofthe immune system more analytically and quantitatively thinking about not only the attackeffect, but also the enough reliability for the protection ofhealthy cells, and may lead the situationof tumor-

immune

systeminteractionto the direction of thecure

more

preciselyIf the results oftumor-immune interactionanalysisarecoherentto the necessity from theresultofthereliabihty analysis,we canbemuchmoreconfident to theanalysisresult.

(2)

It is known that there

are cases

where the $\infty$stimulation of$CD4T$ and $CD8T$ aocompanies the

$CD8T$

activation

to lead to its development to a R. Here forthe simplicity of the discussion, we

analyse onlythe$T$cellbehaviorswhich include the

cases

of thecostimulasion.

While the consideration ofthe$\infty$stimulation will beexpectedtoheightenthe precision to detect

an

antigencorrectly,it willlower thecontactprobability shownin[1].

Thereliabihtydiscussed here

is

verysimilarto thetest of$g\infty$dness of fit in

statistics.

There is necessity forhealthy cells to be protected from thelocal ignitionwith

an

enough high reliabihty.

Fromthesebackgrounds,thefollowings

are

shown.

(1) qkegcell$\infty$ntributes to themechanismto make clearer the separationabilitybetweenbodycells and atumor peptide likein neural networks. in section2.1.

(2) Themechanismof thecell-wise detectionby$T$cellsandthelocalignition

as

the

mass

action ofthe

immunesysteminalocalarea nearthe tumormassis shown insection2.2.

(3) Themodeltocalculatethereliabihtyisshownin section 3.

(4) The calculation example ofthe value which leads to the reliabihtyis conducted and shown in section4

Common

notation

usedisshown here.

[Common notation]

Th helperTcell

$?b$ cytotoxic Tcell

Tact

.

activated$T$cell

$lkgL2$ regulatory Tcellor suppressorTcell

interleukin 2

2.Possibilityof local ignitionincytotoxicimmunesystem Thefollowingtwo stepsarenecessaryto

cause

local ignition.

(1) $T$cellactivation

(2) Necessityof thelocal ignition in

mass

actionof multiple$T$cells

2.1 Activation of$T$cells

.

Itis known thata$T$oell becomes a?$kg$cell in itsdevelopmentinthymuswhenthe$T$cellhasabig

affimitywithapeptideofhealthy bodycells.

Inmanycases,especiallyin tumortherapyreportsforvaccinetherapy, ‘llregcellstendto work to weakenthe therapyeffect [3, 4],butatthe

same

tmehereit isexpectedthat they notonlyprotect healthycells but alsothey

can

make clearer the differencebetween atumorpeptideandpeptidesof healthycells lowerin$g$the$\infty mmon$partlike showninFig.2.

$T$cellshaveaspectstoactivate based onprobabilityand have its distribution function.

Sohereweassumethat theactivationprobabilityofa$T$cellis determinedbytheaffinitybetween

thereceptorsof the$T$celland the antigen which isatumorpeptide.

Wedo not have the data to express the$T$cellactivation

curve

of probabilityvs.affinity

The examples of the

curves

withexpected characters

are

shown inFig, $i.$

$Fhe$possibihtyof theassumption]

(3)

matchin$g$of thereceptor and the tumor peptide

$2$ Itis knownthat there

are

cases where (a) thereceptorsof$T$cells have anaffimity toan

autoantigen which is not

zero

and is not enough strong (b) the antigen is not enough abundant,butan autoimmuneis not caused[2].

$3$ There isanexampleof

an

autoimmunebyinsulin wherean autoimmuneis causedbythe

abundanceof insuhn[2].

In $2$ and $3$ iftheprobabilityof$T$cellactivationaccordingtoaffimityisnot considered, the$T$cells

haveonlythe threshold ofthe activation to work with the thresholdbytheeigenvalue of the local ignition. Then there will be much less difference between a furious immune response and a mild immuneresponseaslong

as

thereisnodifference among$T$cellactivationlevels.

Theseresponses oftheimmunesystemareexpectedtobe caused by not only$T$cells but alsothe

attack by antibodies, etc., sothe upper discussion may be notenoughto prove the existence of the probabilisticaspectof$T$cell activation, buttbeexistence of theprobabilistic aspectisexpected.

2.2$T$cells

mass

actionandlocalignition

When there is a positive feedback in a system behavior, it canbe expressed as an eigen value problem with eigen values and its eigen vectors. The feedback mechanism is shown in Fig 3. The exampleof this mechanismisshown in detailin[1, 5]whereTh,Tc andIL2areconsidered..

It

is considered that

each$T$cellworks through the probability expressed by $\alpha\cdot\beta\cdot\gamma$ in[1, 5]. $\alpha$

.

contactprobabilityofapathogenwitha$T$cellreceptor

$\beta$ recognition probability of the pathogenbythe$T$cell

$\gamma$ deletionprobability ofthepathogen bytheTc cell with the contact ofatumor cell

For the simplicity of thediscussion, $\alpha$ isassumedtobeconstant inthis article.

3. Model tocalculatereliabilityforseparationbetweenatumorpeptideand apeptideofhealthycells

[Background]

Qualitatively speakin$g$, it is expected that the total number ofTact cells has a main role to

cause the local ignition thinking from the mechanism. In otherwords, this means that local

$igniti6n$does not dependonthelocations ofTact cellsaroundthemassbutthe totalnumber of

Tact cellsaround themass.

We consider$X=$ $(x1^{++}X2 +xN)/N.X$

means

the average ofthesamples$X1,$ $X2,$ $\cdot$,

$XN,$

where $x=1$when $T$cell$i$is activated,$Xi=0$when$T$cell $i$is not activated and

xihas activation

probabihty$p$and statisticalindependence.

The total number$N$varies,and the distribution ofxl,$X2,$ $\cdots,$$XN$in themassalsovaries.These

behaviors

are

expressedas an eigenvalue problem, and Xvariationdependsonthe eigenvalue,

anditsabruptincreasedependsontheeigenvalue$\lambda>1$whichmeansthe localignition, andthe

distributiondependsonthe eigen vector[1].

So$X1+x2+\cdot$ $+xN$ is much concemedwiththe expression of the behavior of the local ignition

as a massaction.

Herewe useequation (1)for X. Because the equation(1)is similar tothenormalizedeigenvector in

an

eigen valueproblem,andwe think the responses of$T$cellmassaction.

(4)

betweenhealthycells and

a

tumorpeptideis determinedby the

average

and the diversion. The

nearer

to 1 the averageis,thehigherthe reliabihtyis.

The smaller thedivergenceis,thehigherthe reliabihtyis.

.

$X=$ $(X1+X2+\cdot \cdot +m)/N$

.

(1)

Theaverage$ofX$ $E(X)=p$

Thediversion$ofX\sigma^{2}(X)=pq/N=p(1-p)/$Nq$=1-p$

$N$

.

‘Ibtal number of$T$cells ina unitvolumeinthe

mass

of tumor

$P.$ $T$cellactivationprobabihtywhenthereceptors$\infty$ntactto

a

kin$d$of tumor peptides..

Here, $\sigma^{2}(X)$becomes smaller when$N$becomeslarger.

4,Evaluation ofreliabihty by comparisonbetweentwocases

Atpresentwecannotcalculatedirectly

an

actualvalueofreliabihty mainlybecauseof the lack of necessary data, so here we inspect the effects ofthe total number ofparticipated$T$ cells

near

the

tumormass,the affimity and the number ofTactcellsby$\infty$mparisonbetweentwo

cases

oneofwhich

has

an

enough high reliabihty to

cause

thelocal ignition without anydamage tohealthy cells, the other

case

which

can

not

cause

thelocalignition.

Here, intheformercase, theaffinitybetween the$T$cellreceptorandthe tumorpeptideisenough

highto

cause

thelocal ignition, inthe othercase,theaffimityislesshigh and

can

not

cause

thelocal ignition.We call the former

case

2 and the lattercase1.

This relationshipisshowninFig.4

The probability$p$of the$T$cellactivationand the total number$N$of$T$cellswhich

means

the total

numberof$T$cells with

an

affinity of the receptors to

a

tumorpeptide

are

givenabout

case

1and

case

2

respectivelyasfollows. Case1: $p_{1}$and$N_{1}$

$E_{1}$ $\cdots$its average

$\sigma\iota^{2}$ its$\cdot$

diversion

$Z1$ $\ldots$itsaffinitybetween theraeeptorandthe tumorpeptide Case2: p2andN2

$E_{2}$ $\cdots$its average

$\sigma 2^{2}$ its diversion

z2 itsaffinitybetween thereceptorandthe tumorpeptide

Here, weconduct the following modifications to make the reliabilityofcase 1 same with that of

case

2.

(1) Modificationofdiversionbychange of$T$cell number inthelocal

area

aroundthetumor

mass

The diversion$ofX$isgiven byequation(2).

$\sigma^{2}\{X\}=p(1-p)/N \cdots\cdot\cdot(2)$

Where$0\leqq X\leqq 1.$

When diversionissmaller,the separationbetween healthy cells and tumor cell isclearer. Because the reliabihtybecomeshigher.

Usually $\sigma 2^{2}$ $<\sigma 1^{2}$becausep2is

nearer

to 1than

$p_{1}.$

We

can

make $\sigma 1^{2}$ nearly

same

with $\sigma^{2_{2}}$byincreasing$N_{1}.$

(2)Modificationof averageby changeof$T$cell numberinthe local

area

around the tumormass

The average ofXisgiven by equation(3).

$E\alpha)=p$ $\cdots\cdot\cdot(3)$

(5)

When the averageislarger,the separation betweenhealthycells and tumor cell isclearer,andthe reliabilitybecomeshigher.

Usually $E_{1}<$ E2 becausep2is

nearer

to 1 than$p_{1}.$

Wecanmake$E_{1}$ nearlysamewithE2byincreasing$N_{1}.$

(1) Modificationto make diversionsequalusing the total number of$T$cells $\sigma 1^{2_{=}}\sigma^{2}\otimes 1)=p_{1}(1-p_{1})/N_{1}$

$\sigma 2^{2_{=}}\sigma^{2}\alpha_{2})=p_{2}(1-p_{2})/N_{2}$

From $\sigma 1^{2_{=}}\sigma 2^{2}$

$p_{1}(1-p_{1})/N_{la}=p_{2}(1-p_{2})/N_{2}$

$N_{la}=N_{2}\cdot p_{1}(1-p_{1})/(p_{2}(1-p_{2}))$ $=N_{2}\cdot(p_{1}/p_{2})\{(1-p_{1})/(1-p_{2})\}$

(2) Modificationusingboth averageanddiversion

We conduct modificationby$E_{1^{2}},/\sigma 1^{2_{=E_{2^{2\int}\sigma 2^{2}}}}$where the dimensionsaremadeequal,because both

average and diversion affect thereliabihtyof the separation and have different dimensions..

$p_{1^{2}}/\sigma^{2}(X_{1})=p_{2^{2}}/\sigma^{2}(X_{2})$

$p_{1^{2}}/(p_{1}(1-p_{1})/N_{lb})=p_{2^{2}}/(p_{2}(1-p_{2})/N_{2})$

$N_{lb}$ $p_{1^{2}}/(p_{1}(1-p_{1}))=N_{2}\cdot p_{2^{2}}/(p_{2}(1-p_{2}))$

$N_{lb}$ $=N_{2}\cdot(p_{2^{2}}/p_{1^{2}})\{(p_{1}(1-p_{1}))\nearrow(p_{2}(1-p_{2}))\}$

$=N_{2}\cdot(p_{2}/p_{1})(1rightarrow p_{1})/(1-p_{2})$

Weadopt$N_{lb}$becauseof$N_{lb}>N_{la}$fromp2 $>p_{1}$

$\lceil The$meanmgs oftheresults$\rfloor$

(1)When$N_{1}$is increased to$N_{lb}$, the reliabilityachievedbyboth$N_{lb}$ andthe affimity of$Z1$ becomes

nearlysamewith thereliabilityachievedbybothN2 andtheaffinityof$Z2.$

$N_{lb} = N_{2}\cdot(p_{2}/p_{1})\{(1-p_{1})/(1-p_{2})\}$

(2)When$N_{1}$of$T$cells with theaffinityof

$Z1$isincreased to$N_{lb}$,the number ofTact

can

be calculated inequation(4).

$N_{lb}\cdot p_{1}=(N_{2}\cdot p_{2})\cdot\{(1-p_{1})/(1-p_{2})\} \cdots\cdot(4)$

Equation (4) means that the number ofTact cells with the affimity $Z1$ is nearly same with the

number ofTact with theaffimity$Z2$although$\{(1-p_{1})/(1-p_{2})\}$ismultiplied.

$\lceil the$modified number of Tact withaffimity$Z1\rfloor=\lceil$(thenumberofTact incase2)$\cdot$$\{(1-p_{1})/(1-p_{2})\}\rfloor$

This

means

that the numbers of Tact inthetwocases

are

almost equal each other to a certain

extent after theequahzationofthereliabilities.

The meanin$g$of$(1-p_{1})/(1-p_{2})$isthat when$p_{2}=1,$ $\sigma 2^{2_{=}}p2(1-p_{2})/N_{2}=0$There isnoerrorin the

ditection,so$N_{lb}$must be $\infty.$

$\{(1-p_{1})/(1-p_{2})\}arrow\infty$ when$p_{2}arrow 1$because $\sigma 2^{2_{=}}p2(1-p_{2})/Narrow 0$ when$p_{2}arrow 1.$

5.Conclusion

The results of theanalysisconducted herecanbe arrangedasfollows.

(1) The recogmitionof a tumor peptide by $T$ cells is considered as a separation problem between

healthy cells andtumorcells through the tumor peptide

(2) lkeg has a function to expand and clarify a difference between healthy cells and tumor cells

throughthetumorpeptide.

(3) The reliability of the separation between healthy cells and tumor cells by probability

can

be expressedusingbinomial distributionalthoughitisdivided by thetotalsample number$N.$

(4) Even if theaffimity of$T$cellreceptorstoatumor peptideis weakor$T$cells

are

repressedby qkeg

(6)

increasesgetting

an

enough reliabihtyto be able to achieve the local ignition from the point

view

of reliabilitysafely

(5)Whentheaffimityof$T$cellreceptorstoatumorpeptideisweakeror$T$cells

are more

repressed by

Tregsin comparison with the$\infty$ndition of$T$cells where

an

enoughhighreliabilityis achieved and

thelocal ignition

can

becaused,the number of the$T$oelk with the

same

receptor, whichhas the

weakaffimity,must beincreasedtoaverylargeextent to have thesamereliability. References

1. Takase, M. (2010) Induction and application of

an

equation to analyze

a

local ignition of the immune system for

a

$\infty$mplete deletion of a cancer mass. Theory of Biomathematics and its

applications VI.RISM1704,53-60Kyoto University.

2.Janeway,C.$A$,Jr.etal.Immunobiology:the imnmunesystemin health and disease. Garland. 3. Zhang,$Y$ etal. (2006)Thl celladjuvant therapycombined with tumor

vaccination.

Intemational

Immunology 19,151-161

4. $N\infty rth,$ $R$ $J$., and Bursuker, I. (1984) Generation and decay of the immmune response to a

progressive fibrosarcoma $J.$$Exp$

.

Med. 159,$12\Re$)$-1311$

5.Takase,M.(2009)Cancer and immunesysteminteractionmodel like a neural network

model, analysis of

cancer

mass effect and meaningofvaccine. Theoryof Biomathematics and its applications V.RISM 1663,35-40Kyoto University.

$T$cellactivation $T$cellactivation

Probability$p$ Probability$p$

receptorandantigen receptorandantigen

Fig.1 Thegraphsof$T$cellactivationprobabihtyvs.affinitybetween$T$cellreceptorandantigen

Asmeaningsof $\alpha$ the$f_{0}u_{ow\dot{m}g}$twocasesare$\infty$nsidered.

(1) When theaffimityis1and

near

toapeptideofhealthycells, $\alpha$

means

theinhibitionby Ikegcells.

(2) $\alpha$

means

alittledisplacementof$T$cellreceptorfrom themaximumfittingwhenwe assumethe

(7)

ylocation $y$location

A schematic tumor cell peptide A schematic healthy cell peptide pattemwith a mutationon MHCI or patteminhibitedstrongly bylbeg MHCII

ylocation

A pattem made by mutation thought to be extractedthrough Treg

Fig. 2 Mechanism to make the difference between a tumor peptide and that ofhealthy cells by deleting thecommonandsimilarpart

(8)

Aboutwhyeigenvalue

can

beapplied

Fig.3Anexampleofmechamism to

cause

thelocalignition Thact activated$CD$4$T$cell

qbact activated$CD$8$T$cell

Thm

.

memory$CD$4$T$cell

lbm memory$CD$8$T$cell

It

is

considered that

each$T$cell worksthroughtheprobabihty expressed by $\alpha\cdot\beta\cdot\gamma$

(9)

?keg Tcell Whenan antigen becomes more similar to ahealthy cell peptide,the mhibitionby‘Ilregbecomesstronger.

$\square$

Fig. 4 Relationshipof$T$cellactivationprobabihties by affimity$Z1$,

z2

$T$ cell activation probabilities by affimity $Z1$, z2 are

$p_{1}$ and p2 respectively, but the contact

Fig. 1 The graphs of $T$ cell activation probabihty vs. affinity between $T$ cell receptor and antigen As meanings of $\alpha$ the $f_{0}u_{ow\dot{m}g}$ two cases are $\infty$ nsidered.
Fig. 2 Mechanism to make the difference between a tumor peptide and that of healthy cells by deleting the common and similar part
Fig. 4 Relationship of $T$ cell activation probabihties by affimity $Z1$ , z2

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