Information thermodynamics reveals the robustness of biochemical signal transduction

Bridging he gp beween er nd life! -Discussion wih Prof. Alber Libchber- 3/6/214 Inforion herodynics revels he robusness of biocheicl signl rnsducion Sosuke Io!! Dep. of Phys., he Univ. of Tokyo, Ph D suden (In collborion wih Tkhiro Sgw*) *Dep. of Bsic Science, he Univ. of Tokyo, Associe professor! Our previous work: S. Io nd T. Sgw, Phys. Rev. Le. 111, 1863 (213).

Bridging he gp beween er nd life! -Discussion wih Prof. Alber Libchber- 3/6/214 Inforion herodynics revels he robusness of biocheicl signl rnsducion Sosuke Io!! Dep. of Phys., he Univ. of Tokyo, Ph D suden Ouline 1. An rificil signl rnsducion in inforion heory 2. Biocheicl signl rnsducion of E. Coli cheois 3. Inroducion of inforion herodynics 4. (Min resul) Inforion herodynics revels! he robusness of biocheicl signl rnsducion

1. Inforion heory C. E. Shnnon (1948) A heicl heory of counicion Bell Syse Technicl Journl 27 (3), 379-423 (1948). Shnnon s heore (Noisy-chnnel coding heore) The Shnnon s heore ses h for! ny given noisy counicion chnnel,! i is possible o counice discree d! wihou error. The ighes upper bound! of rchivble inforion re R is given! by he chnnel cpciy C.

1. Shnnon s heore: Mheicl seen Seing: Inpu essge: Ŵ 2 {1,...,M} Ŵ W Oupu essge: W 2 {1,...,M} n y n Encoding funcion: n (Ŵ ):={ 1(Ŵ ),..., n(ŵ )} Decoding funcion: W (y n ):=W (y 1,...,y n ) Error (Gussin chnnel): p(y i i )= 1 p 2N ep pple (i y i ) 2 2N def: Inforion re: R := ln M n def: A re R is sid o be chievble if! here eiss sequence of codes such! h he il probbiliy of error! ends o zero.!

1. Shnnon s heore: Mheicl seen Seing: Inpu essge: Ŵ 2 {1,...,M} Ŵ W Oupu essge: W 2 {1,...,M} n y n Encoding funcion: n (Ŵ ):={ 1(Ŵ ),..., n(ŵ )} Decoding funcion: W (y n ):=W (y 1,...,y n ) Error (Gussin chnnel): p(y i i )= 1 p 2N ep pple (i y i ) 2 2N def: Inforion re: R := ln M n def: A re R is sid o be chievble if! here eiss sequence of codes such! h he il probbiliy of error! ends o zero.! The chievble inforion re R describes how long encoding lengh is needed o! rnsi inforion wihou error.

1. Shnnon s heore: Mheicl seen Shnnon s heore: The ighes upper bound of chievble! inforion re R is given by he chnnel! cpciy C. C def: chnnel cpciy! (wih power consrin P)! C := R sup p():h 2 ipplep I( : y) Shnnon-Hrley heore: Ŵ n def: uul inforion! I( : y) := Z y n W ddyp(, y)ln p(, y) p()p(y) In he cse of Gussin chnnel, he chnnel cpciy is given by he power-noise rio.! C = 1 2 ln 1 P N

1. Shnnon s heore: Sury Shnnon s heore: The ighes upper bound of chievble! inforion re R is given by he chnnel! cpciy C. C R Ŵ n y n W Shnnon-Hrley heore: C = 1 2 ln 1 P N in he cse of Gussin chnnel The Shnnon s heore ses h he chnnel cpciy C chrcerizes! how uch inforion cn be rnsied correcly (chievble re R).! The chnnel cpciy C is given by he supreu of he uul inforion! beween (inpu signl) nd y (oupu signl).

2. Biocheicl signl rnsducion: E. Coli cheois Signl rnsducion in bio-cheicl cheicl nework e.g. signl rnsducion in E. Coli cheois ehylion level -CH 3 lignd recepor flgellr oor kinse civiy : ehyl (CH 3 ) : lignd The signl rnsducion in E. Coli cheois hs been invesiged s siple! odel orgnis for he dpive biocheicl signl rnsducion. To undersnd! is working echnis, such signl rnscion hve been odeled s noisy! inforion processing device. N. Brki nd S. Leibler, Nure 387, 913 (1997). P. Meh e. l., Mol. Sys. Biol. 5, 826 (24). Y. Tu e. l., Proc. Nl. Acd. Sci. USA 15, 14855 (28). F. Tosevin nd P. R. en Wolde, Phys. Rev. Le. 12, 21811 (29).

2. Biocheicl signl rnsducion: Coprison beween inforion device nd biocheicl nework Signl rnsducion in rificil inforion device The chievble inforion re R is well-defined! by he encoder nd decoder.!! The Shnnon s heore ses h he upper bound! of chievble inforion re R is given by he! chnnel cpciy C. Signl rnsducion in bio-cheicl cheicl nework The inforion re R is ill-defined in living cells! wihou he encoder nd decoder.! lignd ehylion level recepor -CH 3 flgellr oor No heore corresponding o he Shnnon s heore eiss.! kinse civiy : ehyl (CH 3 ) : lignd

2. Biocheicl signl rnsducion: Coprison beween inforion device nd biocheicl nework Signl rnsducion in rificil inforion device The chievble inforion re R is well-defined! by he encoder nd decoder.!! The Shnnon s heore ses h he upper bound! of chievble inforion re R is given by he! chnnel cpciy C. Signl rnsducion in bio-cheicl cheicl nework The inforion re R is ill-defined in living cells! wihou he encoder nd decoder.! lignd ehylion level recepor -CH 3 kinse civiy flgellr oor : ehyl (CH 3 ) : lignd No heore corresponding o he Shnnon s heore eiss.! Inforion herodynics revels he! robusness of biocheicl signl rnsducion.!

3. Inforion herodynics: The proble of he Mwell s deon The Mwell s deon J.C. Mwell Theory of he (1871). In 19h cenury, J. C. Mwell creed he hough eperien describing he ppren violion of he second lw.! Therodynics nd inforion processing L. Szilrd, Z. Phys. 53, 84 (1929). L. Szilrd invesiged he relionship beween! he ppren violion of he second lw nd! he inforion enropy. L. Brillouin, J. Appl. Phys. 22, 334 (1951).! R. Lnduer, IBM J. Res. Dev. 5, 183 (1961).! C. H. Benne, In. J. Theor. Phys. 21, 95 (1982).! ec.

3. Inforion herodynics: Recen progress of he Mwell s deon Theoreicl progress H. Touchee nd S. Lloyd, Phys. Rev. Le. 84. 1156 (2). F. Co, L. Denis nd J. M. R. Prrondo Phys. Rev. Le. 93. 463 (24). T. Sgw nd M. Ued, Phys. Rev. Le. 1. 843 (28). T. Sgw nd M. Ued, Phys. Rev. Le. 14. 962 (21). J. M. Horowiz nd S. Vikunnhn, Phys. Rev. E. 82. 6112 (21). D. Abreu nd U. Seifer, Europhys. Le., 95. 15 (211). D. Mndl nd C. Jrzynski, Proc. Nl. Acd. Sci. USA 19. 11641 (212). S. Io nd T. Sgw, Phys. Rev. Le., 111. 1863 (213). The eperienl relizion of he Mwell s deon! (Toybe s eperien, 21) Eperienl progress S. Toybe, T. Sgw, M. Ued, E. Muneyuki nd M. Sno, N. Phys. 6, 988 (21). A. Beru, A. Arkelyn, A. Perosyn, S. Ciliber, R. Dillenschneider nd E. Luz, Nure 483, 187 (212).

3. Inforion herodynics: Recen progress of he Mwell s deon Scheic of inforion herodynics feedbck eory syse 1 esureen inforion flow A flucuing herl syse (i.e., Brownin pricle) The Mwell s deon perfors feedbck conrol o flucuing herl syse (i.e., Brownin pricle).!! The Mwell s deon reduce he enropy producion of Brownin pricle using! inforion of he eory oucoe. he flow For coupled Brownin pricles second lw of herodynics inforion herodynics inforion flow he flow he flow he flow We re he rge syse ( green pricle) s! conrolled syse, nd noher Brownin pricle ( blue pricle) s eory syse (corresponding o he Mwell s deon). S. Io nd T. Sgw, Phys. Rev. Le., 111. 1863 (213).

3. Inforion herodynics: Coprison beween herodynics nd inforion herodynics Coprison beween he second lw of herodynics nd inforion herodynics he second lw of herodynics (Clusius): J ds J Shnnon enropy difference: Z ds := d d d d p[,, d T T he flu: ie: d d d, d ] ln J eperure: ( =, ) second lw of herodynics p[, ] p[d, d ] T kb = 1 inforion herodynics inforion flow he flow he flow he flow

3. Inforion herodynics: Coprison beween herodynics nd inforion herodynics Coprison beween he second lw of herodynics nd inforion herodynics he second lw of herodynics (Clusius): J ds J ds J d T T T d inforion herodynics (our resul): dir d ds d J T condiionl Shnnon enropy difference: ds inforion flow (rnsfer enropy): dir := Z J T p[d, ] d d dd p[,, d ] ln p[d ] := Z d d dd dd p[,, d, d ] ln p[ ] p[d d ] S. Io nd T. Sgw, Phys. Rev. Le., 111. 1863 (213). second lw of herodynics inforion herodynics inforion flow he flow he flow he flow

3. Inforion herodynics: Coprison beween herodynics nd inforion herodynics coprison beween he second lw of herodynics nd inforion herodynics he second lw of herodynics (Clusius): ds J J T d T inforion herodynics (our resul): dir d ds d J T As upper bound of he he flu J,! inforion herodynics gives igher bound.! inforion flow (rnsfer enropy): dir := Z p[d, ] d d dd p[,, d ] ln p[d ] Inforion herodynics do no cre! bou he dissipion of noher syse. second lw of herodynics inforion herodynics inforion flow he flow he flow he flow

3. Inforion herodynics: Inforion flow (rnsfer enropy) Trnsfer enropy is non-preric sic esuring he oun of direced rnsfer of inforion beween wo rndo processes. T. Schreiber, Phys. Rev. Le., 85. 461 (2). If coupled dynics is given by he Mrkov chin, we hve he ph probbiliy such s p[ d, ]p[ d, ]p[, ] rnsiion probbiliy of for single ie sep. If dynics of is independen fro dynics of, we hve he ph probbiliy such s p[ d ]p[ d ]p[, ] rnsiion probbiliy of for single ie sep. Inforion flow (he rnsfer enropy) fro o is given by he difference beween condiionl Shnnon enropy of hese rnsiion probbiliies. di r := hln p[ d, ]i hln p[ d ]i h i :enseble verge

4. Min resul: Applicion of inforion herodynics o biocheicl signl rnsducion Cheois ehylion level -CH3 lignd recepor flgellr oor We re he signl rnsducion of E. coli cheois s he Mwell s deon proble.! kinse civiy : ehyl (CH3) : lignd feedbck feedbck inforion herodynics: dir d ds d J T ehylion level ehylion level lignd lignd recepor recepor CH CH3 3 esureen esureen flgellr oor flgellr oor : negive feedbck loop : negive feedbck loop

4. Model: The coupled Lngevin odel of biocheicl signl rnsducion The coupled Lngevin odel of bio-cheicl signl rnsducion 1 h i = 2T ( ) (,, L ) = h i = 1 = (,, L ) = L Y. Tu e. l., Proc. Nl. Acd. Sci. USA 15, 14855 (28)., : posiive consn! (linerizion of Monod-Chndeu-Wyn odel)! feedbck feedbck > : ie consn ehylion level ehylion level lignd lignd recepor recepor CH CH3 3 esureen esureen flgellr oor flgellr oor : negive feedbck loop : negive feedbck loop

4. Model: The coupled Lngevin odel of biocheicl signl rnsducion coupled Lngevin odel of bio-cheicl signl rnsducion 1 h i = 2T ( ) (,, L ) = h i = 1 = (,, L ) = L Y. Tu e. l., Proc. Nl. Acd. Sci. USA 15, 14855 (28). feedbck feedbck ehylion level ehylion level lignd lignd ie recepor recepor CH CH3 3 esureen esureen flgellr oor flgellr oor : negive feedbck loop : negive feedbck loop A relion of κ chrcerizes he degree of dpion of he syse! o he environenl chnge.

4. Model: The coupled Lngevin odel of biocheicl signl rnsducion coupled Lngevin odel of bio-cheicl signl rnsducion 1 h i = 2T ( ) (,, L ) = h i = 1 = (,, L ) = L A relion of κ chrcerizes he degree! of dpion of he syse! o he environenl chnge. ie Men squre error: h 2 i This quniy chrcerizes he ROBUSTNESS! of he signl rnsducion gins he! environenl noise induced by ξ. feedbck feedbck ehylion level ehylion level lignd lignd recepor recepor CH CH3 3 esureen esureen flgellr oor flgellr oor : negive feedbck loop : negive feedbck loop

4. Min resul: Inforion herodynics for he biocheicl signl rnsducion Inforion herodynics dir d He dissipion: J T 1 = T T 1 2 h i Inforion flow: dir 1 P = ln 1 d 2 N ds J T d The lrger J is, he ore robus he dpion is! gins he environenl noise.! (The ROBUSTNESS of signl rnsducion)! ie 2 : he inensiy of he effecive signl! ( ) V P := h is rnsied fro o 2 V := h2 i N := h i 2T 2 h i p := h ih i V V : he inensiy of he noise feedbck feedbck ehylion level ehylion level lignd lignd recepor recepor CH CH3 3 esureen esureen flgellr oor flgellr oor : negive feedbck loop : negive feedbck loop

4. Min resul: Anlogicl siilriy beween he Shnnon s heore nd inforion herodynics inforion device biocheicl nework robusness in signl rnsducion gins noise chievble inforion re (ccurcy of inforion rnsission gins noise) noise inpu encoder decoder oupu inforion flow chnnel cpciy in sionry se Shnnon-Hrley heore: 1 P C = ln 1 2 N The chievble inforion re R describes! how uch inforion is rnsied correcly! gins he counicion chnnel noise. Inforion herodynics: dir 1 P = ln 1 d 2 N! The he flu J describes! how uch robus dpion is chieved! gins he environenl noise.

4. Nuericl resuls : inforion herodynics VS he second lw of herodynics Lignd signl L nd noise ξ re induced by!. sep funcion, b. sinusoidl funcion, nd c. liner funcion. b c 8 7 6 5 4 3 2 1 4 2-2 -4 12 1 8 6 4 2-2 -1.1.2.3.4.5.6 ie.1.2.3.4.5.6 ie -4.1.2.3.4.5.6 ie Inforion herodynics Convenionl herodynics Info := dir d ds d J T vs SL := J T ds d J T Inforion herodynics is ore useful o discuss! he biocheicl signl rnsducion hn he convenionl herodynics.

SUMMARY We hve found novel ehod o chrcerize he robusness of he biocheicl signl rnsducion in ers of inforion herodynics. We hve esblished he nlogy beween inforion herodynics nd he convenionl inforion heory : he inforion flow dir/d chrcerizes he biologicl counicion in living cells, prllel o he chnnel cpciy C in rificil counicion. Our resul would open up novel qunificion of he biocheicl signl rnsducion in living cells. In he fuure, we could invesige he role of ech coponen on biocheicl cople signl neworks in quniive robusness in signl rnsducion gins noise wy. chievble inforion re (ccurcy of inforion rnsission gins noise) noise inpu encoder decoder oupu chnnel cpciy inforion flow in sionry se