On the principles of multicellular organism development

David Frigola,* José M. Sancho, Marta Ibañes

Departament de Física de la Matèria Condensada, Universitat de Barcelona, Barce­lo­na,­ Catalonia.­ Universitat­ de­ Barcelona­ Institute­ of­ Complex­ Systems­ (UBICS),­Barcelona, Catalonia

Summary. Non-equilibrium­physics­has­traditionally­dealt­mostly­with­inanimate­matter.­Yet,­in­the­last­decades­there­has­been­increasing­interest­in­understanding­living­systems­from­this­perspective.­One­example­is­using­the­framework­and­tools­of­non-equilibrium­statistical­mechanics­and­nonlinear­physics­to­study­how­living­organisms­composed­of­many­differentiated­cells­develop­from­a­single­initial­cell.­The­dynamic­process­of­multicellular­organism­development­is­out­of­equilibrium,­in­that­ it­ consumes­ and­ dissipates­ energy.­ It­ also­ involves­ the­ formation­ of­ many­precise­ and­ complex­ structures.­ Herein­we­ review­ some­ of­ the­ paradigms­ being­used­ that­ focus­ on­ how­ these­ multicellular­ structures­ initially­ emerge­ at­ the­molecular­level.­[Contrib Sci­11(2):215-223­(2015)]

*Correspondence: David­FrigolaDepartament de Física de la Matèria CondensadaUniversitat de BarcelonaMartí­i­Franqués­108028­Barcelona,­Catalonia

E-mail:­frigola@ecm.ub.edu

A historical overview

An­example­of­ the­beauty­and­complexity­of­Nature­ is­ the­development­of­multicellular­organisms.­Animals­and­plants­develop­from­a­single­cell,­which­through­division­gives­rise­to­all­ the­cells­of­the­organism.­During­development,­these­cells­ become­ distinct­ in­ an­ organized­ and­ precise­ manner­to­ robustly­ form­ complex­ structures­ such­ as­ organs.­ How­does­ this­ occur?­What­ are­ the­ principles­ behind­ it?­Many­physicists­are­now­engaged­in­investigations­of­multicellular­

organism­development,­with­the­aim­of­understanding­how­it­proceeds­and­finding­its­fundamental­principles.­Resolving­these­ questions­ is­ expected­ to­ help­ shed­ light­ on­ more­applied­ challenges­ ranging­ from­biomedical­ issues,­ such­as­embryonic­malformations­and­cancer,­to­agricultural­issues,­such­as­the­optimization­of­crop­growth.­However,­the­quest­for­underlying­principles­is­still­in­its­own­early­developmental­stage,­and­an­immense­universe­of­knowledge­lies­ahead.­In­the­following,­we­consider­some­of­the­ideas­and­insights­that­appeared­early­on­and­that­have­influenced­current­research.

O P E N A A C C E S S

BIOPHYSICS Institut d’Estudis Catalans, Barcelona, Catalonia

www.cat-science.cat

CONTRIBUTIONS to SCIENCE 11:215-223 (2015) ISSN (print): 1575-6343 e-ISSN: 2013-410X

Keywords: development­·­morphogen­·­attractor­·­non-equilibrium­·­differentiation­·­self-organization­·­lateral­inhibition­·­multistep­signaling­·­ultrasensitive­response­·­bistability­·­fluctuation

CONTRIB SCI 11:215-223 (2015)doi:10.2436/20.7010.01.233

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Over­ 70­ years­ ago,­ Conrad­ H.­ Waddington­ used­ the­metaphor­that­during­development­cells­roll­down­through­valleys­that­bifurcate­[25],­having­to­choose­what­to­become­at­each­bifurcation.­This­metaphor­for­cell­differentiation­is­now­commonly­used,­with­a­free­energy­landscape­of­which­Waddington’s­valleys­are­the­minima.

Alan­ Turing,­ very­ well­ known­ for­ his­ contributions­ to­computer­science,­proposed,­in­a­seminal­work­published­in­1952,­that­patterns­arising­during­development­might­be­the­natural­output­of­chemical­reactions­between­molecules­that­diffuse­with­different­diffusion­coefficients­across­space­[23].­That­ chemical­ systems­ can­ form­ spatiotemporal­ patterns,­in­which­ the­ concentrations­ of­molecules­ are­ organized­ in­space­and­time,­was­proven­later­in­well-controlled­chemical­and­ physical­ assays.­ However,­ these­ were­ not­ linked­ to­multicellular­ organism­ development,­ but­ instead­ drove­intense­ research­ in­ the­ field­ of­ nonlinear­ dynamics.­ The­relevance­of­ this­mechanism,­ known­as­ Turing’s­ instability,­in­ the­ context­of­ development­ is­ now­appreciated­but­ still­debated­[15,18,19].

In­1970,­the­Nobel­Laureate­Francis­Crick­proposed­that­the­ diffusion­ of­ molecules­ could­ create­ gradients­ across­developing­tissues­[4].­These­gradients­could­convey­to­the­cells­the­positional­information­that­Lewis­Wolpert­had­already­proposed­ [26],­ guiding­ them­ in­ their­ further­development.­This­ is­ the­ morphogen­ gradient­ paradigm,­ which­ has­dominated­research­on­patterning­in­developmental­biology.­The­finding­that­numerous­molecules­form­gradients­during­development­and­that­the­gradients­themselves­are­relevant­for­ the­ development­ of­ different­ tissues­ has­ led­ to­ many­other­complex­questions:­How­does­the­gradient­form?­How­is­ it­ sensed?­And­what­ information­from­the­gradient­does­the­cell­use?

Stuart­Kauffmann­showed­that­the­interactions­between­genes­strongly­restrict­the­possible­cell­types­[14].­In­this­case,­cell­types­are­understood­as­the­attractors­of­the­dynamics­of­genetic­interactions.­At­present,­deciphering­the­large­gene­regulatory­ and­ signaling­ networks­ and­ their­ dynamics­ in­ a­developing­cell­is­an­intense­field­of­research.

These­ conceptual­ frameworks,­ i.e.,­ bifurcations­ to­produce­ changes­ of­ cell­ types,­ self-organization­ out­ of­equilibrium­and­cell­types­as­attractors,­were­mathematically­formulated­and­developed.­However,­ in­the­last­decades­of­the­20th­ century,­ the­use­of­mathematical­ formulations­ to­understand­ development­ became­ unpopular­ because­ they­failed­at­describing­and­predicting­patterns.­The­ result­was­a­ split­ between­ developmental­ biologists­ and­ physicists/mathematicians­[16].­More­recently,­however,­knowledge­of­

which­biological­molecules­participate­ in­development,­ the­ability­ to­manipulate­ them,­ and­ their­ spatial­ and­ temporal­resolution,­ have­ increased­ dramatically.­ At­ the­ same­ time,­important­ progress­ has­ been­ made­ in­ non-equilibrium­statistical­mechanics,­dissipative­systems,­complex­systems,­nonlinear­ dynamics,­ and­ networks,­ accompanied­ by­ an­extraordinary­ increase­ in­computational­power.­As­a­result,­interdisciplinary­ research­ involving­ both­ physicists­ and­biologists­has­become­more­common­and­the­advantages­to­this­approach­are­now­acknowledged­[20].­Thus,­we­are­in­an­exceptional­position­to­embrace­the­challenge­to­understand­development­and­the­principles­behind­it.

Patterning the embryo

A­crucial­step­in­understanding­how­multicellular­organisms­develop­ is­ to­ unravel­ how­ cells­ become­ distinct­ in­ a­coordinated­ and­ organized­ manner.­ In­ the­ language­ of­developmental­biology,­this­can­be­rephrased­as­how­a­cell­attains­a­specific­fate­into­which­it­ultimately­differentiates.­Two­main­mechanisms­have­been­proposed­for­coordinated­cell­ differentiation­ in­ tissues.­ One­ mechanism­ is­ through­positional­ information,­ proposed­ by­ Lewis­ Wolpert­ as­mentioned­above­ [26]:­ the­ fate­of­a­cell­ is­a­ readout­of­ its­spatial­localization­from­a­reference­system­(Fig.­1A).­Cells­read­the­information­of­where­they­are­located­and­differentiate­accordingly.­ Gradients­ of­molecules,­ previously­ referred­ to­as­morphogens­(we­retain­this­term­here­for­convenience),­have­been­proposed­ to­confer­ such­positional­ information.­The­origin­of­the­reference­system­is­the­source­where­the­morphogen­is­produced.­The­amount­or­concentration­of­the­morphogen­decays­as­the­distance­from­the­source­increases­and­thereby­conveys­positional­information­to­the­cell.­This­information­ can­ be­ conferred­ to­ cells­ through­ molecules­that­become­activated­at­distinct­thresholds­of­morphogen­concentrations­ (Fig.­ 1A).­ There­ are­ multiple­ proteins­ that­have­been­shown­to­be­distributed­along­gradients­in­different­developing­ embryos­ and­ that­ seem­ to­ convey­ positional­information.­ Specifically,­ if­ the­ gradient­ is­ altered,­ the­ fate­of­ the­ cells­ changes­ accordingly­ (Fig.­ 1B).­ This­ is­ the­ case,­for­ instance,­ for­ the­protein­Bicoid,­which­forms­a­gradient­along­ the­ anterior-posterior­ axis­ of­ the­ embryo­during­ the­very­early­stages­of­insect­development,­including­that­of­the­fruit­fly­Drosophila [10].­The­region­where­Bicoid­ is­at­high­concentration­becomes­the­head­of­the­fly.­

The­ other­ proposed­ mechanism­ is­ that­ cells­ become­distinct­ only­ because­ of­ coupling.­ This­ is­ an­ example­ of­

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self-organization­ in­ which­ a­ structure­ or­ order­ emerges­spontaneously­ because­ of­ the­ interactions­ between­elements.­ Coupled­ dynamics­ enable­ the­ emergence­ of­robust­ proportions­ and­ periodic­ distributions­ of­ cell­ types.­In­ contrast­ with­ the­ positional­ information­ mechanism,­coupling­does­not­drive­a­specific­cell­type­in­a­given­spatial­position.­In­a­developing­organism­this­self-organization­can­happen­ in­different­ways.­ The­first­one­ corresponds­ to­ the­dynamics­Alan­Turing­studied­[23].­When­chemicals­initially­distributed­homogeneously­throughout­a­given­space­react­and­diffuse,­they­form­heterogeneous­distributions.­Because­the­reactants­diffuse­with­different­diffusion­coefficients,­tiny­small­ random­ fluctuations­ in­ the­ reactant­ concentrations­become­ amplified,­ such­ that­ the­ homogeneous­ state­becomes­ destabilized.­ This­ happens­ for­ a­ wide­ range­ of­diffusion­coefficients­and­reaction­kinetics.­ It­ is­an­example­of­a­non-equilibrium­pattern­formation­process,­in­which­the­

balance­between­antagonistic­processes,­such­as­driving­and­dissipation,­ results­ in­ the­ formation­ of­ non-homogeneous­structures­ [5].­ Thus,­ for­ instance,­ periodic­ stationary­distributions­ of­ the­ molecules­ can­ emerge.­ Cells­ produce­proteins,­which­react­and­diffuse­in­the­extracellular­space.­Accordingly,­when­a­periodic­pattern­of­protein­distributions­emerges­from­these­dynamics,­some­cells­end­up­producing­or­ sensing­ large­ amounts­ of­ proteins­while­ others­ do­ not.­Therefore,­ cells­ become­ distinct­ (Fig.­ 1C,D).­ A­ change­ in­the­ spatial­ interactions,­ as­ in­ the­ diffusion­ coefficient,­results­ in­ relevant­ changes­ of­ the­molecular­ pattern­ being­formed.­Accordingly,­the­pattern,­ if­periodic­and­stationary,­can­change­its­periodicity­(Fig.­1D).­Empirical­evidence­that­such­ a­mechanism­ can­drive­ the­ formation­of­ the­digits­ in­vertebrates­has­recently­been­provided­[21].­The­digits­form­from­an­initial­rather­two-dimensional­round­palette.­In­this­palette­a­stripe-like­pattern­emerges­that­divides­it­into­two­

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Fig. 1. Pattern­formation­mechanisms­that­rely­on­diffusion­along­an­extracellular­medium. (A–B)­Morphogen gradient mechanism.­The­leftmost,­green­cell­generates­a­molecule­that­acts­as­a­morphogen.­The­molecule­diffuses­to­the­right­and­generates­a­gradient,­as­shown­in­the­curve­above­the­cells.­The­amount­of­morphogen­sensed­by­each­cell­conveys­positional­information­to­it.­There­are­two­thresholds,­one­at­concentration­1­and­another­at­concentration­10,­and­cells­differentiate­depending­on­whether­the­concentration­is­above­or­below­these­thresholds.­In­A­there­is­extensive­diffusion,­as­indicated­by­the­larger­curvy­arrow.­In­B,­there­is­less­diffusion,­altering­the­gradient­and­the­position­of­cell­types­accordingly.­(C–D)­Turing pattern mechanism.­Two­or­more­chemicals­that­diffuse­and­react­are­needed­to­establish­a­pattern.­In­C­the­pattern­for­certain­values­of­the­parameters­is­shown.­In­D,­when­diffusion­is­modified­so­is­the­pattern­and­the­corresponding­cell­fates.­(Note­that­the­term­“morphogen”­is­no­longer­used­with­the­mechanism­shown­in­C­and­D.­We­use­the­term­here­because­it­was­introduced­by­Turing­precisely­in­this­context).

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intercalating­ regions:­ interdigital­and­digital­ regions.­ In­ this­specific­ case­ the­ reaction-diffusion­ mechanism­ does­ not­act­ in­ isolation­but­ it­ is­coupled­to­a­positional­ information­mechanism.

Another­ way­ that­ could­ drive­ the­ differentiation­ of­cells­ in­ a­ self-organized­ manner­ but­ does­ not­ require­ the­transport­of­a­molecule­is­through­direct­cell-to-cell­contact.­In­this­case,­cells­interact­through­molecules­present­on­the­cell­membrane­ that,­ upon­binding,­ send­ signals­ to­ the­ cell­nucleus.­An­example­of­this­is­lateral­inhibition­with­feedback­[3].­In­this­case,­the­signal­a­cell­receives­arises­from­protein­ligands­ in­ adjacent­ cells­ and­ it­ decreases­ the­ amount­ of­ligand­ in­the­cell.­Thus,­a­cell­ that­has­more­ ligand­than­ its­neighboring­ cells,­ even­ if­ the­ difference­ is­ very­ small,­ will­reduce­their­amount­of­ligand­and,­at­the­same­time,­increase­its­own­ligand­production­by­preventing­inhibition­by­those­neighbors.­ Ultimately,­ the­ cell­ with­ an­ initially­ very­ small­excess­of­ligand­will­end­up­with­a­relatively­large­amount­of­that­ ligand,­while­ ligand­ in­neighboring­cells­will­be­almost­completely­eliminated.­This­type­of­interaction­underlies­the­specification­of­neurons,­for­instance.­

In­ the­ 1970s,­Meinhardt­ and­Gierer­ proposed­ a­ theory­for­ biological­ pattern­ formation­ based­ on­ two­ elements:­(1)­ self-activation­ and­ (2)­ long-range­ inhibition­ [9].­ Turing-like­ reaction-diffusion­ dynamics­ and­ lateral­ inhibition­ with­feedback­ can­ both­ be­ understood­ in­ terms­ of­ these­ two­elements.­Moreover,­ self-activation­evidences­ a­ key­ aspect­in­ the­dynamics­of­coupled­elements­that­drive­patterning:­nonlinearities.­All­these­self-organizing­interacting­dynamics­drive­ the­ emergence­ of­ robust­ proportions­ and­ periodic­distributions­ of­ cell­ types.­ In­ this­ mechanism­ based­ on­coupling,­ the­cell­ types­arise­ in­a­coordinated­manner­but,­unlike­in­the­positional­information­mechanism,­it­does­not­enable­the­robust­specification­of­a­cell­type­in­a­given­spatial­position.­Nevertheless,­if­spatial­asymmetric­cues­are­added­to­ interacting­dynamics,­ then­ spatial­ precision­ can­ arise­ as­well.

It­ is­worth­noting­that­how­the­pattern­will­be­modified­when­ the­ elements­ driving­ it­ are­ altered­ can­ be­ predicted­by­ constructing­ mathematical­ and­ computational­ models­of­the­dynamics.­The­resulting­predictions­can­then­be­used­to­ test­ whether­ assumptions­ regarding­ the­ mechanism­ of­patterning­ are­ correct,­ by­ comparing­ the­ predicted­ results­with­ the­ empirically­ derived­ data.­ This­ task­ is­ nowadays­common­routinely­done­but­it­has­not­always­been­so­easily­possible.­Now­we­can­propose­which­specific­molecules­are­acting­and,­in­several­cases,­we­can­experimentally­see­how­their­distribution­changes­over­time­and­space­with­detailed­

resolution.­Manipulations­of­ the­ interactions­and­ reactions­and­how­the­molecular­distribution­changes­accordingly­can­now­be­done­and­the­results­measured.­

The­ mechanisms­ described­ herein­ assume­ that,­ in­terms­of­ their­ patterning,­ cells­ can­be­described­by­only­ a­few­ relevant­molecules.­ The­ role­ of­ cell­ dynamics­ and­ the­particular­mechanical­forces­that­are­active­are­not­taken­into­account.­ This­ simplification­ is­ valid­ in­ some­ circumstances,­especially­ when­ the­ dynamics­ that­ control­ the­ molecular­concentrations­are­much­faster­than­those­of­the­cell.­Many­efforts­ are­being­done­on­ the­ role­ of­mechanical­ forces­ in­shaping­ developing­multicellular­ organisms,­ which­ are­ not­reviewed­herein.­A­ challenge­ that­ remains­ is­ to­determine­how­mechanical­ forces­and­ the­dynamics­of­ the­molecular­components­ that­ direct­ cell­ signaling­ or­ impinge­ on­ gene­regulation­are­coupled­to­each­other.

Nonlinear responses

We­ have­ discussed­ how­ molecular­ gradients­ can­ confer­positional­information,­in­which­each­cell­type­is­dictated­by­a­threshold,­cell-type-dependent,­morphogen­concentration.­In­ Fig.­ 1A,­ cell­ type­ “blue”­ is­ induced­ above­ a­morphogen­concentration­ of­ 10­ (arbitrary­ units),­ whereas­ cell­ type­“white”­ is­ induced­above­a­morphogen­concentration­of­1.­Yet,­ is­this­type­of­threshold­response­possible­in­biological­systems?­ It­ is,­ thanks­ to­ ultrasensitivity.­ As­ opposed­ to­ a­gradual­ or­ linear­ response,­ in­ which­ the­ relative­ changes­in­ input­ (signal)­ and­ output­ (response)­ are­ equal,­ an­ultrasensitive­response­is­that­in­which­a­small­relative­change­in­the­signal­generates­a­very­large­(relative)­response.­Since­a­ cellular­ response­ usually­ saturates­ (i.e.,­ when­ the­ input­signal­is­large­enough,­the­response­no­longer­changes),­an­ultrasensitive­response­in­cells­can­translate­to­a­threshold­or­“all-or-nothing”­response­(Fig.­2A).­

But­how­is­this­ultrasensitivity­achieved­by­cells?­A­variety­of­mechanisms­have­been­elucidated­through­mathematics­and­then­experimentally­demonstrated­[27].­A­few­of­them­are­summarized­ in­Fig.­2­and­reviewed­ in­[27].­“Zero-order­ultrasensitivity”­ was­ the­ first­ of­ these­ mechanisms­ to­ be­proposed,­ in­ 1981­ [11].­ In­ this­ mechanism,­ an­ enzyme­covalently­ modifies­ a­ protein­ (covalent­ modification­ is­ a­common­ regulatory­ mechanism­ in­ which­ a­ molecule­ such­as­a­phosphate­or­methyl­group­is­bound­to­a­protein­by­an­enzyme),­ and­ an­ opposing­ enzyme­ restores­ the­ protein­ to­its­ unmodified­ state.­ When­ both­ enzymes­ are­ working­ at­saturation,­a­small­change­in­the­amount­of­one­of­them­can­

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produce­ a­ large­ change­ in­ the­ proportion­ of­ the­modified­and­ unmodified­ proteins,­ thus­ enabling­ an­ ultrasensitive­response­(Fig.­2B).

Another,­very­common­mechanism­is­multistep­signaling,­in­which­an­element­representing­the­signal,­or­proportional­to­ the­ signal­ intensity,­ acts­ on­ two­or­more­ elements­ that­independently­ affect­ the­ strength­ of­ the­ response.­ An­example­ is­ a­ signal­ that­ acts­ on­ two­different­ steps­ of­ the­modification­of­a­protein­that­will­ultimately­turn­ it­ into­ its­active­ form.­ The­ multiplied­ effect­ elicits­ an­ ultrasensitive­behavior.­Mathematically,­the­repeated­effect­of­the­signal­is­represented­as­multiplicative­terms­that­can­raise­it­up­to­the­power­of­the­number­of­points­at­which­the­signal­affects­the­system­independently­(Fig.­2D).

Direct­or­ indirect­ self-activation,­also­known­as­positive­feedback,­can­drive­ultrasensitive­responses­as­well.­Positive­feedback­ occurs,­ for­ instance,­ when­ a­ protein­ binds­ to­ its­own­ DNA­ promoter­ to­ boost­ its­ own­ transcription­ (auto-activation),­or­when­a­protein­ inhibits­the­production­of­ its­inhibitor­(mutual­inhibition)­(Fig.­2F).

Bistability

A­positive­feedback­loop­can­also­enable­bistability,­i.e.,­two­different­ responses­ to­ the­same­ input­ (mathematically,­ the­equation­that­represents­the­system­has­two­stable­solutions­instead­ of­ one).­ In­ other­ words,­ genetically­ identical­ cells­exposed­ to­ the­ same­ environmental­ conditions­ can­ be­ in­two­ different­ states­ and­ hence­ become­ two­ distinct­ cell­types.­ An­ example­ of­ bistability­ in­ development­ occurs­ in­the­ vulval­ development­ of­ the­ hermaphroditic­ nematode­worm­Caenorhabditis elegans­[10,12].­Before­this­egg-laying­organ­is­formed,­two­adjacent­cells,­which­can­be­labeled­1­and­2,­for­instance,­become­distinct­from­each­other­based­on­ their­ position­ in­ the­ embryo.­ One­ becomes­ an­ anchor­cell­(AC)­and­the­other­a­ventral­uterine­(VU)­cell.­Each­cell­has­a­50%­probability­of­becoming­an­AC.­Hence,­under­the­same­conditions­ two­states­can­arise,­with­50%­probability­each:­ (AC,VU)­ or­ (VU,AC),­ in­ which­ the­ first­ term­ within­the­ parentheses­ denotes­ the­ type­ acquired­ by­ cell­ 1,­ and­the­second­term­refers­to­cell­2.­ In­this­case,­the­bistability­of­ these­ two­ states­ arises­ through­ a­ positive­ feedback­that­ involves­ the­ above-described­ lateral­ inhibition­ with­feedback.­ Nonlinearities­ are­ essential­ for­ this­ bistability.­Figure­ 3­ provides­ an­ example­ of­ this­ case­ and­ shows­ how­a­ mathematical­ model­ of­ the­ interactions­ can­ help­ us­ to­understand­and­visualize­this­process.­

Fluctuations

As­we­have­seen,­cells­have­mechanisms­to­process­signals­coming­from­neighboring­cells­and­from­their­surroundings­that­can­yield­precise­results.­However,­these­signals­cannot­

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Fig. 2.­ Mechanisms­ that­ generate­ ultrasensitivity.­ (A)­ A­ signal-response­function­showing­ultrasensitivity­and­an­all-or-nothing­response,­as­shown­in panels B, D, E, F.­ (B)­ Zero-order ultrasensitivity.­ As­ explained­ in­ the­text,­the­purple­enzyme,­corresponding­to­signal­S,­enhances­the­covalent­modification­of­the­red­protein,­while­the­yellow­enzyme­mediates­ its­de-modification.­The­modified­protein­amount­corresponds­to­the­response­R.­(C)­Molecular titration.­ Free­molecule­A­ (corresponding­ to­or­ activating­a­response­R)­can­be­sequestered­by­B,­which­is­present­in­very­large­amounts.­Molecule­A­exhibits­an­ultrasensitive­response­to­changes­in­its­production.­There­are­free­A­molecules­only­when­their­production­level­surpasses­the­sequestering­effect.­At­this­threshold,­the­amount­of­A­suddenly­increases.­This­ behavior­ is­ not­ like­ that­ shown­ in­ A,­ because­ its­ response­ does­ not­saturate.­(D)­Multistep signaling.­The­signal­S,­or­some­element­proportional­to­ it,­ aids­ in­ two­different­ steps­of­ the­modification­of­a­protein­ that­will­ultimately­assume­its­active­form,­which­then­enacts­response­R.­Its­effect­is­multiplied­and­can­elicit­an­ultrasensitive­response.­(E)­Cooperative binding. A­ receptor,­ in­ green,­ has­ several­ binding­ sites­ for­ the­ same­ ligand,­ the­amount­of­which­corresponds­to­signal­strength­S.­If­full­occupancy­of­the­receptor’s­binding­sites­is­needed­to­elicit­a­response­R,­or­if­each­occupied­site­increases­the­chance­that­a­new­ligand­will­bind­(thicker­arrows­indicate­larger­amounts­of­bound­ligand),­ultrasensitivity­arises.­(F)­Positive feedback loop.­ A­ signal­ S­ (here­ a­ blue­ enzyme)­ activates­ a­ protein­ (in­ red).­ This­active­protein­elicits­response­R,­but­it­can­also­bind­to­DNA­and­enhance­the­production­of­ its­own­unmodified­ form.­This­ increases­ the­amount­of­substrate­upon­which­the­signal­can­act,­multiplying­ its­effect­and­making­the­response­ultrasensitive.­These­mechansims­are­reviewed­in­[27].

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be­ sensed­ with­ perfect­ precision­ due­ to­ the­ physical­ laws­that­govern­molecular­dynamics­[2].­These­signals,­and­the­proteins­ that­ process­ them,­ consist­ of­ discrete­ molecules­that­ jiggle­ around,­ embedded­ in­ the­ thermal­ bath­ of­ the­cytoplasm.­ This­ aqueous­ medium­ is­ crowded­ with­ many­moving­molecules­ such­as­proteins.­ Some­molecules­move­stochastically­ without­ a­ preferred­ direction,­ because­ of­thermal­forces­coming­from­collisions­with­water­molecules.­Others,­such­as­molecular­motors,­move­directionally­using­electrochemical­ forces.­ Several­ of­ these­ electrochemical­reactions­ have­ associated­ energies­ (such­ as­ the­ energy­required­for­some­reactions­to­start,­or­the­energy­required­to­break­specific­chemical­bonds)­comparable­to­the­thermal­energy­ of­ the­ medium.­ Therefore­ the­ stochastic­ “jiggling”­of­molecules­can­spontaneously­activate­reactions­or­break­chemical­bonds.­

These­ fluctuations­ also­ affect­ the­ production­ and­degradation­of­different­proteins­in­the­cell,­which­stochas-tically­ vary­ in­ time.­ This­ could­ not­ be­ directly­ observed­until­ the­ very­ recent­ advances­ in­ the­ spatial­ and­ temporal­resolution­ of­ fluorescence­ microscopy­ techniques.­ Before­that­ (but­ also­ only­ recently),­ temporal­ fluctuations­ in­ the­

amount­of­specific­molecules­could­only­be­inferred­from­the­heterogeneous­ amounts­ found­ among­ genetically­ identical­cells­in­the­same­environment.­Even­though­fluctuations­are­a­common­object­of­study­in­non-equilibrium­and­statistical­physics,­our­direct­knowledge­of­the­motion­and­fluctuations­of­particles­embedded­in­the­crowded­medium­of­a­cell­is­still­incipient.­ Yet,­with­ the­advent­of­nanotechnologies­we­are­entering­a­new­era­in­which­it­will­be­possible­to­characterize­the­motions­of­and­fluctuations­in­cellular­components.

Fluctuations and cell decisions

Because­ fluctuations­ are­ ubiquitous­ in­ the­ cell,­ they­ must­somehow­ be­ relevant­ to­ an­ understanding­ of­ all­ cellular­processes,­including­those­in­the­previously­mentioned­examples­of­ morphogen­ diffusion,­ cellular­ sensing­ of­ these­molecules,­and­the­related­signaling­processes.­The­exquisite­precision­and­regularity­of­developmental­processes­ indicates­ that­cells­ can­cope­with­this­variability,­or­perhaps­even­profit­from­it.

One­ obvious­ way­ of­ avoiding­ the­ effect­ of­ fluctuations­is­ by­ producing­ large­ amounts­ of­ molecules­ to­ minimize­

Fig. 3. Bistability.­(A)­Lateral­inhibition­with­feedback.­The­ligand­in­cell­1­inhibits­the­ligand­in­neighboring­cell­2­and­vice­versa,­establishing­positive­feedback.­Inhibition­is­represented­by­the­blunt­arrows.­There­is­a­50%­probability­for­the­(AC,VU)­outcome­and­50%­for­the­(VU,CA)­outcome,­determined­by­which­cell­achieves­a­high­or­low­amount­of­ligand.­(B)­The­equation­that­governs­the­temporal­evolution­of­a­ligand­in­cell­ i ­(1­or­2).­ /idl dt ­is­the­time­derivative­of­concentration­

il ­and­represents­its­changes­over­time.­The­production­term­ ( )jg l decreases­nonlinearly­when­ jl ­(the­ligand­in­the­other­cell)­increases.­(C)­Phase­diagram­of­this­two-cell­system.­Each­point­corresponds­to­a­unique­pair­of­

1 2l l− ­values.­The­evolution­of­either­one­is­fully­determined­and­shown­by­the­blue­arrows­of­the­vector­field.­The­red­and­blue­dashed­lines­are­called­nullclines­and­correspond­to­the­points­at­which­the­time­derivative,­i.e.,­the­rate­of­change,­for­the­ligand­at­one­of­the­cells­(blue­for­cell­1­and­red­for­2)­is­zero.­At­the­points­where­the­nullclines­cross­both­derivatives­have­the­values­of­zero,­so­the­system,­if­unperturbed,­will­not­move­away­from­them.­Because­of­the­nonlinearity­of­the­nullclines,­there­are­three­of­these­points;­if­they­were­not­nonlinear,­there­would­only­be­one­such­point.­Of­these,­the­black­points­are­stable­states:­when­the­system­is­at­one­of­them,­it­will­return­to­it­after­a­small­perturbation­(this­state­is­therefore­also­called­an­attractor).­Indeed,­all­trajectories­starting­in­the­purple­half­of­the­portrait­(called­the­basin­of­attraction)­will­evolve­towards­the­(AC,VU)­stable­state­at­the­bottom­left­(one­such­trajectory­is­shown­in­black).­Similarly,­the­green­area­is­the­basin­of­attraction­for­the­(VU,AC)­stable­state.­The­orange­point­represents­a­state­with­intermediate­values­of­ligand­for­both­cells,­as­shown­in­gray,­that­is­not­stable.­A­small­perturbation­from­this­state­can­lead­the­system­away­from­it­and­to­one­of­the­stable­solutions.­The­scenario­in­B­was­obtained­from­simulations­performed­by­Juan­Camilo­Luna-Escalante,­Dept.­of­Condensed­Matter­Physics,­University­of­Barcelona).­The­data­are­used­with­permission.

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their­ effects.­ This­ is­ not­ always­ worthwhile,­ or­ possible.­For­ instance,­when­ a­ cell­ receives­ a­ fluctuating­ signal­ that­it­ cannot­ control,­ how­ can­ it­ cope­ with­ the­ fluctuations?­One­way­to­buffer­fluctuations­is­to­respond­to­the­amount­of­ signaling­ molecules­ received­ only­ during­ an­ interval­ of­time­ [2].­ This­ corresponds­ to­ an­ integration­ over­ time­ of­the­number­of­molecules,­ the­ result­of­which­ is­much­ less­variable­ than­ the­ number­ of­molecules­ at­ any­ given­ time.­Therefore,­ the­ input­ to­which­ the­ cell­ responds­ is­ not­ the­highly­fluctuating­number­of­molecules­but­the­much­more­constant­total­number­of­molecules­received­per­unit­time.­This­time­integration­is­performed,­for­instance,­by­bacteria­to­ sense­ the­ level­ of­ nutrients­ in­ their­ environment­ [2].­ It­also­is­the­mechanism­proposed­for­developing­embryos,­in­the­ cellular­ response­ to­morphogen­ gradients­ [6].­ In­ cases­in­ which­ cells­ respond­ too­ rapidly­ compared­ to­ the­ time­interval­ that­would­be­ required­ for­ integration­ to­filter­out­fluctuations,­ the­ additional­ interactions­ of­ neighboring­

cells­may­reinforce­the­correct­cell­decision­and­increase­its­robustness­[13].

There­ are­ several­ examples­ of­ biological­ systems­ that­profit­from­fluctuations­[7].­Most­of­them­are­in­unicellular­rather­ than­ multicellular­ developing­ organisms,­ but­ their­existence­ can­ suggest­ that­ fluctuations­ may­ also­ be­ used­during­ development.­ For­ instance,­ fluctuations­ enable­wide-ranging­ heterogeneity­ between­ genetically­ identical­cells­ in­ the­ same­ environment.­ This­ heterogeneity­ can­ be­beneficial­ when­ the­ environment­ changes­ rapidly­ and­ the­cellular­ response­ is­ heterogeneous.­ If­ this­ heterogeneous­population­of­cells­comprises­different­cell­types­that­respond­differently,­then­when­the­environment­changes­some­of­the­cell­types­may­die­while­others­will­prevail.­Because­of­this­heterogeneous­ response­ to­environmental­ change,­ the­cell­population­persists,­providing­a­benefit.­This­is­known­as­bet-hedging­(the­colony­of­cells­hedges­its­bets­instead­of­putting­“all­ of­ its­ eggs­ in­ one­ basket”)­ and­ has­ been­ described­ in­

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Fig. 4.­Stochastic­switching.­(A)­A­model­of­a­bistable­system.­The­black­continuous­line­is­the­energy­landscape­of­the­system;­the­bottoms­of­the­two­wells­are­the­stable­states.­The­blue­circle­represents­the­system­at­one­of­these­states,­and­the­blue­arrows­the­fluctuations,­which­can­drive­the­system­to­higher­energies.­If­the­fluctuations­are­large­enough,­or­the­energy­barrier­ ( )U∆ ­low­enough,­the­system­can­jump­to­the­leftmost­well­and­switch­states.­(B)­Time­evolution­of­the­amounts­of­a­protein­for­a­single­cell­in­two­different­cases.­There­are­two­clearly­defined­states,­a­high­concentration­state­at­270­protein­copies­and­a­low­concentration­state­at­50­copies.­The­cells­switch­from­one­state­to­the­other.­Note­that­the­transitions­are­very­fast­and­that­the­system­spends­most­of­its­time­around­one­of­the­two­stable­states.­(C)­Evolution­over­time­of­the­concentrations­of­a­protein­of­interest­in­a­cell­culture,­as­shown­in­a­histogram.­When­a­subpopulation­in­one­of­the­states­from­an­originally­bistable­population­is­separated­and­left­to­evolve­over­time,­stochastic­switching­allows­the­recovery­of­the­two­states.­The­cells­in­a­population­are­shown­on­the­right.­Note­how­one­cell­may­switch­states­more­than­one­time.­Panels­B and C­are­simulations­of­a­mutual­inhibition­system,­simulated­through­the­Gillespie­algorithm,­which­allows­exact­simulations­based­on­the­theoretical­description­of­discrete­stochastic­systems­in­the­form­of­master­equations.

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different­types­of­bacteria­[24].­Fluctuations­ in­the­number­of­ molecules­ can­ drive­ large­ heterogeneities­ among­ cells­in­ different­ways.­ One­ is­ through­ positive­ feedback,­which­can­drive­ the­molecule­ to­be­present­at­either­high­or­ low­concentrations.­ These­ two­ concentration­ states­ can­ be­understood,­at­least­conceptually,­as­free­energy­minima­and­are­separated­by­an­energy­barrier­[1].­Dissipation­drives­the­molecular­concentration­to­reach­one­of­these­two­states­and­remain­there­forever­after.­Which­concentration­is­achieved­depends­on­the­initial­state;­that­is,­on­which­concentration­was­present­initially.­This­scenario­changes­when­we­take­into­account­that­there­are­fluctuations.­They­provide­the­energy­required­ to­ surpass­ the­ energy­ barrier­ that­ separates­ the­states,­allowing­a­switch­from­a­low­to­a­high­concentration­or­vice­versa­(Fig.­4).

An­ example­ of­ heterogeneous­ cell­ populations­ comes­from­experiments­using­mouse­embryonic­stem­cells­ (ESC),­which­ in­ culture­ express­ pluripotency­ factor­ NANOG­ in­ a­highly­ stochastic­ manner­ [8].­ NANOG­ allows­ ESCs­ to­ self-renew­ and­ to­ maintain­ their­ pluripotency.­ When­ NANOG­levels­ of­ individual­ ESCs­ in­ a­ culture­ are­ measured,­ the­distribution­of­values­is­very­broad.­If­cells­with,­for­instance,­low­ NANOG­ expression­ are­ selected,­ separated­ from­ the­others,­and­allowed­to­divide­over­time,­measurements­show­that­ the­ very­ broad­ distribution­ of­ NANOG­ concentrations­is­ eventually­ recovered.­Hence,­ some­cells,­ despite­ initially­being­ in­ the­ low­NANOG­ concentration­ state,­ have­ clearly­switched­ and­ now­ express­ very­ high­ concentrations­ of­NANOG.­ Whether­ this­ stochastic­ switching­ corresponds­to­ bistable­ or­ other­ type­ of­ dynamics­ is­ a­ current­ topic­ of­research.

A­ role­ of­ fluctuations­ in­ multicellular­ development­ has­been­proposed­for­cells­that­need­to­establish­a­pattern­that­is­not­ spatially­ordered­but,­ instead,­only­needs­ to­preserve­certain­proportions­of­different­types­of­cells,­randomly­spaced­around­the­tissue.­A­stochastic­decision­mechanism­has­been­proposed­for­processes­such­as­the­differentiation­of­different­photoreceptors­in­the­retina­of­humans­and­flies,­or­of­olfactory­cells­ in­ the­mouse­ [17].­Mice­ have­ 1000­ olfactory­ proteins,­with­ only­ one­ expressed­ in­ any­ given­ cell­ to­ avoid­ sensory­confusion.­ Hence,­ initially­ equivalent­ cells­ become­ distinct,­reaching­one­of­1000­different­states.­This­has­been­proposed­to­be­accomplished­by­the­activation­of­one­olfactory­protein­type­stochastically­and­subsequent­inhibition­of­all­the­other­remaining­types­of­olfactory­proteins.­In­addition,­fluctuations­of­molecular­components­can­be­expected­to­trigger­patterns­arising­ from­ interacting­ self-organizing­ dynamics­ such­ as­reaction-diffusion­and­lateral­inhibition.

Our­knowledge­on­the­effect­and­role­of­fluctuations­ in­developmental­processes­ is­ still­ limited.­However,­ research­in­ physics­ over­ the­ last­ few­ decades­ has­ evidenced­ that­nonlinear­ systems­ can­ take­ advantage­ of­ fluctuations­[22].­ Thus,­ it­ is­ to­ be­ expected­ that­ developing­organisms,­which­exhibit­highly­nonlinear­dynamics­and­are­subject­to­fluctuations,­profit­from­them­as­well.­The­concepts­and­tools­to­study­this­topic­have­already­been­developed­by­physicists­and­biologists,­and­the­results­should­soon­be­available.

Conclusions

The­ development­ of­ multicellular­ organisms­ is­ subject­ to­the­ physical­ laws­ that­ govern­Nature.­ It­ is­ indeed­ because­cells­live­out­of­equilibrium­that­they­are­able­to­create­the­myriad­of­rich­and­complex­structures­that­form­multicellular­organisms.­ Insights­ have­ been­ gained­ into­ some­ of­ the­molecular­ gene­ regulatory­ and­ signaling­mechanisms­ used­by­cells­in­the­spatially­and­temporally­coordinated­processes­that­ allow­ them­ to­ become­ distinct­ in­ an­ organized­ and­reproducible­ manner.­ These­ processes­ require­ nonlinear­responses­and­dynamics.­Previously,­development­was­mostly­understood­ as­ a­ succession­ of­ stationary­ states­ and­many­aspects­were­ described­ through­ averages­ over­many­ cells.­However,­we­ now­have­ strong­ evidence­ that­ development­is­ a­ highly­ dynamic­process­ and­ that­ cellular­ dynamics­ are­strongly­ stochastic.­ Although­many­ technical­ limitations­ to­advancing­ our­ knowledge­ remain,­ new­ data­ are­ expected­that­will­ reveal­ the­ highly­ complex­ and­ dynamic­ nature­ of­developing­organisms.­As­physicists,­we­expect­ to­continue­to­work­together­with­biologists­to­define­the­principles­that­govern­multicellular­organism­development.­

Competing interests. None declared

Acknowledgements. The­ authors­ acknowledge­ financial­ support­ by­project­ FIS2012-37655-C02-02­ by­ the­ Spanish­ Ministry­ de­ Economy­ and­Competitiveness­and­2014SGR878­from­the­Generalitat­de­Catalunya.

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About the image on the first page of this article. This­photograph­was­made­by­Prof.­Douglas­Zook­(Boston­University)­for­his­book­Earth Gazes Back [www.douglaszookphotography.com].­See­the­article­“Reflections:­The­enduring­symbiosis­between­art­and­science,”­by­D.­Zook,­on­pages­249-251­of­this­ issue­ [http://revistes.iec.cat/index.php/CtS/article/view/142178/141126].­This­ thematic­ issue­on­ “Non-equilibrium­physics”­ can­be­unloaded­ in­ISSUU­format­and­the­individual­articles­can­be­found­in­the­Institute­for­Catalan­Studies­journals’­repository­[www.cat-science.cat;­http://revistes.iec.cat/contributions].