The Informational Architecture Of The Cell

Sara Imari Walker1,2,3, Hyunju Kim1 and Paul C.W. Davies1 1 Beyond Center for Fundamental Concepts in Science, Arizona State University, Tempe, AZ 86281, USA

2 School of Earth and Space Exploration, Arizona State University, Tempe, AZ 86281, USA 3Blue Marble Space Institute of Science, Seattle WA USA

Keywords: Informational architecture, Boolean network, Yeast cell cycle, Top-down causation, Integrated Information, Information dynamics.

 

Abstract   In his celebrated book “What is Life?” Schrödinger proposed using the properties of living systems to constrain unknown features of life. Here we propose an inverse approach and suggest using biology  as  a  means  to  constrain  unknown  physics.  We  focus  on  information  and  causation,  as  their  widespread  use  in  biology  is  the  most  problematic  aspect  of  life  from  the  perspective  of  fundamental  physics.  Our proposal is cast as a methodology  for  identifying  potentially  distinctive  features  of  the  informational  architecture  of  biological  systems,  as  compared  to  other  classes  of  physical  system.  To  illustrate  our  approach,  we  use  as  a   case   study   a   Boolean   network   model   for   the   cell   cycle   regulation   of   the   single-­‐‑celled   fission   yeast  (Schizosaccharomyces   Pombe)   and   compare   its   informational   properties   to   two   classes   of   null  model   that  share   commonalities   in   their   causal   structure.   We   report   patterns   in   local   information   processing   and  storage   that   do   indeed   distinguish   biological   from   random.   Conversely,   we   find   that   integrated  information,   which   serves   as   a   measure   of   “emergent”   information   processing,   does   not   differ   from  random   for   the   case   presented.   We   discuss   implications   for   our   understanding   of   the   informational  architecture  of  the  fission  yeast  cell  cycle  network  and  for  illuminating  any  distinctive  physics  operative  in  life.    

1.  Introduction   Although   living   systems   may   be   decomposed   into   individual   components   that   each   obey   the   laws   of  physics,  at  present  we  have  no  explanatory  framework  for  going  the  other  way  around  -­‐‑  we  cannot  derive  life  from  known  physics.  This,  however,  does  not  preclude  constraining  what  the  properties  of  life  must  be  in  order  to  be  compatible  with  the  known  laws  of  physics.  Schrödinger  was  one  of  the  first  to  take  such  an  approach  in  his  famous  book  “What  is  Life?”  [1].  This  account  was  written  prior  to  the  elucidation  of  the  structure   of   DNA,   but   by   considering   the   general   physical   constraints   on   the  mechanisms   underlying  heredity,  he  reasoned  that  the  genetic  material  must  necessarily  be  an  “aperiodic  crystal”.  His  logic  was  two-­‐‑fold.  Heredity  requires  stability  of  physical  structure  –  hence  the  genetic  material  must  be  solid,  and  more   specifically   crystalline,   since   those   are   the  most   stable   solid   structures   known.  However,   normal  crystals  display  only  simple  repeating  patterns  and  thus  contain  little  information.  Schrödinger  therefore  correctly  reasoned  that  the  genetic  material  must  be  aperiodic  in  order  to  encode  sufficient  information  to  specify  something  as  complex  as  a  living  cell  in  a  (relatively)  small  number  of  atoms.  In  this  manner,  by  simple  and  general  physical  arguments,  he  was  able  to  accurately  predict  that  the  genetic  material  should  be  a  stable  molecule  with  a  non-­‐‑repeating  pattern  –  in  more  modern  parlance,  that  it  should  be  a  molecule  whose  information  content  is  algorithmically  incompressible.      Today  we  know  Schrödinger’s  “aperiodic  crystal”  as  DNA.  Watson  and  Crick  discovered  the  double  helix  just   under   a   decade   after   the   publication   of   “What   is   Life?”   [2].   Despite   the   fact   that   the   identity   of  

*Author for correspondence: Sara Imari Walker (sara.i.walker@asu.edu). †Present address: School of Earth and Space Exploration, Arizona State University, Tempe, AZ 86281, USA

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Schrödinger’s  aperiodic  crystal  was  discovered  over  sixty  years  ago,  we  are  in  many  ways  no  closer  today  than  we  were  before  its  discovery  to  having  an  explanatory  framework  for  biology.  While  we  have  made  significant  advances   in  understanding  biology  over   the   last  several  decades,  we  have  not  made  comparable  advances  in  physics  to  unite  our  new  understanding  of  biology  with  the  foundations  of  physics.  Schrödinger  used  physical   principles   to   constrain   unknown   properties   of   biology   associated   with   genetic   heredity.   One  might  argue  that  this  could  be  done  again,  but  now  at  the  level  of  the  epigenome,  or  interactome,  and  so  on  for  any  level  of  biological  organization,  and  not  just  for  the  genome  as  Schrödinger  did.  But  this  kind  of  approach  only  serves  to  predict  structures  consistent  with  the  laws  of  physics;  it  does  not  explain  why  they  should  exist  in  the  first  place.1  An  alternative  approach,  and  one  which  we  advocate  in  this  paper,  is  that  we  might  instead  use  insights  from  biology  to  constrain  unknown  physics.  That  is,  we  suggest  a  track  working  in  the  opposite  direction  from  that  proposed  by  Schrödinger:  rather  than  using  physics  to  inform  biology,  we  propose  to  start  by  thinking  about  biology  as  a  means  to  inform  potentially  new  physics,  if  indeed  such  missing  physics  exists.      The  divide  between  physics  and  biology  is  most  acute  with  regard  to  the  ontological  status  of  information  [3].   Progress   in   illuminating   the   physics   underlying   biology   is   therefore   most   likely   to   derive   from  studying  the  informational  properties  of  living  systems.    In  biology,  information,  in  its  many  forms,  is  a  core-­‐‑unifying  concept  [4;  5;  6;  7]  and  was  central  to  the  genomic  revolution  [5].  It  has  even  been  suggested  by   Maynard   Smith   that   in   the   absence   of   drawing   on   informational   analogies,   our   discoveries   in  molecular  biology  over  the  last  60  years  would  certainly  have  been  slower,  and  may  not  have  been  made  at  all  [8].  This  stands  in  stark  contrast  to  the  situation  in  physics,  which  in  our  current  formulation  has  no  natural   place   for   information   as   a   fundamental   concept.2  Physics   is   cast   in   terms   of   initial   states   and  (deterministic)  laws3.  Information,  on  the  other  hand,  by  its  very  definition,  requires  a  system  to  take  on  at  least  two  possible  states,  e.g.,  to  say  something  carries  1  bit  of  information  means  that  it  might  have  been  in   one   of   two   states.   Fixed   deterministic   laws   only   allow   one   outcome   for   a   given   initial   state,   and  therefore   do   not   allow   for   more   than   one   possibility.   Thus   “information”   cannot   be   instantiated   in  anything  but  a  trivial  sense  in  current  physics4.  A  key  open  question  is  whether  the  concept  of  information  as   applied   to   living  matter   is  merely   a   descriptive   tool   or   if   it   is   actually   intrinsic   to   life,   implying   the  existence   of   deeper   principles   currently  missing   from  our   formulation   of   fundamental   physics   [10;   11].  The   work   reported   here   should   cast   light   on   this   issue   by   providing   insights   into   how   information  operates  in  biological  systems.    

1 There is an interesting point here in the kind of prediction made by Schrödinger – he was able to accurately predict the general structure of the genetic material, but only by first taking as an axiom that hereditary structures exist that can reliably propagate sufficient “information” to specify an organism from one generation to the next. So his “first principles” argument already makes strong assumptions about physical reality and the ontological status of information. 2 Exceptions include a few oddities such as an exact informational interpretation of the von Neumann density matrix in quantum theory and the correspondence between the mathematical functions describing information and entropy. It remains open question whether the latter correspondence is physical or merely a mathematical coincidence. 3 Although there is evidence for quantum effects in biology here and there [9] we take the view here that life is fundamentally a classical phenomenon and as such do not worry about the issues of indeterminism that arise in quantum physics. 4 By trivial, here we mean that a system may of course be described externally in terms of information, operationally we do this with Shannon information all the time. One might therefore argue that information can make sense as currently described, so long as it is an open system. One could then take a larger and larger volume V of space-time, such that “external” is defined in order to describe “information” in larger and larger systems. However, in the limit 𝑉 → ∞ this approach inevitably breaks down. Our current physics therefore says nothing about how a system with a sender and receiver, communicating counterfactual statements, could possibly be physically instantiated in a universe where each may only ever be in one state, e.g., in the absence of an external observer. We operate under the assumption that all “observers” must be part of physical reality.

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In   what   follows,   we   propose   a   methodology   to   identify   potentially   distinctive   features   of   the  informational  architecture  of  biological  systems,  as  compared  to  other  classes  of  complex  physical  system.  We   utilize   the   terminology   “informational   architecture”   rather   than   “information”   or   “informational  pattern”  herein,  as  architecture  implies  physical  structure  whereas  mere  patterns  are  not  necessarily  tied  to  their  physical  instantiation  [12].  Our  approach  is  to  compare  the  informational  architecture  of  biological  networks  to  randomly  constructed  null  network  models  with  similar  physical  (causal)  structure  (e.g.,  size,  topology,  thresholds).  In  this  paper,  we  analyse  the  Boolean  network  model  for  the  cell  cycle  regulatory  network  of   the  single  celled  organism  fission  yeast  Schizosaccharomyces  Pombe   [13]  as  a  case  study  of   the  methodology  we  propose.  We  calculate  a  number  of  information-­‐‑theoretic  quantities  for  the  fission  yeast  cell   cycle   network   and   contrast   the   results  with   the   same   analysis   performed   on   null   random  network  models   of   two   different   classes:   Erdos-­‐‑Renyi   (ER)   random   networks   and   Scale-­‐‑Free   (SF)   random  networks.   The   latter   share   important   topological   properties   with   the   fission   yeast   cell   cycle   network,  including  degree  distribution.  By  comparing  results  for  a  biological  network  to  two  classes  of  null  models,  we  are  able  to  distinguish  which  features  of  the  informational  architecture  of  biological  networks  arise  as  a   result   of   common   causal   structure   (topology)   and   which   are   peculiar   to   biology   (and   presumably  generated   by   the   mechanism   of   natural   selection).   We   avoid   introducing   new   information-­‐‑theoretic  measures,  since  there  already  exist  a  plethora  of  useful  measures  in  the  literature.  Instead  we  choose  a  few  known  measures   relevant   to   biological   organization   –   including   those   of   integrated   information   theory  [14;  15]  and  information  dynamics  [16;  17].  Although  we  focus  here  on  gene  regulatory  networks,  we  note  that  our  analysis  is  not  level  specific.  The  formalism  introduced  is  intended  to  be  universal  and  may  apply  to  any   level   of   biological   organization   from   the   first   self-­‐‑organized   living   chemistries   to   technological  societies.  Although  this  special  theme  issue  is  focused  on  the  concept  of  “DNA  as  information”,  we  do  not  explicitly   focus   on   DNA   per   se.   We   instead   consider   distributed   information   processing   as   it   operates  within   the   cell,   as  we   believe   such   analyses   have   great   potential   to   explain,   for   example,  why  physical  structures  capable  of  storage  and  heredity,  such  as  DNA,  should  exist  in  the  first  place.  Thus,  we  regard  an  explanation  of  “DNA  as  information”  to  arise  only  through  a  proper  physical  theory  that  encompasses  living  processes.  We  hope  that  the  work  reported  here  will  be  a  step  towards  constraining  such  a  theory.    

2.   A   Phenomenological   Procedure   for   Mapping   the  Informational  Landscape  of  Non-­‐‑life  and  Life   A  widely  accepted  perspective  holds  that  while  information  is  useful  to  describe  biological  organization,  ultimately  all  of  biological  complexity  is,  at  least  in  principle,  fully  reducible  to  known  physics.  This  view  thus   regards   biology   as   being   complex,   but   does   not   view   this   complexity   as   requiring   new   physical  principles  to  explain  it  [18].  Information  can  therefore  only  be  descriptive.  A  contrasting  viewpoint  holds  that  information  is  more  than  merely  descriptive,  and  is  intrinsic  to  the  operation  of  living  systems  [4;  19]  thus  necessitating  a  fundamental  status  for  “information”  in  physical  theory,  which  is  not  accommodated  by   our   current   formulations   of   fundamental   physics.   A   pre-­‐‑requisite   for   the   latter   viewpoint   is   that  biological  systems  must  be  quantitatively  different  in  their  informational  architecture,  as  compared  to  other  classes  of  physical  systems,  as  proposed  by  us  in  [12],  hereafter  referred  to  as  KDW.      Herein   we   take   a   different,   but   complementary,   perspective   and   ask   how   the   patterns   uncovered   by  contrasting   the   informational   architecture   of   biological   systems   to   other  physical   systems  might   inform  the   physics   underlying   life,   if   such   a   new   frontier   in   physics   indeed   exists.   Like   in  KDW  we   focus   on  Boolean   network  models   for   a   real   biological   system   –   specifically   the   cell   cycle   regulatory   network   of  yeast  –  which  have  been  used  in  numerous  cases  as  accurate  models  of  biological  function  (see  e.g.,  [13;  20;  21]).   To   compare   the   properties   of   biological   networks   to   those   of   non-­‐‑living   physical   systems,   a   first  requirement   is   to  have  a   sample  of  non-­‐‑living  physical   systems  with  which   to  make  ready  comparison.  We   sampled   networks   that   are   randomized   relative   to   a   biological   network   under   the   constraint   of  retaining  certain  features  of  the  biological  network’s  causal  structure,  as  described  in  detail  in  KDW  [12]  (see  also  [22;  23;  24]).  We  then  compared  the  informational  properties  of  the  biological  network  with  that  

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of  random  ensembles  of  networks  generated.  Table  1  shows  the  relationships  between  the  biological  case  and  the  two  classes  of  random  causal  graphs  used  for  our  null  models  in  this  study:  Erdos-­‐‑Renyi  (ER)  and  Scale-­‐‑Free  (SF)  networks5.      Since  a  network’s  causal  structure  may  be  regarded  as  a  representation  of  its  physical  implementation,  we  view  manipulation  of  topological  features  as  manipulations  of  physical  structure.  We  thus  expect  that   if  biological  systems  are  “wired  differently”  with  regard  to  their  causal  structure  in  a  manner  that  enables  information  to  play  a  non-­‐‑trivial  role  in  their  dynamics,  this  will  be  uncovered  by  the  kinds  of  analyses  we  propose  herein.    2-­‐‑1)  Fission  Yeast  Cell-­‐‑Cycle  Regulatory  Network:  A  Case  Study  in  Informational  Architecture  

A  motivation  for  choosing  the  Boolean  network  model  of  the  fission  yeast  S.  Pombe  cell  cycle  in  this  study  is   that   it   is   relatively   small,  with   just   nine  nodes.   It   is   therefore   computationally   tractable   for   all   of   the  information   theoretic   measures   we   implement   in   our   study   –   including   the   computationally-­‐‑intensive  integrated   information   [14;   15].  The   fission  yeast   cell   cycle  network  also   shares   important   features  with  other  Boolean  network  models  for  biological  systems,  including  the  shape  of  its  global  attractor  landscape  and   resultant   robustness  properties   [13],   and   the  presence   of   a   control   kernel   (described  below)   [20].   It  therefore  serves  as  a  sort  of  “hydrogen  atom”  for  complex  information-­‐‑rich  systems.      The  dynamics  of  the  fission  yeast  cell  cycle  network  are  tightly  connected  to  its  function,  and  have  been  shown   to   accurately   track   the   phases   of   cellular   division   in   S.   Pombe   [13].  Although  many   of   the   fine  details  of  biological  complexity  (such  as  kinetic  rates,  signalling  pathways,  noise,  asynchronous  updating)  are   jettisoned   by   resorting   to   such   a   coarse-­‐‑grained   model,   it   does   retain   the   key   feature   of   causal  architecture.  The  network   is  shown  in  Fig.  1.  Each  node  corresponds   to  a  protein  needed  to  regulate  cell  cycle  function.  Nodes  may  take  on  a  value  of  1  or  0,  indicative  of  whether  the  given  protein  is  present  or  not.  Edges  represent  causal  biomolecular  interactions  between  two  proteins,  which  can  either  activate  or  inhibit  the  activity  of  individual  proteins.  The  successive  states  Si  of  node  i  are  determined  in  discrete  time  steps  by  the  updating  rule:    (1)  

   

where  aij  denotes  weight  for  an  edge  (i  ,  j)  and  θi  is  threshold  for  a  node  i.  The  threshold  for  all  nodes  in  the   network   is   0,   with   the   exception   of   Cdc2/13   and   Cdc2/13*,   which   have   thresholds   of   −0.5   and   0.5,  respectively.   For   each   edge,   a  weight   is   assigned   according   to   the   type   of   the   interaction:   aij(t)   =   −1   for  

5 In our study, Scale-Free (SF) network, unlike its common definition, does not mean that sample networks in the ensemble exhibit power-law degree distributions. Instead, the name emphasizes that the sample networks have the same exact degree sequence as the fission yeast cell cycle and hence the networks share the same bias in terms of global topology as the biological fission yeast cell-cycle network. For scale-free biological networks, the analogous random graphs would therefore be scale free.

DISCOVERY OF SCALING LAWS FOR INFORMATIONAL TRANSFER IN BIOLOGICAL NETWORKS 5

Si(t+ 1) =

8<

:

1,P

j aijSj(t) > ✓i.

0,P

j aijSj(t) < ✓i.

Si(t),P

j aijSj(t) = ✓i.

(1)

where aij is the edge weight between node i and j (e.g. aij = �1 for inhibition links andaij = 1 for activation links), and ✓i is the threshold for node i. The threshold for all nodesin both networks is 0, with the exception of Cdc2/Cdc13 and Cdc2/Cdc13*, which havethresholds of �0.5 and 0.5, respectively.

Although there are many other biological systems for which Boolean network modelshave been studied, we specifically focus on these two networks because they are both smalland both accurately model biological function. The small network size, with ⇠ 10 nodeseach, permits statistically more reliable comparison of results for the biological networksto the average properties of randomized networks of the same size, due to the relativelysmall ensemble size of random networks. Additionally, for both cell-cycle networks, thereis a direct connection between the dynamics on these networks and the correspondingbiological function, which is not the case for all Boolean network models of similar size.Iterating the set of rules in Eq. 1 correctly reproduces the sequences of protein statescorresponding to the phases of the cell-cycle for both networks (see [8, 9] for details). Thedynamics of the attractor landscape, which is associated with the execution of function, hasalso previously been rigorously quantified for both networks [8, 9]. The cell-cycle networkstherefore allow us to identify connections between distinctive features of informationalarchitecture uncovered in our analysis, and more traditional approaches to the dynamicsof biological systems.

An important feature of our analysis connecting informational architecture to dynamicsincludes detailing information transfer through the control kernel nodes for both networks.In Fig. 1, the nodes colored with red represent the control kernel for each network, whichwas recently discovered by Kim et al [10] in a number of Boolean models for biologicalnetworks. Their analysis uncovered a small fraction of nodes in a given biological Booleannetwork that can lead the network to one particular attractor state by fixing the valuesof those nodes to their values in the attractor state. The “control kernel” is then definedas the minimal set of nodes such that pinning their value to that of the primary attractorguarantees the convergence of the network state to the primary attractor associated withbiological function (See Fig. 1). Kim et al concluded that the control kernel acts tolocally regulate the global behavior the network. In our analysis, we address whether thisregulatory function is also associated with distinct patterns in information processing inbiological networks that are attributable to the presence of control kernel nodes.

2.2. Construction of Random Boolean Networks with Biological Constraints.To identify any features distinctive to information processing in biological networks, wecompare results of our information–theoretic analysis on both cell-cycles to the same anal-ysis performed on random networks. We utilize two classes of random networks in ourcomparative analysis: Erdos-Renyi (ER) networks and Scale-Free (SF) networks. Both

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inhibition   and   aij(t)   =   1   for   activation,   and   aij(t)   =   0   for   no  direct   causal   interaction.  This   simple   rule   set  captures   the   causal   interactions   necessary   for   the   regulatory   proteins   in   the   fission   yeast   S.   Pombe   to  execute  the  cell  cycle  process.  

2-­‐‑2)  The  Causal  Structure  of  the  Fission  Yeast  Cell-­‐‑Cycle  Network  

In   the  current  study,  we  will  make  reference   to   informational  attributes  of  both   individual  nodes  within  the  network,  and  the  state  of  the  network  as  a  whole,  which  is  defined  as  the  collection  of  all  Boolean  node  values   (e.g.,   at   one   instant   of   time).   We   study   both   since   we   remain   open-­‐‑minded   about   whether  informational   patterns   potentially   characteristic   of   biological   organization   are   attributes   of   nodes   or   of  states   (or   both).  When   referring   to   the   state-­‐‑space   of   the   cell   cycle   we  mean   the   space   of   the  2!   = 512  possible  states  for  the  network  as  a  whole.  We  refer  to  the  global  causal  architecture  of   the  network  as  a  mapping  between  network  states,  and  the  local  causal  architecture  as  the  edges  within  the  network.  

Time  Evolution  of  the  Fission  Yeast  Cell-­‐‑Cycle  Network.  Iterating  the  set  of  rules  in  Eq.  1  reproduces  the  time  sequence  of  network  states  corresponding  to  the  phases  of  the  fission  yeast  cell  cycle,  as  measured  in  vivo  by  the  activity  level  of  proteins  [13].  When  the  rules  in  Eq.  (1)  are  evolved  starting  from  each  one  of  the   512   possible   initial   states   (e.g.,   starting   from   every   state   in   the   state-­‐‑space),   one   can   obtain   a   flow  diagram  of  network  states  that  details  all  possible  dynamical  trajectories  for  the  network,  as  shown  in  Fig.  2.   The   flow   diagram   highlights   the   global   dynamics   of   the   network,   including   its   attractors.   In   the  diagram,  each  point  represents  a  network  state:  two  network  states  are  connected  if  one  is  a  cause  or  effect  of  the  other.  More  explicitly,  two  network  states  G  and  G’  are  causally  connected  when  either  𝐺 → 𝐺′  (G  is  a  cause  for  G’)  or  𝐺′ → 𝐺  (G’  is  the  cause,  and  G  the  effect)  when  the  update  rule  in  Eq.  1  is  applied  locally  to  individual  nodes.  The  notion  of  network  states  being  a  cause  or  an  effect  may  be  accommodated  within  integrated  information  theory  [14;  15],  which  we  implement  below  for  the  fission  yeast  cell  cycle  network.  We   note   that   because   the   flow   diagram   contains   all   mappings   between   network   states   it   captures   the  global  causal  structure  of  the  network,  encompassing  any  possible  state  transformation  consistent  with  the  local  rules  in  Eq.  1.    

Each  initial  state  flows  into  one  of  sixteen  possible  attractors  (15  stationary  states  and  one  limit  cycle).  The  network   state   space   can   be   sub-­‐‑divided   according   to  which   attractor   each   network   state   converges   to,  represented  by  the  colored  regions  in  the  left-­‐‑hand  panel  of  Fig.  2.  About  70%  of  states  terminate   in  the  primary   attractor   shown   in   red.     This   attractor   contains   the   biological   sequence   of   network   states  corresponding   to   the   four   stages   of   cell-­‐‑cycle   division: G1—S—G2—M,   which   then   terminates   in   an  inactive  G1  state  [13].  

The  Control  Kernel.  An  interesting  feature  to  emerge  from  previous  studies  of  the  fission  yeast  cell  cycle  network,  and  other  biologically  motivated  Boolean  network  models,   is   the  presence  of  a  small  subset  of  nodes  –  called  the  control  kernel  –  that  governs  global  dynamics  within  the  attractor  landscape  (see  Kim  et.  al.  [20]).  When  the  values  of  the  control  kernel  are  fixed  to  the  values  corresponding  to  the  attractor  the  entire  network  converges  to  that  attractor  (see  Fig.  2,  right  panel).  The  control  kernel  of  the  fission  yeast  network   is   highlighted   in   red   in   Fig.   1.    We   conjecture   that   the   control   kernel   has   evolved   to   play   an  important  role  in  the  informational  architecture  by  linking  local  and  global  causal  structure.  We  therefore  pay  particular  attention  to  information  flows  associated  with  the  control  kernel  in  what  follows.  

3.  Quantifying  Informational  Architecture      Information-­‐‑theoretic   approaches   have   provided   numerous   insights   into   the   properties   of   distributed  computation  and  information  processing  in  complex  systems  [14;  15;  16;  17;  25;  26;  27;  28].    Since  we  are  interested   in   level   non-­‐‑specific   patterns   that  might   be   intrinsic   to   biological   organization,  we   investigate  both   local   (node-­‐‑to-­‐‑node)  and  global   (state-­‐‑to-­‐‑state)   informational   architecture.  We  note   that   in  general,  

6    

biological  systems  may  often  have  more  than  two  “levels”,  but  focusing  on  two  for  the  relatively  simple  cell  cycle  network  is  a  tractable  starting  point.  To  quantify  local   informational  architecture  we  appeal  to  the   information   dynamics   developed   by   Schreiber   [28]   and   Lizier   et   al.   [16;   17;   25;   26].   For   global  architecture,   we   implement   integrated   information   theory   (IIT),   which   quantifies   the   information  generated   by   a   network   as   a   whole   when   it   enters   a   particular   state,   as   generated   by   its   causal  mechanisms   [14;   15].   In   this   section  we   illustrate  an  application  of   these  measures.    We  note   that  while  both  formalisms  have  been  widely  applied  to  complex  systems,  they  have  thus  far  seen  little  application  to  direct  comparison  between  biological  and  random  networks,  as  we  present  here.  We  also  note  that  this  study   is   the   first   to   our   knowledge   to   combine   these   different   formalisms   to   uncover   informational  patterns  within  the  same  biological  network.    

3-­‐‑1)  Information  Dynamics      Information   dynamics   is   a   formalism   for   quantifying   the   local   component   operations   of   computation  within  dynamical  systems  by  analysing  time  series  data  [16;  17;  25;  26].  We  focus  here  on  transfer  entropy  (TE)   [28]   and   active   information   (AI),   which   are   measures   of   information   processing   and   storage  respectively  [16;  17].  For  the  fission  yeast  cell  cycle  network,  time  series  data  is  extracted  by  applying  Eq.  1  for  20  time  steps,  for  every  possible  initial  state,  thus  generating  all  possible  trajectories  for  the  network.  Time  series  data  were  similarly  generated  for  each  instance  of  null  model  network.  The  trajectory  length  of   t=20   time   steps   is   chosen   to   be   sufficiently   long   to   capture   transient   dynamics   for   trajectories   before  converging   on   an   attractor   for   the   cell   cycle   and   for   the   vast  majority   of   random  networks   in   our   null  network  modes.  Using  this  time  series  data  we  then  extracted  the  relative  frequencies  of  various  temporal  patterns  expressed  as  a  fraction  of  all  possible  patterns.  We  use  these  relative  frequencies  as  probabilities  in  the  measures  TE  and  AI  discussed  below.  

Transfer  entropy  (TE)  is  the  (directional)  information  transferred  from  a  source  node  Y  to  a  target  node  X,  defined  as   the  reduction  in  uncertainty  provided  by  Y  about  the  next  state  of  X,  above  the  reduction  in  uncertainty  due  to  knowledge  of  the  past  states  of  X.  Formally,  TE  from  Y  to  X  is  the  mutual  information  between  the  previous  state  of  the  source  yn  and  the  next  state  of  the  target  xn+1  ,  conditioned  on  k  previous  

states  of  target,  x(k):    

(2)  

 

where  A0  indicates  the  set  of  all  possible  patterns  of  sets  of  states  (x(k),  xn+1,    yn).  The  directionality  of  TE  arises  due  to  the  asymmetry  in  the  computed  time  step  for  the  state  of  the  source  and  the  destination.  Due  to   this   asymmetry,   TE   can   be   utilized   to   measure   “information   flows”6,   absent   from   non-­‐‑directed  

6 Herein  we  use  the  term  “information  flow”  to  refer  to  the  transfer  of  information  between  nodes.  We  use  this  interchangeably  with  the  terminology  “information  transfer”  and  “information  processing”.  To  avoid  confusion  with  the  concept  of  information  flow  as  proposed  by  Ay  and  Polani,  which   is  a  measure  of   causation   [27],  we  refer   to   the  Ay  and  Polani  measure  as  “causal   information  flow”.  

T

Y!X

(k) =

P

(x(k)n ,xn+1,yn)2A0

p(x

(k)n

, x

n+1, yn) log2p(xn+1|x(k)

n ,yn)

p(xn+1|x(k)n )

7

measures  such  as  mutual  information.  

Active   information   (AI)   quantifies   the   degree   to   which   uncertainty   about   the   future   is   reduced   by  knowledge  of  the  past  from  examining  its  time  series.  Formally:    

(3)  

 

where  A1   indicates   the   set   of   all   possible   patterns   of   (x(k),  xn+1).   Thus,   AI   is   a  measure   of   the  mutual  information  between  a  given  node’s  k  previous  states  and  its  next  state.    

Both  TE  and  AI  depend  on  the  history  length,  k,  which  specifies  how  much  knowledge  of  the  past  states  of  a   given   node   affects   the   immediate   future.   Typically   one   considers  𝑘 → ∞  (see   e.g.,   [25]   for   discussion).  However,  we   suggest   that   short   history   lengths   can   provide   insights   into   how   a   biological   network   is  processing  information  more  locally  in  space  and  time.  In  particular,  we  note  that  it   is  unlikely  that  any  physical  system  would  contain  infinite  “knowledge”  of  past  states  [29;  30]  and  thus  that  the  limit  𝑘 → ∞  may  be  unphysical.   In   truncating   the  history   length  k,  we   treat  k  as  a  physical  property  of   the  network  related   to   knowledge   (memory)   about   past   states   as   stored   in   its   causal   structure.   This   enables   us   to  quantify  how  much  information  is  transferred  between  two  distinct  nodes  at  adjacent  time  steps,  and  thus  provides  insights  into  the  spatial  distribution  of  the  information  being  processed.  By  contrast,  we  view  AI  as  a  measure  of  local  information  processed  through  time.    

We   analyzed   information   storage   for   all   nodes   in   the   fission   yeast   network,   shown   in   Fig.   3,   which  includes   history   lengths  𝑘 = 1, 2… 10.   A   clear   trend   is   evident   in   the   dependency   of   AI   on   the   history  length:   larger   k  corresponds   to   larger   values   of  AI   for   every  node   in   the  network.  The   exception   is   the  node  SK,  which  functions  to  initiate  the  cell  cycle  process  (triggered  by  an  external  signal)  and  therefore  has  no  information  storage  for  any  k.  Control  kernel  nodes  are  highlighted  in  red.  For  every  history  length  k,  the  control  kernel  nodes  store  the  most  information  about  their  own  future  state,  as  compared  to  other  nodes  in  the  network.  Information  storage  can  arise  locally  through  a  node’s  self-­‐‑interaction  (self-­‐‑loop),  or  be   distributed   via   causal   interactions   with   neighboring   nodes.   Control   kernel   nodes   have   no   self-­‐‑interaction,   so   their   information  storage  must  be  distributed  among  direct  causal  neighbors.  Nodes   that  are  self-­‐‑interacting  (in  this  case  self-­‐‑degradation  of  the  protein  they  represent)  are  highlighted  in  blue  in  Fig.   3,   and   tend   to   have   relatively   low   AI   by   comparison.   This   suggests   that   the   distribution   of  information  storage  in  the  fission  yeast  cell  cycle  arises  as  a  result  of  distributed  storage  embedded  within  the  network’s  causal  structure  that  acts  primarily  to  reduce  uncertainty  in  the  future  state  of  the  control  kernel   nodes.  We   note   that   the   patterns   in   information   storage   reported   here   are   consistent   with   that  reported  in  [31]  since  our  networks  have  inhibition  links,  which  detract  from  information  storage.    

A  pressing  question  is  whether  the  above-­‐‑reported  features  are  specific  to  biology,  or  would  arise  in  many  similar  networks  with  no  biological  connection.  To  test  this,  we  compared  the  distribution  of  TE  and  AI  for  the  fission  yeast  cell  cycle  network  with  two  types  of  randomization  (ER  and  SF),  averaging  over  an  ensemble  of  1,000  networks.  The  results  are  shown  in  Fig.  4.    

We  first  consider  the  distribution  of  TE  among  pairs  of  nodes  (right  panel),  which  significantly  differs  for  the   yeast   cell   cycle   network   (red,   right   panel   Fig.   4)   as   compared   to   both   the   ER   and   SF   network  ensembles.   ER   networks   are   homogenous   in   their   degree   distribution   and   do   not   share   topological  features   with   either   the   biological   or   SF   networks,   except   for   nodes   with   self-­‐‑   degradation   loops.   The  relatively   low   information   transfer   among   nodes   observed   on   average   in   these   networks   (green,   right  panel  Fig.   4)   therefore   suggests   that  heterogeneity   in   the  distribution  of   edges  among  nodes   –   as   is   the  case  for  the  SF  and  biological  networks,  but  not  the  ER  networks  –  plays  as  significant  role  in  increasing  information  transfer  within  a  network.  However,  heterogeneity  alone  does  not  account  for  the  high  level  

A

X

(k) =

P

(x(k)n ,xn+1)2A1

p(x

(k)n

, x

n+1) log2p(xn+1|x(k)

n )p(xn+1)

8    

of   information   transfer   observed  between  nodes   in   the   biological   networks.  The   biological  networks   differ  from   the   SF   networks   despite   common   topological   features.  We   note   that   although   scale   free   networks  with  power  law  degree  distributions  have  been  much  studied  in  the  context  of  metabolic  networks,  signaling  networks,   protein-­‐‑protein   networks,   social   networks   and   the   internet   [32],   there   has   been   very   little  attention  given  to  how  actual  biological  systems  might  stand  out  as  different.  Here  both  the  biological  and  SF   networks   share   similarities   in   causal   structure   (inclusive   of   degree   distribution).   However,   the   cell  cycle   network   exhibits   statistically   significant   differences   in   the   distribution   of   information   processing  among  nodes.  The  excess  TE  observed  in  the  biological  networks  in  Fig.  4  deviates  between  1σ  to  5σ  from  that  of  the  SF  random  networks  (blue,  right  panel  Fig.  4),  with  a  trend  of  increasing  divergence  from  the  SF  ensemble  for  lower  ranked  node  pairs  that  still  exhibit  correlations  (e.g.,  where  TE  >  0).  As  reported  in  KDW,  this  biologically  distinct  regime  appears  to  be  correlated  with  the  presence  of  the  control  kernel  (see  KDW   for   extended   discussion   [12]).   The   biological   network   is   also   an   outlier   in   the   total   quantity   of  information  processed  by  the  network,  processing  more  information  on  average  than  either  the  ER  or  SF  random  null  models.      

For   the   present   study  we   also   computed   the   distribution   of  AI   for   the   biological   network   and   random  network   ensembles   (left   panel   in   Fig.   4).   Our   results   for   the   distribution   of   AI   do   not   distinguish   the  biological   network   from   random.  Nodes   in   the   biological   network   have   similar   information   storage   on  average   to   those   in   the  null  model  ensembles.  Why   is   this?   Information  storage   is  attributable   to  causal  structure,  as  described  above.  Our  null  models  are  constructed  to  maintain  features  of  the  fission  yeast  cell  cycle  network’s  causal  structure  (see  Table  1).  It  is  therefore  not  so  surprising  that  the  null  models  should  share   common   properties   in   their   information   storage.   However,   regulation   of   control   kernel   nodes   is  associated  with  the  specific  attractor  landscape  unique  to  the  biological  network,  which  in  general  will  not  be   shared   by   null  models   drawn   from   the   ensembles   SF   and  ER  networks   (e.g.,   these   networks   do   not  share   the   same   attractor   landscape   as   shown   in   Fig.   2).   The   interpretation   of   the   distribution   of  information   storage   among   nodes   is   therefore   very   different   for   the   null   models   than   the   biological  network.  For  the  biological  network,  control  kernel  nodes  contribute  the  most  to  information  storage  and  are  also  associated  with   regulation  of   the  dynamics  of   the  network   through  causal   intervention.  For  ER  and   SF   ensembles,   these   nodes   likewise   store   a   large   amount   of   information   due   to   their   local   causal  structure,  but  do  not  have  a  special  role  in  the  global  causal  structure  of  the  network.    

3-­‐‑2)  Effective  and  Integrated  Information    Whereas   information   dynamics   quantifies   patterns   in   local   information   flows,   integrated   information  theory  (IIT)  quantifies  information  arising  due  to  the  uncertainty  reduced  by  knowledge  of  the  network’s  causal  mechanisms  (defined  by  its  state-­‐‑to-­‐‑state  transitions)  [14;  15].  As  such,  it  captures  important  aspects  of   global   causal   structure.   IIT  was  developed  as   a   theory   for   consciousness,   for   technical  details   on   the  theory  see  e.g.,   [33].  Herein,  we  use  IIT  to  quantify  effective  information   (EI)  and   integrated  information   (φ)  (defined  below)  for  the  fission  yeast  cell  cycle  and  the  ensembles  of  random  ER  and  SF  networks.  Unlike  information  dynamics,  both  EI  and  φ  characterize  entire  network  states,  rather  than  individual  nodes,  and  they  do  not  require  time  series  data  to  compute.  One  can  calculate  EI  or  φ  for  all  causally  realizable  states  of   the   network   (states   the   network   can   enter   via   its   causal   mechanism)   independent   of   calculating  trajectories  through  state-­‐‑space.  These  measures  may  in  turn  be  mapped  to  the  global  causal  structure  of  the  network’s  dynamics  through  state  space,  e.g.  for  the  fission  yeast  cell  cycle,  the  structure  of  the  flow  diagram  in  Fig.  2.    

Effective   information   (EI)  quantifies   the   information  generated  by   the  causal  mechanisms  of  a  network,  when  it  enters  a  particular  network  state  G’  (as  defined  by  its  edges,  rules  and  thresholds  –  see  e.g.,  Eq.  (1)).   More   formally,   the   effective   information   for   each   realized   network   state   G’,   given   by   EI(G’),   is  calculated  as  the  relative  entropy  of  the  a  posteriori  repertoire  with  respect  to  the  a  priori  repertoire:    

(4)      𝐸𝐼 𝐺   → 𝐺! = 𝐻 𝑝!"# 𝐺 − 𝐻(𝑝(𝐺   → 𝐺!)  

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The  a  priori  repertoire  is  defined  is  as  the  maximum  entropy  distribution,  𝑝!"# 𝐺 ,  where  all  network  states  are   treated  as  equally   likely.  The  a  posteriori  repertoire,  𝑝(𝐺   → 𝐺!),  is  defined  as   the  repertoire  of  possible  states   that  could  have   led   to   the  state  G’   through  the  causal  mechanisms  of   the  system.   In  other  words,  EI(G’)  measures  how  much  the  causal  mechanism  reduces   the  uncertainty  about   the  possible  states   that  might  have  preceded  G’,  e.g.,  its  possible  causes.  For  the  fission  yeast  cell  cycle,  the  EI  of  a  state  is  related  to  the  number  of  in-­‐‑directed  edges  (causes)  in  the  flow  diagram  of  Fig.  2.    

Integrated  information   (φ)  quantifies  by  how  much  the  “sum  is  more  than  the  parts”  and  is  quantified  as  the  information  generated  by  the  causal  mechanisms  of  a  network  when  it  enters  a  particular  state  G’,  as  compared   to   sum   of   information   generated   independently   by   its   parts.   More   specifically,   φ   can   be  calculated  as  follows:  1)  divide  the  network  entering  a  state  G’  into  distinctive  parts  and  calculate  EI  for  each  part,  2)  compute  the  difference  between  the  sum  of  EIs  from  every  part  and  EI  of  the  whole  network,  3)  repeat  the  first  two  steps  with  all  possible  partitions.  φ  is  then  the  minimum  difference  between  EI  from  the  whole  network  and  the  sum  of  EIs  for  its  parts  (we  refer  readers  to  [14]  for  more  details  on  calculating  φ).  If  φ(G’)  >  0,  then  the  causal  structure  of  network  generates  more  information  as  a  whole,  than  as  a  set  of  independent  parts  when  it  enters  the  state  G’.  For  φ(G’)  =  0,  there  exist  causal  connections  within  the  network  that  can  be  removed  without  leaking  information.  

The  distribution  of  EI  for  all  accessible  states  (states  with  at  least  one  cause)  in  the  fission  yeast  cell-­‐‑cycle  network   is   shown   in   Fig.   5   where   it   is   compared   to   the   averaged   EI   for   the   ER   and   SF   null   network  ensembles.  For  the  biological  network,  most  states  have  EI  =  8,  corresponding  to  two  possible  causes  for  the  network  to  enter  that  particular  state.  Comparing  the  biological  distribution  to  that  of  the  null  model  random   network   ensembles   does   not   reveal   distinct   differences   (the   biological   network   is   within   the  standard   deviation   of   the   SF   and   ER   ensembles),   as   it   did   for   information   dynamics.   Thus,   the   fission  yeast  cell  cycle’s  causal  mechanisms  do  not  statistically  differ  from  SF  or  ER  networks  in  their  ability  to  generate   effective   information.   Stated   somewhat   differently,   the   statistics   over   individual   state-­‐‑to-­‐‑state  mappings   within   the   attractor   landscape   flow   diagram   for   the   fission   yeast   cell   cycle   (Fig.   2)   and   the  ensembles  of  randomly  sampled,  null  model  networks  are  statistically  indistinguishable.    

Fig.  6  shows  φ  for  all  network  states  that  converge  to  the  primary  attractor  of  the  fission  yeast  network  (all  network   states   within   the   red   region   in   the   left   panel   of   Fig.   2).   Larger   points   denote   the   biologically  realized  states,  which  correspond  to  those  that  a  healthy  S.  Pombe  cell  will  cycle  through  during  cellular  division  (e.g.,  the  G1—S—G2—M  phases).  Initially,  we  expected  that  φ  might  show  different  patterns  for  states  within   the   cell   cycle   phases,   as   compared   to   other   possible   network   states   (e.g.,   that   biologically  functional   states  would   be  more   integrated).   However,   the   result   demonstrates   that   there   are   no   clear  differences   between   φ   for   biologically   functional   states   and   other   possible   network   states.   We   also  compared  the  average  value  of   integrated   information,  𝜙!"#,  taken  over  all   realizable  network  states   for  the  fission  yeast  cell  cycle  network,  to  that  computed  by  the  same  analysis  on  the  ensembles  of  ER  and  SF  random  networks.  We  found  that  there  is  no  statistical  difference  between  𝜙!"#  for  the  biological  network  and   the   random  networks  generated  based  on  our  null  models:   as   shown   in  Table   2,   all  models   in  our  study  show  statistically  similar  averaged  network  integration.    At first, we were surprised that neither EI nor φ  successfully  distinguished  the  biological  fission  yeast  cell  cycle  network  from  the  ensembles  of  ER  or  SF  networks.  It  is  widely  regarded  that  a  hallmark  of  biological  organization  is  that  “more  is  different”  [34]  and  that  it  is  the  emergent  features  of  biosystems  that  set  them  apart  from  other  classes  of  physical  systems  [35].  Thus,  we  expected  that  global  properties  would  be  more  distinctive  to  the  biological  networks  than  local  ones.  However,  for  the  analyses  presented  here,  this  is  not  the   case:   the   local   informational   architecture,   as   quantified   by   TE,   of   biological   networks   is   markedly  distinct   from  random  ensembles:  yet,   their  global  structure,  as  quantified  by  EI and φ,   is  not.  There  are  several  possible  explanations  for   this  result.  The  first   is   that  we  are  not   looking  at   the  “right”  biological  network  to  observe  the  emergent  features  of  biological  systems.  While  this  may  be  the  case,   this  type  of  argument  is  not  relevant  for  the  objectives  of  the  current  study:  if  biological  systems  are  representative  of  

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a   class   of   physical   system  distinguished   by   its   informational   structure,   than   one   should   not   be   able   to  cherry-­‐‑pick  which  biological   systems  have   this  property   –   it   should  be   a  universal   feature  of  biological  organization.  Hence  our  loose  analogy  to  the  “hydrogen  atom”  –  given  the  universality  of  the  underlying  atomic  physics  we  would  not  expect  helium  to  have  radically  different  physical  structure  than  hydrogen  does.  Thus,  we  expect   that   if   this   type  of  approach   is   to  have  merit,   the  cell   cycle  network   is  as  good  a  candidate  case  study  as  any  other  biological  system.  We  therefore  consider  this  network  as  representative,  given   our   interest   in   constraining   what   universal   physics   could   underlie   biological   organization.     One  might  further  worry  that  we  have  excised  this  particular  network  from  its  environment  (e.g.,  a  functioning  cell,  often  suggested  as  the  indivisible  unit,  or  “hydrogen  atom  of  life”).    However,  this  kind  of  excision  should  diminish   emergent   informational  properties   of   biological   organization.   It   is   then   surprising   that  the  local  signature  in  TE  is  so  prominent,  while  EI  and  φ  are  not  statistically  distinct.    

Another  explanation   for  our  results   is   that  we  have  defined  our  random  networks   in  a  way   that  makes  them  too  similar  to  the  biological  case,  thus  masking  some  of  the  differences  between  life  and  non-­‐‑life.  It  is  indeed   likely   that   biologically-­‐‑inspired   “random”   networks   will   mimic   some   features   of   biology   that  would  be  absent   in  a  broader  class  of  random  networks.  However,   if  our  random  graph  construction   is  too  similar  to  the  biological  networks  to  pick  up  important  distinctive  features  of  biological  organization,  than  it  does  not  explain  the  observed  unique  patterns  in  TE  nor  that  AI  is  largest  for  control  kernel  nodes  which  play  a  prominent   role   in   regulation  of   function.  We   therefore  proceed  by  considering   the   lack  of  distinct  global  patterns  in  information  generated  due  to  causal  architecture  as  a  real  feature  of  biological  organization  and  not  an  artifact  of  our  construction,  at  least  for  the  fission  yeast  cell  cycle  network.    

4. Characterizing  Informational  Architecture     The   forgoing   analyses   indicate   that   the   “emergent”   features   of   biological   organization   should   be  characterized   as   a   combination   of   topology   and   dynamics.   In   our   example,   what   distinguishes   biology  cannot  be  global   topological   features  alone,  since  the  SF  networks  differ   in  their  patterns  of   information  processing  from  the  fission  yeast  cell  cycle  despite  sharing  common  topological  features,  such  as  degree  distribution.  Similarly,  it  cannot  be  dynamics  alone  due  to  the  lack  of  a  distinct  signature  in  EI  or  φ  for  the  biological  network,  as  both  are  manifestations  of  global  causal  structure  (e.g.,  the  structure  of  the  attractor  landscape).  We  therefore  conjecture   that   the  signature  of  biological   function   lies   in  how  correlations  are  distributed  through  space  and  time  in  biological  networks.  In  other  words,  that  biology  is  distinguished  as  a  physical  system  not  by  its  causal  structure,  which  is  set  by  the  laws  of  physics,  but  in  how  the  flow  of  information  directs  the  execution  of   function.  This   is  consistent  with   the  distinct  scaling   in   information  processing   (TE)  observed  for  the  fission  yeast  cell  cycle  network  as  compared  to  ER  and  SF  ensembles.        An  important  question  opened  by  our  analysis  is  why  the  biological  network  exhibits  distinct  features  for  TE  and  AI  when  global  measures  yield  a  null  result.  We  suggest  that  the  separate  analysis  of  two  distinct  levels  of   informational  patterns   (e.g.  node-­‐‑node  or   state-­‐‑state)   as  presented  above  misses  what  arguably  may  be  one  of  the  most  important  features  of  biological  organization  –  that  is,  that  distinct  levels  interact.  The  interaction  between  distinct  levels  of  organization  is  typically  described  as  ‘top-­‐‑down’  causation  and  has  previously  been  proposed  as  a  hallmark  feature  of  life  [35;  36;  37]  We  hypothesize  that  the  lower  level  patterns   observed  with   TE   and  AI   arise   because   of   the  manner   in  which   network   integration   operates  through   the   dynamics   of   the   fission   yeast   cell   cycle   network.   Instead   of   studying   individual   levels   of  architecture  as   separate  entities,  we  must   therefore  additionally   study   informational  patterns  mediating  the   interactions   between   different   levels   of   informational   structure   as   distributed   in   space   and   time.  We  next  assess  some  aspects  of  this  hypothesis  by  analysing  the  spatiotemporal  and  inter-­‐‑level  architecture  of  the  fission  yeast  cell  cycle.     4-­‐‑2)  Spatiotemporal  Architecture  

11

 We  may  draw  a  loose  analogy  between  information  and  energy.  In  a  dynamical  system,  there  will  be  two  characteristic   time  scales:  a  dynamical  process   time   (e.g.,   the  period  of  a  pendulum)  and   the  dissipation  time  (e.g.,  time  to  decay  to  an  equilibrium  end  state  or  attractor.)  TE  and  AI  are  calculated  utilizing  time  series  data  of  a  network’s  dynamics  and,  as  with  energy,  there  are  also  two  distinct  time  scales  involved:  the   history   length   k   and   the   convergence   time   to   the   attractor   state(s),   which   are   characteristic   of   the  dynamical   process   time   and   dissipation   time,   respectively.   For   example,   the   dissipation   of   TE   may  correlate  with  biologically  relevant  time  scales  for  the  processing  of  information,  which  can  be  critical  for  interaction   with   the   environment   or   other   biological   systems.   In   the   case   study   presented   here,   the  dynamics  of  TE  and  AI  can  provide   insights   into   the   timescales  associated  with   information  processing  and  memory  within  the  fission  yeast  cell  cycle  network.      The  TE  scaling  relation  for  the  fission  yeast  cell  cycle  network  is  shown  in  Fig.  7  for  history  lengths  k  =  1,  2  …   10   (note:   history   length   k   =   2   is   compared   to   null   models   in   Fig.   2).   The   overall   magnitude   and  nonlinearity  of  the  scaling  pattern  decreases  as  knowledge  about  the  past  increases  with  increasing  k.  The  patterns  in  Fig.  7  therefore  indicate  that  the  network  processes  less  information  in  the  spatial  dimension  when   knowledge   about   past   states   increases.     In   contrast,   the   temporal   processing   of   information,   as  captured  by  information  storage  (AI)  increases  for  increased  k  (See  Fig.  3),  as  discussed  above.      To  make  explicit  this  trade-­‐‑off  between  information  processing  and  information  storage  we  define  a  new  quantity.  Consider   in-­‐‑coming  TE,  and  out-­‐‑going  TE   for  each  node  as   the   total  sum  of  TE  from  the  rest  of  network  to  the  node  and  the  total  sum  of  TE  from  the  node  to  the  rest  of  network,  respectively.  We  then  define  the  Preservation  Entropy  (PE),  as  follows:    

(5)              where    AX(k)  ,TIX(k)  and  TOX(k)  denote    AI,    in-­‐‑coming  TE,    and  out-­‐‑going  TE.    PE  therefore  quantifies  the  difference   between   the   information   stored   in   a   node   and   the   information   it   processes.   For   PE(X)   >   0,   a  node  X’s  temporal  history  (information  storage)  dominates  its  information  dynamics,  whereas  for  PE(X)  <  0,   the   information   dynamics   of   node  X   are   dominated   by   spatial   interactions  with   rest   of   the   network  (information  processing).    Preservation  entropy  is  so  named  because  nodes  with  PE  >  0  act  to  preserve  the  dynamics  of  their  own  history.    Fig.  8  shows  PE  for  every  node  in  the  fission  yeast  network,  for  history  length  k  =  1,  2  …  10.  As  the  history  length  increases  the  overall  PE  also   increases,  with  all  nodes  acquiring  positive  values  of  PE  for   large  k.  For  all  k,  the  control  kernel  nodes  have  the  highest  PE,  with  the  exception  of  the  start  node  SK  (which  only  receives  external  input).  Stepping  from  k  =  3  to  k  =  4,  it  is  clearly  evident  that  the  cell  cycle  network  has  a  transition   from   being   space   (processing)   dominated   to   time   (storage)   dominated   in   its   informational  architecture.  When  the  dynamics  of  the  cell  cycle  network  is  initiated,  knowledge  about  future  states  can  only  be  stored  in  the  spatial  dimension.    The  transition  to  temporal  storage  first  occurs  for  the  four  control  kernel  nodes,  which  for  k  =  4  have  positive  values  for  PE,  while  others  nodes  have  negative  PE  (self-­‐‑loops  nodes)  or  PE  close  to  0  (these  nodes  make  the  transition  to  PE  >  0  by  k  =  5).  This  suggests  that  the  early  time  evolution  of   the  cell   cycle   is  process  dominated  and   transitions   to  being  storage  dominated  with  a  characteristic  time  scale  associated  with  the  knowledge  that  past  states  of  the  control  kernel  contain  about  their  future  state.      4-­‐‑3)  Inter-­‐‑level  Architecture    We  pointed   out   in   Section   3   the  peculiar   fact   that   the   local   level   informational   architecture  picks   out   a  clear   difference   between   biological   and   random   networks,   whereas   the   global   measures   do   not.   We  conjecture   that   this   is   because   in   biological   systems   the   information   flows   that  matter   to  matter   occur  

PX(k) = AX(k)� 12

�T IX(k) + TO

X (k)�

12    

between   levels,   i.e.  from  local  to  global  and  vice-­‐‑versa.  In  the  case  study  presented  here,  this  means  there  should   be   information   flow   arising   from   node-­‐‑state,   or   state-­‐‑node   interactions.   To   investigate   this,   we  treat  φ  itself  as  a  dynamic  time  series  and  ask  whether  the  dynamical  behavior  of   individual  nodes   is  a  good  predictor   of  φ   (i.e.,   of   network   integration),   and   conversely,   if   network   integration   enables   better  prediction  about  the  states  of  individual  nodes.  To  accomplish  this  we  define  a  new  Boolean  “node”  in  the  network,  named  the  Phi-­‐‑node,  which  encodes  whether  the  state  is  integrated  or  not,  by  setting  its  value  to  1  or  0,  respectively  (in  a  similar  fashion  to  the  mean-­‐‑field  variable  in  [35]).  We  then  measure  TE  between  the   state  of   the  Phi-­‐‑node  and   individual  nodes   for   all  possible   trajectories  of   the   cell   cycle,   in   the   same  manner  as  was  done  for  calculating  TE  between  local  nodes.  Although  the  measure  was  not  designed  for  such  analyses,   there   is  actually  nothing  about   the  structure  of   these   information  measures   that  suggests  they  must   be   level   specific   (see   e.g.,   [38]   for   an   example   of   the   application   of   EI   at   different   scales   of  organization).  Indeed,  higher  transfer  entropy  from  global  to  local  scales  than  from  local  to  global  scales  has  previously  been  put  forward  as  a  possible  signature  of  collective  behavior  [35;  39;  40;  41].  We  note  that  since  TE  is  correlative,  we  do  not  need  to  make  any  assumptions  about  whether  φ  is  causally  driving  the  dynamics  of  the  network  or  not,  e.g.  about  whether  top-­‐‑down  causation  is  operative7.    

The  results  of  our  analysis  are  shown  in  Fig.  9.  The  total  information  processed  (total  TE)  from  the  global  to  local  scale  is  1.6  times  larger  than  total  sum  of  TE  from  the  local  to  global  scale.  That  is,  the  cell  cycle  network  tends  to  transfer  more  information  from  the  global  to  local  scales  (top-­‐‑down)  than  from  the  local  to  global  (bottom-­‐‑up),  indicative  of  collective  behavior  arising  due  to  network  integration.  Perhaps  more  interesting  is  the  irregularity  in  the  distribution  of  TE  among  nodes  for  information  transfer  from  global  to  local   scales   (shown   in  purple   in  Fig.  9),   as   compared   to  a  more  uniform  pattern   in   information   transfer  from  local  to  global  scales  (shown  in  orange  in  Fig.  9).  This  suggests  that  only  a  small  fraction  of  nodes  act  as  optimized  channels   for   filtering  globally   integrated   information   to   the   local  scale.  This  observation   is  consistent  with  what  one  might  expect  if  global  organization  is  to  drive  local  dynamics  (e.g.,  as  is  the  case  of  top-­‐‑down  causation),  as  this  must  ultimately  operate  through  the  causal  mechanisms  at  the  lower  level  of   organization.   Our   analysis   suggests   a   promising   line   of   future   inquiry   characterizing   how   the  integration  of  biological  networks  may  be   structured   to   channel  global   state   information   through  a   few  nodes   to  regulate   function  (e.g.,  such  as   the  control  kernel).  Future  work  will   include  comparison  of   the  biological  distribution  to  null  network  models  to  gain  further  insights  into  if  and  how  this  feature  may  be  unique   to   biology.   We   note,   however,   that   this   kind   of   analysis   requires   one   to   regard   the   level   of  integration   of   a   network   as   a   “physical”   node.   But   perhaps   this   is   not   too   radical   a   step:     effective  dynamical  theories  often  treat  mean-­‐‑field  quantities  as  physical  variables.    

5. Discussion      

7  We  note  that  TE  and  AI  quantify  correlations  within  a  dynamical  system  and  are  agnostic  to  the  local  causal  mechanisms  that  underlie  those  correlations  (e.g.,  edges  in  the  network  graph).  They  thus  differ  from  local  causal  measures,  such  as  the  causal  information  flow  introduced  by  Ay  and  Polani,  which  measures  direct  causal  effect  [27]  using  the  intervention  calculus  of  Pearl  [43].  In  the  analysis  presented  herein,  we  have  considered  information  dynamics  as  capturing  more  distributed  causal  properties  of  networks,  which  emerge  from  a  network’s  global  topological  structure  and  dynamics,  rather  than  strictly  through  nearest  neighbour  interactions  between  nodes  (as  is  captured  by  a  direct  causal  effect  (edge)  and  measured  by  causal  flow).  Thus,  our  analysis  is  suggestive  of  distributed  causation,  or  common  causes,  operative  in  the  fission  yeast  cell  cycle,  but  does  not  explicitly  measure  causation  in  the  strict  formal  sense  of  Ay  and  Polani  or  Pearl.  

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It  is  obvious  that  living  systems  are  significantly  distinct  from  non-­‐‑living  systems,  perhaps  constituting  a  separate  state  of  matter.  However,  it  is  notoriously  hard  to  pin  down  precisely  what  it  is  that  makes  living  organisms  so  special.  We  suggest  that  the  way  forward  is  to  focus  on  the  informational  properties  of  life,  as   it   is   these   that   so   sharply   distinguish   biological   explanations   from   those   of   physics   and   chemistry.  Given   that   efficient   information  management   is   so   crucial   a   feature   of   biological   functionality,   it   seems  clear   that  natural  selection  will  have  operated  to  evolve  certain  distinctive   informational  signatures   (i.e.,  biological   systems  will   be   “wired   differently”).   To  make   progress   in   testing   this   assumption,   it   is   first  necessary   to   find   quantitative   formalisms   to   describe   the   informational   organization   of   living   systems.  Several  measures  of  both  information  storage  and  information  flow  have  been  studied  in  recent  years,  and  we  have  applied  these  measures  to  a  handful  of  tractable  biological  systems.  Here  we  have  reported  our  results  on  the  regulatory  gene  and  protein  network  that  controls  the  cell  cycle  of  fission  yeast,  treated  as  a  simple  Boolean  dynamical  system.  We  confirmed  that  there  are  indeed  informational  signatures  that  pick  out   the   biological   versus   suitably  defined   random  comparison  networks.   Intriguingly,   the   relevant   bio-­‐‑signature   measures   are   those   that   quantify   local   information   transfer   and   storage   (TE   and   AI  respectively),   whereas   global   information  measures   such   as   Tononi’s   integrated   information  φ   do   not.  Should   our   results   turn   out   to   be   generic   for   a   wide   class   of   living   systems,   we   face   the   challenge   of  explaining  why  this  is  the  case.  Given  that  biological  systems  seem  to  exemplify  the  dictum  that  “more  is  different”    [34]  one  might  expect φ  to  manifest  some  form  of  emergence.  However,  given  that  φ  is  defined  in  terms  of  lower  level  causal  mechanisms  it  must  be  reducible  to  the  underlying  physics.  That  is,  φ  is  an  attribute  emerging  from  causal  architecture  and  not  informational  architecture.  For  this  reason  we  do  not  expect  φ  to  distinguish  other  biological  networks  from  similarly  constructed  causal  graphs.    In  our  view,  the  distinctive  signature  of  biological  information  management  lies  not  with  the  underlying  causal   structure   (the   network   topology)   but  with   the   informational   architecture,   and   in   particular   how  that   informational  architecture  “interacts”  with   the  causal   structure.  Evidence   for   this  view  comes   from  the  fact  that  the  integration  of  the  network  is  a  better  predictor  of  the  states  of  individual  nodes,  than  vice  versa   (see   Fig.   9),   an   asymmetry   perhaps   related   to   biological   functionality   of   the   network.   Further  support   comes   from  the  distinct   scaling   relation  we   found   for   information   transfer  as   compared   to  null  models.  The   ensemble  of   SF  networks   in  our   study   share   common   topological   features  with   the   fission  yeast   cell   cycle   network,   including   degree   distribution,   yet   they   statistically   differ   in   the   scaling   of  information   transfer   between   nodes.   In   other   words,   although   the   biological   fission   yeast   cell   cycle  network  and  ensemble  of  SF  null  model  networks  share  commonalities  in  causal  structure,  the  pattern  of  information   flows   for   the  biological  network   is  quite  distinct.  While   it   is  conceivable   that   these  patterns  are  a  passive,  secondary,  attribute  of  biological  organization  arising  via  selection  on  other  features  (such  as   robustness,   replicative   fidelity   etc),   we   think   the   patterns   are   most   likely   to   arise   because   they   are  intrinsic  to  biological  function  –  that  is,  they  direct  the  causal  mechanisms  of  the  system  in  some  way,  and  thereby  constitute  a  directly  selectable  trait  [5;  42].      If  the  patterns  of  information  processing  observed  in  the  biological  network  are  indeed  a  joint  product  of  information   architecture   and   causal   structure,   as   we   conjecture,   they   may   be   regarded   as   an   emergent  property   of   topology   and   dynamics.   In   this   respect,   biological   information   organization   differs   from   other  classes   of   collective   behaviour   commonly   described   in   physics   [44].   In   particular,   the   distribution   of  correlations   indicates   that   we   are   not   dealing   with   a   critical   phenomenon   in   the   usual   sense,   where  correlations  at  all  length  scales  exist  in  a  given  physical  system  at  the  critical  point,  without  substructure.  Instead,   we   find   “sub-­‐‑critical”   collective   behavior,   where   a   few   network   nodes   regulate   collective  behavior  through  the  global  organization  of  information  flows.  The  question  arises  of  what  features  of  cell  cycle  organization  lead  to  this  emergent  property.  We  believe  that  the  answer  lies  with  the  control  kernel,  which  has  the  requisite  property  of  connecting  informational  architecture  to  the  causal  mechanisms  of  the  network.  The   control   kernel  nodes  were   first  discovered  by  pinning   their  values   to   that   of   the  primary  attractor   –   e.g.,  by   causal   intervention   (see   e.g.,   [43]).  Causal   intervention  by   an   external   agent  does  not  occur   “in   the   wild”.   We   therefore   posit   that   the   network   is   organized   such   that   information   flowing  through  the  control  kernel  performs  an  analogous  function  to  an  external  causal  intervention.  Indeed,  this  

14    

is  corroborated  by  the  manner  in  which  the  network  transitions  from  being  information  “processing”  to  “storage”  dominated,  where  the  control  kernel  nodes  play  the  dominant  role  in  information  storage  for  all  history   lengths   k   and   are   the   first   nodes   to   transition   to   storage-­‐‑dominated   dynamics.  We   additionally  note   that   the   control  kernel   state   takes  on  a  distinct  value   in  each  of   the  network’s  attractor   states,   and  thus   is   related   to   the  distinguishability  of   these  states   [20].   In   the   framework  presented  here,  a  consistent  interpretation   of   this   feature   is   that   the   control   kernel   states   provide   a   coarse-­‐‑graining   of   the   network  state-­‐‑space  relevant  to  its  function  that  is  intrinsic  to  the  network  (not  externally  imposed  by  an  observer).  Storing  information  in  control  kernel  nodes  therefore  provides  a  physical  mechanism  for  the  network  to  manage   information   about   its   global   state   space.   This   interpretation   is   consistent   with   Kim   et   al’s  observations  that  the  size  of  the  control  kernel  scales  both  with  the  number  and  size  of  attractors  [29].  In  short,  one  interpretation  of  our  results  is  that  the  network  is  organized  such  that  information  processing  that  occurs  in  early  times  “intervenes”  on  the  state  of  the  control  kernel  nodes,  which  in  turn  transition  to  storage  dominated  dynamics  that  regulate  the  network  states  along  the  biologically  functional  trajectory.    The  observed  scaling  of   transfer  entropy  may  be  a  hallmark  of   this  very  organization.  Such  “scaling  for  information  processing”  may  be  a  universal  signature  of  life.      Acknowledgements  This  project  was  made  possible   through  support  of  a  grant   from  Templeton  World  Charity  Foundation.  The   opinions   expressed   in   this   publication   are   those   of   the   author(s)   and  do  not   necessarily   reflect   the  views   of   Templeton  World   Charity   Foundation.   The   authors  wish   to   thank   Larissa   Albantakis,   Joseph  Lizier,  Mikhail   Prokopenko,   Paul   Griffiths,   Karola   Stotz   and   the   Emergence@ASU   group   for   insightful  conversations  regarding  this  work.  

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physics,  74(1),  47.    33.   Tononi,  G.  (2008).  Consciousness  as  integrated  information:  a  provisional  manifesto.  The  Biological  

Bulletin,  215(3),  216-­‐‑242.    34.   Anderson,  P.  W.  (1972).  More  is  different.  Science,  177(4047),  393-­‐‑396.    35.   Walker,   S.   I.,   Cisneros,   L.,   &   Davies,   P.   C.   W.   (2012).   Evolutionary   transitions   and   top-­‐‑down  

causation.  arXiv  preprint  arXiv:1207.4808.    36.   Walker,  S.,  &  Davies,  P.  (2013).  The  Algorthmic  Origins  of  Life.  J.  Roy.  Soc.  Interfac,  10.    37.   Auletta,  G.,  Ellis,  G.  F.  R.,  &  Jaeger,  L.  (2008).  Top-­‐‑down  causation  by  information  control:  from  a  

philosophical   problem   to   a   scientific   research   programme.   Journal   of   the   Royal   Society   Interface,  5(27),  1159-­‐‑1172.    

38.   Hoel,  E.  P.,  Albantakis,  L.,  &  Tononi,  G.  (2013).  Quantifying  causal  emergence  shows  that  macro  can  beat  micro.  Proceedings  of  the  National  Academy  of  Sciences,  110(49),  19790-­‐‑19795.    

39.   Ho,  M.-­‐‑C.,   &   Shin,   F.-­‐‑C.   (2003).   Information   flow   and   nontrivial   collective   behavior   in   chaotic-­‐‑coupled-­‐‑map  lattices.  Physical  Review  E,  67(5),  056214.    

40.   Cisneros,   L.,   Jiménez,   J.,   Cosenza,   M.   G.,   &   Parravano,   A.   (2002).   Information   transfer   and  nontrivial  collective  behavior  in  chaotic  coupled  map  networks.  Physical  Review  E,  65(4),  045204.    

41.   Paredes,  G.,  Alvarez-­‐‑Llamoza,  O.,  &  Cosenza,  M.  G.  (2013).  Global  interactions,  information  flow,  and  chaos  synchronization.  Physical  Review  E,  88(4),  042920.    

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42.   Donaldson-­‐‑Matasci,   M.   C.,   Bergstrom,   C.   T.,   &   Lachmann,   M.   (2010).   The   fitness   value   of  information.  Oikos,  119(2),  219-­‐‑230.    

43.   Pearl,  J.  (2000).  Causality:  models,  reasoning  and  inference  (Vol.  29):  MIT  press  Cambridge.  44.     Goldenfeld,  N.,  &  Woese,  C.  (2011).  Life  is  physics:  Evolution  as  a  collective  phenomena  far  from  

equilibrium.  Ann.  Rev.  Cond.  Matt.  Phys.  2,  375-­‐‑399.        Tables    

  Erdös-­‐‑Rényi  (ER)  networks   Scale-­‐‑Free  (SF)  networks  

Size  of  network    (Total  number  of  

nodes,  inhibition  and  activation  links)  

Same  as  the  cell-­‐‑cycle  network   Same  as  the  cell-­‐‑cycle  network  

Nodes  with  a  self-­‐‑loop   Same  as  the  cell-­‐‑cycle  network   Same  as  the  cell-­‐‑cycle  network  

The  number  of  activation  and  

inhibition  links    for  each  node  

NOT  the  same  as  the  cell-­‐‑cycle  network  (è  no  structural  bias)  

Same  as  the  cell-­‐‑cycle  network  (è  Same  degree  distribution)  

 Table  1.  Constraints  for  constructing  random  network  graphs  that  retain  features  of  the  causal  structure  of  a  reference  biological  network,  which  define  the  two  null  model  network  classes  used  in  this  study:  Erdos-­‐‑Renyi  (ER)  networks  and  Scale-­‐‑Free  (SF)  networks.    

Network  Type   𝝓𝒂𝒗𝒈   Δ  𝝓  Fission  Yeast  Cell-­‐‑Cycle  Network   0.151   0  ER  Random  Networks   0.099   0.008  SF  Random  networks   0.170   0.004  

 Table  1.  Comparison  of  the  state-­‐‑averaged  integrated  information  for  the  fission  yeast  cell-­‐‑cycle  network  and   two   types   of   null   model   networks   reveals   no   statistically   significant   differences   for   the   biological  network.        

17

 Figures  

 Figure  1.  Boolean  network  model  representing  interactions  between  proteins  in  the  fission  yeast  S.  Pombe  cell   cycle   regulatory   network.   Nodes   and   edges   represent   regulatory   proteins   and   their   causal  interactions,  respectively.  Figure  adopted  from  [12].    

Figure  1.  Global  causal  structure  of  the  fission  yeast  cell  cycle  network,  showing  the  dynamics  of  causal  mappings  between  network  states.  Left  panel:  Attractor  landscape  for  the  fission  yeast  cell  cycle  network.  Regions  are  coloured  by  the  attractor  to  which  states  within  that  region  converge.  Right  panel:  Attractor  landscape  after  global  regulation  caused  by  pinning  the  values  of  control  kernel  notes  to  their  state  in  the  primary  attractor.  Figure  adopted  from  [20].  

 Figure  3.  Information  stored  (as  measured  by  AI)  for  each  node  in  the  fission  yeast  cell  cycle  network  as  a  function  of  history   length,  k,   for  k=1,2  …  10.  Nodes   coloured   in   red  or  blue  are   control  kernel  nodes  or  nodes  with  a  self-­‐‑loop,  respectively.    

a desired attractor because cellular phenotypes are known to berobust to external perturbations22. Surprisingly, however, we foundthat we needed to regulate only a small fraction of network nodes todrive the network state to the primary attractor (36% for the S. cere-visiae cell cycle network, 44% for the S. pombe cell cycle network, 10%for the GRN underlying mammalian cortical area development, 6.7%for the GRN underlying A. thaliana development, 30% for the GRN

underlying mouse myeloid development, 5% for the mammalian cellcycle network, 7.8% for the CREB signaling network, and 8.6% forthe human fibroblast signaling network (Table 1)).

The size of the control kernel. We found that the control kernel is, ingeneral, relatively small in view of the total number of network nodes,but we also found that its size varies depending on the network. This

Figure 2 | Five biomolecular regulatory networks and their state transition diagrams before and after regulating the nodes of the control kernel.(a) Saccharomyces cerevisiae cell cycle network. (b) Schizosaccharomyces pombe cell cycle network. (c) Gene regulatory network (GRN) underlyingmammalian cortical area development. (d) GRN underlying Arabidopsis thaliana development. (e) GRN underlying mouse myeloid development.In (a)–(e), the left, center, and right panels show the original network with the control kernel denoted by red nodes, the state transition diagram of theoriginal network, and the state transition diagram of the controlled network, respectively. In each state transition diagram, the same colored dotsrepresent inclusion in the same basin of attraction.

www.nature.com/scientificreports

SCIENTIFIC REPORTS | 3 : 2223 | DOI: 10.1038/srep02223 3

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Figure   4.   Information   storage   for   individual   nodes   (measured  with   AI)   and   the   scaling   distribution   of  information  processing  (measured  with  TE)  among  nodes  for  a  biological  network  and  ensembles  of  null  models.  Contrasted  are  results  for  the  fission  yeast  cell  cycle  regulatory  network  (red)  and  ensembles  of  Erdos-­‐‑Renyi   networks   (green)   and   Scale   Free   networks   (blue).   Left   Panel:   AI   for   all   nodes   for   history  length  k  =  5.  Ensemble  statistics  are  taken  over  a  sample  of  500  networks.    Control  kernel  nodes  and  their  analogs  in  the  null  network  models  are  highlighted  in  red  on  the  x-­‐‑axis  and  nodes  with  a  self-­‐‑loop  in  blue.  Right  panel:  Scaling  of  information  processing  for  history  length  k=  2.  Ensemble  statistics  are  taken  over  a  sample  of  1,000  networks.  The  y-­‐‑axis  and  x-­‐‑axis  are  the  TE  between  a  pair  of  nodes  and  its  relative  rank.  Right  panel  adopted  from  Kim  et  al.  [12].  

Figure  5.  Distribution  of  effective  information  for  the  fission  yeast  cell-­‐‑cycle  regulatory  network  and  null  models.  Contrasted  are  results  for  the  fission  yeast  cell  cycle  regulatory  network  (red)  and  ensembles  of  Erdos-­‐‑Renyi  networks  (green)  and  Scale  Free  networks  (blue).  

Figure  6.  Diagram  illustrating  the  dynamics  of  network  states  for  the  primary  basin  of  attraction  for  the  fission  yeast  cell-­‐‑cycle  regulatory  network.  Colours  indicate  the  value  of  integrated  information  for  each  state.  Large  points  represent  states  in  the  functioning  (healthy)  cell  cycle  sequence.  

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Figure  7.  Scaling  of  information  transfer  (as  measured  by  TE)  for  every  pair  of  nodes  in  the  fission  yeast  cell-­‐‑cycle  network,  shown  for  history  lengths  k  =  1,  2,  …  10  (results  for  k=2  are  shown  in  Fig.  4  contrasted  with  ensembles  of  Erdos-­‐‑Renyi  and  Scale  Free  null  model  networks).  

 

Figure  8.  Preservation  entropy  (PE)  for  every  node  in  the  fission  yeast  network  for  history  lengths  k  =  1,  2,  …   10.     Red   and   blue   coloured   nodes   represent   control   kernel   nodes   and   nodes   with   a   self-­‐‑loop,  respectively.    

 

Figure  9.   Information   transfer  between  network   integration,  as  quantifed  by  φ   for   individual  states   (the  “Phi-­‐‑node”),   and   individual   nodes   in   the   fission   yeast   cell   cycle   regulatory   network.   The   orange   line  shows  the  TE  from  node  →  network  state  (individual  nodes  to  the  Phi-­‐‑node)  and  the  purple  line  shows  TE  from  network  state  →  node  (Phi  node  to  individual  nodes).  

   

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