complexityandevolution - 11/28/11 !   The human...

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Unformatted text preview: 11/28/11 !   The human genome project in the 1990s offered the promise that once the complete DNA sequence of the human genome was known, scienCsts would be able to solve many of the problems of biology and medicine !   The sequence of genomes of humans and many other species has provided biology with powerful new tools, but rather than solving the problems, it has revealed how difficult some problems are !   The components of biological mechanisms have been found to interact with each other in a mulCtude of ways !   Understanding these interacCons requires the development of new tools to show how these interacCons can give rise to the biological phenomena of interest !   Biologists are increasingly turning to mathemaCcal/ computaConal modeling to understand the systems of interest !   Complicated systems are ones with lots of parts and operaCons !   Biological systems certainly count as complicated !   Complex systems involve non ­linear interacCons that oMen give rise to behavior not anCcipated from knowledge of the parts !   May actually arise from just a few parts !   Linear equaCons such as !   Ax + By = 0 can be graphed as straight lines in Cartesian coordinates !   EquaCons with mulCplicaCve relaCons, powers, etc., can generate much more complex graphs !   LogisCc map: xt+1=rxt(1 ­xt) !   For different values of x, fixed value, oscillaCons, chaos 1 11/28/11 !   Both mechanisCc science and philosophical accounts of mechanism have emphasized decomposing mechanisms to idenCfy the parts and operaCons that contribute to the phenomena of interest !   Biologists have tended to be skepCcal of computaConal modeling, preferring laboratory research !   PopulaCons geneCcs the excepCon, but it mostly employed linear models !   But increasingly biologists are discovering systems which can only be represented in non ­linear equaCons which generate complex (emergent) behavior !   Amending the characterizaCon of mechanism to include dynamics !   A mechanism is a structure performing a funcCon in virtue of its component parts, component operaCons, and their organizaCon. The orchestrated funcConing of the mechanism, manifested in pa[erns of change over Cme in properCes of its parts and operaCons, is responsible for one or more phenomena. !   To understand how a complex mechanism will behave !   To suggest manipulaCons that can be made to test the proposed account experimentally !   To reveal how the mechanism might respond to altered condiCons in the environment !   Following the discovery by Jacob and Monod of a feedback mechanism (the lac operon) whereby bacteria suppress transcripCon of specific genes except when they are needed, Brian Goodwin proposed a model of how such a control mechanism might generate oscillaCons !   Three differenCal equaCons, each of which has terms for the generaCon and degradaCon of one component !   Goodwin showed that this system, with appropriate values for n, would generate oscillaCons in the quanCCes of the components !   Provided an exemplar for modeling circadian rhythms k1 dX = − k4 X dt Z n + 1 dY = k 2 X − k 5Y dt dZ = k 3Y − k 6 Z dt 2 11/28/11 !   The first step in developing a mathemaCcal model of a mechanism is to represent it in terms of quanCCes !   Goldbeter took the 1990 Hardin et al. model in which PER inhibits its own transcripCon and represented the components of the system in terms of !   variables idenCfying concentraCons of various parts (M=per mRNA concentraCon, etc.) !   parameters specifying the rates governing various reacCons (vs= maximum rate of transcripCon !   The next step is to write equaCons to characterize how the values of each variable changes dependent upon other variables !   Eq. 1 has one term for the making of new per mRNA and one for its degradaCon !   The equaCon introduces a non ­linearity in the exponent n (which was taken to reflect the assumpCon that mulCple molecules of PER have to interact to suppress transcripCon) !   OMen, as in this case, it is not possible to derive a soluCon to mulCple equaCons analyCcally and so modelers apply them iteraCvely to simulate the operaCon of the mechanism !   Goldbeter showed that the model generated sustained oscillaCons of !   per mRNA !   Total PER (Pt) !   Nuclear PER (PN) !   Cytoplasmic PER whether phosphorylated (P1 & P2) or not (P0) !   When plo[ed in phase space, the results showed a limit cycle 3 11/28/11 !   In addiCon to the negaCve feedback loop whereby PER:CRY inhibits its own transcripCon (via removing the CLK:BMAL1 dimer from its promoter) !   There is a second negaCve feedback loop in which CLK:BMAL1 excites producCon of REV ­ERBα, which then inhibits producCon of BMAL1 !   And a posiCve feedback loop in which CLK:BMAL1 excites PER:CRY, which inhibits REV ­ERBα and stops it from inhibiCng BMAL1 expression !   Will mulCple loops disrupt the 24 ­oscillaCons? From Smolen and Byrne, 2009 !   As more components of the mechanism were discovered, Goldbeter expanded his model (Goldbeter and Leloup, 2003) !   Adding variables for the addiConal parts !   Parameters for the addiConal operaCons !   And many more equaCons (73 in the latest) !   Challenge: models with large numbers of equaCons and many parameters can be fi[ed to data and may not reveal how the mechanism works !   Thus, some modelers prefer abstracCng and employ reduced models. !   Modelers oMen speak of conducCng experiments with their models !   Changing the values of variables or, more typically, parameters, and determining their effects !   One use of such use of models is to determine whether, on the account proposed, experimental manipulaCons of the actual mechanism would be expected to have the effect they have !   Can manipulaCon of appropriate parameters reproduce the effects of various induced mutaCons (e.g., Konopka’s original results) 4 11/28/11 !   We saw that individual SCN neurons exhibit considerable variability in period and phase when cultured !   This variability is radically reduced when neurons interact in a whole network—neurons synchronize their acCvity !   What is the organizaCon of the network that facilitates synchronizaCon? !   Researchers have not yet been able to directly observe the network organizaCon of the SCN !   Instead they have worked by construcCng hypotheses, represenCng them in computaConal models, and evaluaCng how well these models could explain the observed behavior !   Most mathemaCcal analysis of networks in the 20th century focused on !   Regular laqces: High clustering, long characterisCc path length !   Random networks: Low clustering, short characterisCc path length !   These lent themselves to mathemaCcal analysis !   Random networks achieve synchronized behavior quickly !   Regular laqces support regular waves of behavior 14 Gonze et al. (2005) developed a mean field model by assuming that VIP from individual neurons accumulated and distributed equally. !   Adopted Goodwin s oscillator by adding expressions for !   V = neurotransmi[er dX Xi kn KF i = v1 n 1 n − v4 + vc +L whose synthesis is dt Z i + k1 k4 + X i K c + KF induced by X dYi Yi !   F = mean field or = k 2 X i − v5 average dt k5 + Yi concentraCon Zi of neurotransmi[er dZ i = k Y − v 3i 6 dt k6 + Z i !   K = sensiCvity of individual oscillators dVi Vi = k7 X i − v8 to the VIP dt k8 + Vi neurotransmi[er/ ! coupling strength 1 F= N N ∑V i i =1 5 11/28/11 ! Gonze et al. employed 1000 oscillators !   Set K=0 to simulate no VIP release !   Periods were normally distributed with a mean of 23.5 h and an SD of 1.17 h !   Se[ K=0.5 to simulate VIP release !   All cells synchronized to a period of 26.5 h !   To et al. 2007 modeled diffusion based on distance !   Started with the Leloup and Goldbe[er mammalian model and added random perturbaCons in the basal Per transcripCon rate (νsp0) so that ~40% of neurons oscillated !   Other parameters randomly varied to create range of oscillatory periods from 18 to 30h ! i (t ) = a γ i (t ) = 1 ε M P , i (t ) M P , i (t ) + b N ∑ a ρ (t ) ij j =1 j ρi is extracellular concentraCon of VIP produced by neuron i MP is the Per mRNA concentraCon γ is the local VIP concentraCon observed by neuron i !   IntroducCon of VIP resulted in rapid synchronizaCon in Per mRNA oscillaCons !   Results parallel those for SCN aMer prolonged blockage of acCon potenCals with tetrodotoxin (TTX) !   Loss of VIP resulted in desynchronizaCon 6 11/28/11 !   What happens if most connecCons are local, but a few are long ­distant? !   High clustering and short characterisCc path length !   In the 1990s Duncan Wa[s appealed to these characterisCcs to define small ­world networks !   Showed that they are highly suited for informaCon processing !   Local regions can specialize !   Whole network can remain coordinated !   Many networks in the real world turn out to have a small ­world structure: airline route systems, the internet, gene networks, protein networks, the brain 19 !   Wa[s and Strogatz speculated that small world networks would display enhanced signal propagaCon speed, synchronizability and computaConal power, as compared with regular laqces of the same size. The intui)on is that the short paths could provide high ­speed communica)on channels between distant parts of the system, thereby facilita)ng any dynamical process (like synchroniza)on or computa)on) that requires global coordina)on and informa)on flow (Strogatz, 2001) !   PotenCal advantage over random networks: enable different clusters to specialize in different ways !   Without sacrificing the ability to rapidly adapt to acCvity elsewhere ! Vasalou et al. (2009) set out to explore this quesCon in a model. They replaced 1N γ i (t ) = ∑ aij ρ j (t ) ε j =1 !   from the To et al. model with 1N !   γ i (t ) = aij ρ j (t ) ki ∑ j =1 !   Where ki is number of synapCc inputs received by neuron i and ! aij = 1 if there is a connecCon between i and j and 0 otherwise !   Network architectures: A.  Nearest neighbor, VIP expressed in all neurons B.  Small world: AddiConal connecCons added with prob p C.  Mean field or totally connected net D.  Small world with only some neurons producing VIP !   Small world when 0.01 < p < 0.1 7 11/28/11 !   Per mRNA concentraCon of ten randomly selected cells Small Worlds Small World Network Nearest neighbor (laqce) Nearest Neighbor Vassalou et al. (2009) !   Two measures computed from Per mRNA concentraCons: !   SI: SynchronizaCon Index—compares instantaneous phase angle of each oscillator relaCve to a reference cycle, thereby quanCfying the ability of the system to produce a coherent signal [in slice SI = 0.93] !   R: order parameter represents the overall degree of synchrony over a specified Cme period. Small world and totally ­ connected networks are comparable on these measures Vassalou et al. (2009) Vassalou et al. (2009) !   Without VIP, only 30% to 40% of SCN neurons oscillate, but with VIP all do !   A. When coupling increased from nearest neighbor to small ­world, percentage of non ­ oscillaCng neurons dropped !   Without VIP, mean period is approximately 22 hours, whereas with VIP it approximates 24 hours !   B. When coupling increased to small ­world levels, period extends to approximately 24 hours !   Without VIP, there is large variability in periodicity, whereas with VIP oscillators are largely synchronized !   C. When coupling increases to small ­world levels, variability drops dramaCcally 8 ...
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This note was uploaded on 12/12/2011 for the course PHIL 147 taught by Professor Bechtel,w during the Fall '08 term at UCSD.

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