A random walk from molecular dynamics to data science Benedict Leimkuhler University of Edinburgh The Maxwell Institute for Mathematical Sciences and The Alan Turing Institute
Molecular Dynamics (MD) Biomolecules Materials cytochrome C folding: Elber and Cardenas Quasi-continuum method: molecular dynamics with coarse-graining for indentation studies. Ortiz, Phillips and Tadmor
Molecular Dynamics 2017: article in Drug Discovery Today Molecular dynamics-driven drug discovery: leaping forward with confidence developing a market-approved drug costs a staggering $2.6 billion Given that the dynamics of the covalent bonds involving hydrogen atoms are not crucial in biological problems, they are usually constrained using integration algorithms, such as SHAKE, RATTLE, Hence, a time-step value in the range of 1.5 2 fs is possible and has been shown to be suitable for MD simulations of many biological systems." Examples of some 2017 simulations out of around 20000 published Ligand binding in the HIV-integrase enzyme Druggability of membrane bound Rab5 Fission fragment damage in nuclear fuel Patchy particle systems Nanoparticle cholestoral metabolism therapeutics Multiscale protein dynamics in antigen presentation Bottlebrush copolymers Modification of membrane structure by lithium Smectic C semiflexible polymers Graphene reinforced polymer nanocomposites Unsaturated Zwitterionic lipids Identifying the warfarin binding site
HIV-1 Virus Capsid (protective shield around virus) Molecular dynamics 64M atoms (including surroundings) Described by a potential energy function U(x) Chaotic, nonlinear dynamics, ever expanding scale Key questions are of a probabilistic nature
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<latexit sha1_base64="pohpw9humefyccb8iv+v21wdfp8=">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</latexit> invariant measure <latexit sha1_base64="ufox4zita877+ikq+m6ienxmvh0=">aaab6nicbvdlsgnbeoynrxhfuy9ebopgkeykomegf48rzqosjcxozpmhm7pltk8qqj7biwdfvppf3vwbj8kenlggoajqprsrsqww6pvfxmftfwnzq7hd2tnd2z8ohx41bzizxhsskylpr9ryktrvoedj26nhvewst6lr7cxvpxfjraifczzyungbfrfgfj300fvzr1zxq/4czjueoalajnqv/nxtjyxtxcot1npo4kcytqhbwssflrqz5sllizrghuc1vdygk/mpu3lmld6je+nki5mrvycmvfk7vphrvbshdtmbif95nqzj63aidjoh12yxkm4kwytm/iz9ythdoxaemipcryqnqaemxtolf0kw/piqav5ua78a3f9wajd5heu4gvm4hwcuoaz3uicgmbjam7zcmye9f+/d+1i0frx85hj+wpv8av1ejdy=</latexit> <latexit sha1_base64="ypu/zz/rjuesmglf2kjx3atf4h4=">aaab+3icbvbns8naen3ur1q/yj16crahxkoigl6eohepfewhtdfstpt26wytdifsepjxvhhqxkt/xjv/xm2bg7y+ghi8n8pmpd/mtiftfxultfwnza3ydmvnd2//wdysdlsuselbjokr7plyuc4ebqmdtnuxpdj0oe36k9uz332iurfipeaauzfei8ecrjboytor9aknzrxmxnv544cjaflprnknew5rltgfqaeclc/8ggwjkoruaofyqb5jx+bmwaijnoavqajojmkej2hfu4fdqtxsfntunwplaawr1cxamqu/jzickpwgvu4mmyzvsjct//p6cqrxbszenaavzleoslgfktulwhoysqnwvbnmjno3wmsmjsag46roejzll1dj57zh2a3n/qlwvcnikknjdilqyegxqinuuau1euft9ixe0zurgy/gu/gxac0zxcwr+gpj8wdvk5ql</latexit> <latexit sha1_base64="zhzqntoszix3l2pm3wlwblnlzai=">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</latexit> <latexit sha1_base64="piajc1yww/awe6ss5xb0fgy0lp0=">aaacgxicbvbps8mwhe39o+e/qkcvwshmg6mvqs/c0ivhcxybrrwkabafjw1jutko/rpe/cpepcjiuu9+g7otb918epj47/1ifi9igjxksr6nhcwl5zxv0lp5fwnza9vc2w3kobwyodhmswghsbjgi+ioqhhpj4ighjdscgzxy7/1qiskcxsrrgnxoopftesxulrytstzbydh7vk0ojycf/dozztp77njo4dex9drrg6l3na3k1bnmgdoe7sgfvcg4zufbhjjljniyyak7nhworwmcuuxi3nztsvjeb6ghulogifopjdnnsvhovzc2i2fppgce/x3ria4lcme6crhqi9nvbh4n9djvffcy2iupipeeppqn2vqxxbcewypifixksyic6r/cnefcysvlross7bnv54nzzoabdxsm9nk/bkoowt2wqgoahucgtq4bg3gaawewtn4bw/gk/fivbsf0+icuczsgt8wvn4a71ue/g==</latexit> Sampling and MCMC Typical approach: using a Markov Chain with prescribed Example method: Metropolis Monte Carlo
<latexit sha1_base64="ipbhajzhpzzplun7nsgjggxlzim=">aaacdxicbvdltsjafj3ic/fvdelmipq0iwdbmojghojgjsywskcq6tdahokjm1mdifyag3/fjquncevenx/jaf0oejkbnjxzb+69x48zfdkyvrxmyura+kz2m7e1vbo7p+8fvewucexchlgi130kckmhcswvjnrjtldgm1lzbzdtv/zaukbrec9hmfec1atpl2ikldtwt1xjamiraphfdoayxrdlfg2z5cacnozcwtbbet4qwtpazwknja9svnr6v7mt4sqgocqmcdgwrvh6y8qlxyxmcs1ekbjhaeqrhqihcojwxrnvjvbukr3yjbiqumkz+ntijaihrogvogmk+2lrm4r/ey1edi+9mq3jrjiqzxd1ewzlbkfrwa7lbes2ugrhttwtepcrr1iqahmqbhvx5wvsduq2vblvzvpl6zsoldgcx8aanrgazxalksafgdycz/ak3rqn7uv71z7mrrktntkef6b9/gcqezy6</latexit> <latexit sha1_base64="u++gakoc+pkxbqglatnj+qg4cfq=">aaab8hicdvdltsjafj3ic/gfunqzkzjgwqblzy7oxiumfjbqyxqyymj02sxmjathk9y40bi3fo47/8ypykjgt3ktk3puzb33ecgjulnwh5fawv1b30hvzra2d3b3svshlrleahmhbywqhq9jwignjqkkku4ocpi9rtre5clx23desbrwazunieujeaddiphs0g25jc+c/p3prj/nwwalwq4xi9ay7vq5vk9iovsu2da0twuohfii2c++9wybjnzcfwziyq5thcqnkvaumzll9cjjqoqnaes6mnlke+ng84nn8eqrazgmhc6u4fz9phejx8qp7+loh6mx/o0l4l9en1ldmhtthkakclxyniwyvafmvocdkghwbkojwolqwyeei4gw0hlldahfn8l/satg2jqiq1kucb6miw2owdhiaxtuqqncgizwaay+eabp4nkqxqpxyrwuwlpgcuyq/idx9gl2hjaw</latexit> Example: Metropolis Monte Carlo for the Uneven Double Well Gaussian prior: rare events slow convergence too much measure in the right basin
<latexit sha1_base64="9tto/n8ofer8lgngebhgmxsljd8=">aaacjnicbzdnsgmxfiuz/lv/qi7dbitqectmexqjig5cvndaqmcomtrtg5nmmnyrlmgexo2v4safiulorzftr9dwcygh79xlck8qc67btj+tufmfxaxlldxc2vrg5lzxe6euo0rr5tjirkozem0el8wfdoi1y8vigajwco6urn7jgsnni3klw5j5ielj3uwugeht4nnqqrb3sge+x8eejieg2c0pdnhoar9ht98rskvzd5vcjawn7wljrtjjwrpcyuuj5vvrf1+9tkstkemggmjdcuwy/jqo4fswroalmswe3peeaxkpsci0n47xzpcbir3cjzq5evcy/p5isaj1maxmz0igr6e9efzpayxqpfntlumemksth7qjwbdhuwa4wxwjiizgekq4+sumfaiibznswytgtk88k+rvimnxnjut0svlhsck2kp7qiwcdiou0dwqirdr9iie0st6s56sf+vd+pi0zln5zc76u9bxn3xho/g=</latexit> Sampling using SDEs Ito-type Stochastic Differential Equation Brownian Dynamics Wiener measure describes a particle diffusing in a potential U at fixed temperature
Fokker-Planck Equation The stochastic paths of an Itō SDE sample the evolving distribution with probability density satisfying @ @t = L L is the L 2 adjoint of L defined by (L g)(x) = lim t!0 E[g(x( t)) x(0) = x] g(x) t
<latexit sha1_base64="lnwzustj1+34cpnlf+gcgyowa1o=">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</latexit> <latexit sha1_base64="ur9aqlkrsnvuaj45yzb/1dknvmy=">aaab/hicbzdnssnafiun/tb6f+3szwar6sksikaboejgzqxtftpyjtpbdugke2ymygj1vdy4umstd+lot3hazqgtbwy+zr2xe+cemwdko863tbs8srq2xtgobm5t7+zae/snjrjjwaocc9kkialoiva00xxasqqsbhyaweh6um8+gfrmrhc6jcepysbifuajnlbxlnxkufqej/elhvvsxdm47tplp+pmhrfbzagmctw79lenj2gsqqqpj0q1xsfwfkakzptdunhjfmsejsga2gyjeolys+nxy3xknb7uc2lepphu/t2rkvcpnaxmz0j0um3xjuz/txai+xd+xqi40rdr2aj+wrewejie7jejvppuakgsmvsxhrjjqdz5fu0i7vyxf6fxwnwdqnt7vq5d5xeu0ae6rbxkonnuqzeojjxeuyqe0st6s56sf+vd+pi1lln5tan9kfx5a3o/k1o=</latexit> Brownian Dynamics so is a stationary density & geometric ergodicity under mild conditions on U Ergodic limit Just need a reliable way of computing numerical solutions of SDEs (on long intervals)!
SDE Numerics What is a suitable definition for error for an SDE? Examples: weak, strong, ergodic How to get high accuracy in SDE discretisation? How does discretisation affect convergence to the invariant distribution?
Euler-Maruyama Method Numerical Method discrete Brownian path Leimkuhler-Matthews Method [L. & Matthews, AMRX, 2013] [L., Matthews & Stoltz, IMA J. Num. Anal., 2015] [L., Matthews & Tretyakov, Proc Roy Soc A, 2014]
Uneven Double Well ergodic error small stepsize bimodal distribution large stepsize L-M E-M
Invariant measures of numerical methods Sampling Applications Charlie Matthews (UoE PhD 2012-2016) 5 papers + a book and over 300 citations
Charlie Matthews (PhD 2015, now @ Chicago) Langevin (BAOAB) Matthias Sachs (PhD 2017, now @ Duke) Invariant measures Overdamped Limit Constrained LD Xiaocheng Shang (PhD 2015, now @ETH) Langevin/AdL Ergodicity Adaptive Langevin Gabriel Stoltz (ENPC Paris) Michael Tretyakov (Nottingham) Vincent Danos (ENS Paris)
the most recent citations for our 2016 geodesic integrators paper: physical chemistry biomolecules data science biomolecules molecular software data science physics stats & data science stats & data science Why are statisticians reading molecular dynamics papers?
Wind Turbines (with industry partner) Zofia Trstanova (postdoc) Anton Martinsson (PhD student)
Financial giant Goldman Sachs announces that it plans to invest $150 billion in clean energy projects and technology like solar and wind farms, energy efficiency upgrades for buildings, and power grid infrastructure. Problem: Goldman Sachs analysts lack of turbine knowledge Solution: Employ a specialist energy management firm Problem: Each windfarm may have hundreds of turbines each turbine produces data at 10s to 10m intervals on dozens of properties Large data problem [10K turbine years!] Solution 1: Solution 2: Hire engineers to stare at data Get a team of mathematicians to develop a data-driven model to analyse the data.
Ideal power output to windspeed diagram Example data set Learn: to categorise conditions: derating, icing, Goal: Automatic analysis of data streams for maintenance, performance analysis, planning
<latexit sha1_base64="6st+2uni49vtkzhpfrx8lltbni8=">aaab6hicbvbns8naej3ur1q/qh69lbbbu0le0gpri8cw7ae0owy2k3btzhn3n0ij/qvepcji1z/kzx/jts1bwx8mpn6bywzekaiujet+o4w19y3nrej2awd3b/+gfhju0ngqgdzzlglvcahgwsu2dtcco4lcgguc28h4dua3n1bphst7m0nqj+hq8pazaqzueoyxk27vnyosei8nfchr75e/eooyprfkwwtvuuu5ifezqgxnaqelxqoxowxmh9i1vniitz/nd52sm6smsbgrw9kqufp7iqor1pmosj0rnso97m3e/7xuasjrp+mysq1ktlgupokymmy+jgoukbkxsyqyxe2thi2ooszybeo2bg/55vxsuqh6btvrxfzqn3kcrtibuzghd66gbndqhyywqhigv3hzhpwx5935wlqwnhzmgp7a+fwb252m9q==</latexit> <latexit sha1_base64="do2tvhsgwpjem6ngq7vwez4dw5a=">aaacj3icbzdps8mwfmft+wvwx1wpxojd8dtaiuwpytclxwludtzs0jtbwtkkjkkwyv4bl/4rxgqv0ap/idlwudcfhhzzee/x8r5ryqjsrvtplzawv1bxyuv2xubw9o6zu9dwipoytlbgqnyipaijnlq01yx0uklqejfyfw2vjvm7eyivffxwj1isjkjpay9ipa0knytmmpsygey9hufqrz7vfd8uycz6m+jwfqdqayil7ivoxvztgivck0qffnemnrc/fjhlcneyiaw6npvqiedsu8zi2pyzrvkeh6hpukzylbav5nm9x/dikbj2hdshazilvztylcg1sijtmsa9upo5cfwv18107ztiku8zttiedepldgobj6bbmeqcnrszgbck5q8qd5bewbtrbwocn7/yomjxqp5b9w5oko3lwo4yoach4bh4oa4a4bo0qqtg8acewct4sx6tz+vd+pivlqyizx/8cevrg3jrofk=</latexit> Model Models may be artificial or not, but the distinction is often artificial! Ex: convolutional neural network
<latexit sha1_base64="rzlbq3g0+drgaqrply9ijnsmmqu=">aaacc3icbvc9tsmwght4leuvwmhitujqlypbsdbwsdawibarmihyxke16isu7sbvaxcwxowfayryeqe23ganzqatj1k63x2fpt8fnfgplovbwfvf2nzalu2ud/f2dw7no+ootfkbsrsnlbfogcrhncztrrujdhcerqej3wb0k/vdbyiktej7nehei9agpihfsgnjnysup7xx1kldl4ueqwtmgjmd5wl1rdq47ptvq2hnavejxzaqkndyzs+3n+a0irhcdenzsy2uvawjrtejs7kbssirhqeb6wkao4hil5tnmcezrfrhmaj9ygxn6u+ndevstqjat0zidewyl4v/eb1uhvdermoekhljxaewzvbhzoubfsoivmyicckc6r9cpeqcyaxrk+ss7oxiq6rz3rcthn13uw1ef3wuwcmogbqwwsvoglvqam2awsn4bq/gzxgyxox342mxumyuoyfgd4zph56emxs=</latexit> <latexit sha1_base64="qmjny7d3nwzcz0iuuxl57xkwnik=">aaaccnicbvdlssnafj34rpuvdelmtah1uxirdfmounelfewdmhgm00k7ddkjmxoxxk7d+ctuxcji1i9w5984abpq1gmxdufcy733+dgjulnwtze3v7c4tfxyka6urw9smlvbtrklapmgjlgk2j6shffogooqrtqxicj0gwn5g/pmb90riwner9uwjm6iepwgfcoljc/cc2jabj/chsiqdorus2nvht1cwky+96g2plnkvawx4cyxc1icoeqe+ev0i5yehcvmkjqd24qvmykhkgzkvhqsswkeb6hhoppyfblppunxrvbak10yreixv3cs/p5iusjlmpr1z4hux057mfif10lucoqmlmejihxpfgujgyqcws6wswxbig01qvhqfsvefsqqvjq9og7bnn55ljspkrzvsa+os7wzpi4c2ax7oaxscajq4aluqqng8aiewst4m56mf+pd+ji0zhn5za74a+pzb7p9mp8=</latexit> DATA PARAMETERS prior model LIKELIHOOD Bayes Theorem minimise U(q) to determine optimal parameters I m not really me Data Scientist Thomas Bayes, U of Edinburgh, Class of 1721
Wind Turbine Benchmarking With enough data, it is straightforward to replace the engineer with a convolutional neural network for detecting a specific condition, e.g. icing Problems: 1. Mixed data sets 2. Lack of transferability Unsupervised learning to learn classes from data Generalisation error Generative networks 3. Multimodality all require sampling!
Multimodality in NNs With Zofia Trstanova and Frederik Heber (Turing) P(red) isolated local min 9 parameter model, multimodal loss landscape with pretty severe trapping in upper well.
Sampling Metastable Systems Ways forward: 1. EQN. New types of diffusions that more efficiently explore the complex geometry; in particular nonreversible diffusions. w. J. Weare & C. Matthews, Chicago. 2. ISST. Use of temperature and other collective variables to enhance and guide sampling. w. A. Martinsson, J. Lu (Duke) & E. Vanden-Eijnden (NYU). 3. DM. Data-analytic tools based on diffusion maps to perform unsupervised manifold learning from simulated data in order to enhance exploration. w. Z. Trstanova, T. Lelievre (Paris) and R. Banisch (Berlin)
(building on work of Clementi, Kevrekides )
Thermodynamic Analytics Toolkit (TATi) Frederik Heber (Rutherford-Turing Fellow) Benedict Leimkuhler Zofia Trstanova (EPSRC PDRA - Edinburgh) Vision: Exploration tool for loss landscape in ANNs Sampling methods in TensorFlow purposes Develop understanding of NN properties Assist in analysis/redesign of NNs
The Edinburgh Environment Maxwell Institute Union of mathematicians at Edinburgh and Heriot-Watt Universities. Currently unifying graduate programmes under the banner Maxwell Institute Graduate School. International Centre for the Mathematical Sciences Runs workshops, supports small group research and coordinates knowledge exchange and public events. Centres for Doctoral Training e.g. MIGSAA Main mechanism for funding PhD students and coordinating training across the Maxwell Institute 3 proposed in 2018: MAC-MIGS (Modelling, analysis and computation), GLAMS (Glasgow-Maxwell School in algebraic methods), and Data Science & AI (a joint venture with Informatics)
Thomas Bayes Centre
Why Edinburgh? amazing skyline: access to the highlands fringe festival folk music still part of the EU but.
The real reason Scottish Boletus Harvest (Porcini/Ceps)
Thank you for listening!