Sonic Chronicles: The Dark Brotherhood and Mathematical and theoretical biology: Difference between pages

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{{Infobox VG
{{for|use of statistics in Biology|Biostatistics}}
| title = Sonic Chronicles: The Dark Brotherhood
:''For use of basic artimethics in Biology, see relevant topic, such as [[Serial dilution]].''
| image = [[Image:The Dark Brotherhood.jpg]]
| caption = Box art for ''Sonic Chronicles: The Dark Brotherhood''
| developer = [[BioWare]]
| publisher = [[Sega]]
| designer =
| released = {{vgrelease|AUS=September 25, 2008<ref name="release_aus">{{cite web|url=http://palgn.com.au/games.php?id=4985 | title=Sonic Chronicles: The Dark Brotherhood (Nintendo DS) - Game Details | publisher=PALGN | accessdate=2008-09-23}}</ref>}}{{vgrelease|EU=September 26, 2008<ref name="scb_eu" />}}{{vgrelease|NA=September 30, 2008<ref name="playtm_gold" />}}{{vgrelease|JP=TBA 2008}}
| genre = [[Console role-playing game]]
| modes = [[Single-player]]
| ratings =
| platforms = [[Nintendo DS]]
| media =
| requirements =
| input =
}}
'''''Sonic Chronicles: The Dark Brotherhood''''' is a [[Nintendo DS]] [[console role-playing game]], based on the ''[[Sonic the Hedgehog (series)|Sonic the Hedgehog]]'' franchise, developed by Canadian RPG developer [[BioWare]]. The game focuses on the events surrounding the kidnapping of Knuckles, and the disappearance of the [[Chaos Emeralds]] after the defeat of [[Dr. Eggman]].<ref name="igninterview">{{cite web| author=Matt Casamassina|url = http://ds.ign.com/articles/852/852731p1.html| title = Sonic Chronicles: The Dark Brotherhood Interview| date = 2008-02-15| publisher = [[IGN]]| accessdate = 2008-02-16}}</ref>


'''Mathematical biology''', '''biological mathematical modeling''', '''biomathematics''' or '''computational biomodeling''' is an [[interdisciplinary]] field of academic study which aims at modeling natural, [[biology|biological]] processes using [[Applied mathematics|applied mathematical]] techniques and tools. It has both practical and theoretical applications in biological research: In cell biology, protein interactions are typically expressed as “cartoon” models, which, although easy to visualize, do not fully describe the systems studied: to do this, mathematical models are required, which, by describing the systems in a quantitative manner, can better simulate their behavior and hence predict unseen properties.
The game was released in September 2008 in North America, Europe, and Australia.<ref name="playtm_gold" /><ref name="scb_eu">{{cite web|url=http://blogs.sega.com/sonic/2008/08/28/im-off-but-before-i-go/|title=Sonic City Blognik - I'm Off...But Before I Go|author=ArchangelUK|publisher=Sonic City Blognik|accessdate=2008-09-26|date=2008-08-28}}</ref><ref name="release_aus">{{cite web|url=http://palgn.com.au/games.php?id=4985 | title=Sonic Chronicles: The Dark Brotherhood (Nintendo DS) - Game Details | publisher=PALGN | accessdate=2008-09-23}}</ref> It was generally received favourably by critics, who appreciated both the graphics and environments, as well as Sonic's overall transition to the role-playing genre; the story and some gameplay elements received mixed reactions.


==Gameplay==
==Importance==
Applying mathematics to biology has a long history, but only recently has there been an explosion of interest in the field. Some reasons for this include:
[[Image:Battle in Sonic Chronicles.jpg|100px|thumb|left|An early image of the game's battle scenario]]
* the explosion of data-rich information sets, due to the [[genomics]] revolution, which are difficult to understand without the use of analytical tools,
As with many RPGs, two main types of gameplay feature: exploration and combat. Exploration areas see the character being controlled by tapping the stylus where the player wants the character to go, while action buttons are used to traverse certain elements, such as loop-the-loops.<ref name="gamespot200804">{{cite web|url=http://uk.gamespot.com/ds/rpg/sonicrpg/news.html?sid=6189256&om_act=convert&om_clk=newsfeatures&tag=newsfeatures;title;1|author=Ricardo Torres|title=Sonic Chronicles: The Dark Brotherhood Hands-On|date=2008-04-15|publisher=Gamespot}}</ref> Navigation can also require a certain character to be in the lead for progress to continue, taking advantage of that character's special ability, be it flying, climbing, or jumping large gaps.<ref name="ignhandson1">{{cite web|url=http://uk.ds.ign.com/articles/866/866494p1.html|date=2008-04-15|author=Craig Harris|title=Sonic Chronicles: The Dark Brotherhood Hands-on|publisher=IGN UK}}</ref> Puzzle elements also feature, allowing the party to break up to complete the task, such as pressing switches in various areas.<ref name="ignhandson2">{{citeweb|url=http://uk.ds.ign.com/articles/874/874573p1.html|author=Bozon|title=Hands-on Sonic Chronicles: The Dark Brotherhood|publisher=[[IGN]]}}</ref>
* recent development of mathematical tools such as [[chaos theory]] to help understand complex, nonlinear mechanisms in biology,
* an increase in [[computer|computing]] power which enables calculations and [[simulation]]s to be performed that were not previously possible, and
* an increasing interest in [[in silico]] experimentation due to ethical considerations, risk, unreliability and other complications involved in human and animal research.


==Areas of research==
Combat gameplay occurs when enemies are walked into, shifting the view to a close-up for turn-based battles to occur. Standard attacks are available, whilst special attacks, dubbed POW Moves, can be performed by rhythmically tapping the stylus. These will use up ''POW Points'', in a manner similar to ''Magic Points'' in other RPGs.<ref name="gamespot200804" /> Examples of individual special attacks include Sonic's ''Axe Kick'' and ''Whirlwind'', whilst some group-based special attacks require certain characters in the party, such as the ''Blue Bomber'', which requires Tails.<ref name="ignhandson2" /> After combat is complete, loot is available, ranging from healing items to character equipment - equippable in one of three slots per character - whilst experience points boost one of four attributes per character: speed, attack, defense, and luck.<ref name="gamespot200804" />
Below is a list of some areas of research in mathematical biology and links to related projects in various universities. These examples are characterised by complex, nonlinear mechanisms and it is being increasingly recognised that the result of such interactions may only be understood through mathematical and computational models. Due to the wide diversity of specific knowledge involved, biomathematical research is often done in collaboration between mathematicians, [[physics|physicists]], engineers, biologists, [[physiology|physicians]], [[zoology|zoologists]], [[chemistry|chemists]] etc.


===Computer models and automata theory in biology and medicine===
Characters can also collect and equip Chao, each with their own ability, to enhance a team member's status<ref name="ignhandson1" />. These Chao are collectable and can be stored in the Chao Garden, and using the DS's wireless connectivity, players can swap the Chao they have collected.<ref name="cvgngamer">{{cite web|url=http://www.computerandvideogames.com/article.php?id=190471|author=NGamer Staff|title=DS Previews: Sonic Chronicles: The Dark Brotherhood|accessdate=2008-06-09|quote=Fiddling with the item menus revealed that new character Shade will be playable, as will E-123 Omega, the clanking bot star of Sonic Heroes and Shadow The Hedgehog.}}</ref>
A monograph on this topic summarizes an extensive amount of research
in this area up to 1987, including subsections in the following areas: [[computer modeling]] in biology and medicine, arterial system models, [[neural model]]s, biochemical and [[oscillation network]]s, [[neural net]]s, [[genetic network]]s, abstract relational biology, metabolic-replication systems, [[category theory]] applications in biology and medicine, [[automata theory]] and complete [[self-reproduction]], [[chaotic system]]s in [[organism]]s, relational biology and organismic theories.<ref>Baianu, I. C.: 1986--1987a, Computer Models and Automata Theory in Biology and Medicine., in M. Witten (ed.),
''Mathematical Models in Medicine'', vol. '''7'''., Ch.11 Pergamon Press, New York, 1513 -1577. http://www.springerlink.com/content/w2733h7280521632/ </ref>; this published report also includes 390 references to peer-reviewed articles by a large number of authors <ref>http://www.springerlink.com/content/w2733h7280521632/</ref>,
<ref> Also currently available as an updated PDF at: http://cogprints.ecs.soton.ac.uk/archive/00003718/01/COMPUTER_SIMULATIONCOMPUTABILITYBIOSYSTEMSrefnew.pdf</ref>
[http://planetmath.org/?method=l2h&from=objects&id=10746&op=getobj (See also the 300+ References List in: "Bibliography on category theory applications", including mathematical biology applications).]


==Plot==
Sonic is on vacation after defeating [[Doctor Eggman]] some time ago, when he receives a call from Tails, stating that Knuckles has been kidnapped, and that the [[Chaos Emeralds]] are disappearing.<ref name="igninterview"/>


'''Modeling cell and molecular biology'''
The story is split into two acts. The first takes place in Sonic's world, with the team attempting to unravel the situation they are in, whilst also stopping the [[Master Emerald]] from being taken. The second act sees Sonic and the team travelling to another dimension in order to stop a new threat to their own world.<ref name="igninterview"/>


This area has received a boost due to the growing importance of [[molecular biology]].<ref>{{cite web|url=http://www.maths.gla.ac.uk/research/groups/biology/kal.htm |title=Research in Mathematical Biology |publisher=Maths.gla.ac.uk |date= |accessdate=2008-09-10}}</ref>
The game features areas and music from previous Sonic games, such as [[Sonic the Hedgehog (16-bit)|Green Hill Zone]], [[Sonic Adventure|Mystic Ruins]], [[Sonic Battle|Emerald Town]] and [[Sonic Adventure 2|Central City]], while new areas also feature, such as Blue Ridge Zone.<ref name="igninterview" /> Most musics in the game are arrangements of themes from [[Sonic 3D Blast|Sonic 3D Blast]] (both the Sega Genesis and Sega Saturn/PC versions) and [[Sonic the Hedgehog CD|Sonic CD]].
*Mechanics of biological tissues
<ref>http://www.maths.gla.ac.uk/~rwo/research_areas.htm</ref>
*Theoretical enzymology and [[enzyme kinetics]]
*[[Cancer]] modelling and simulation <ref>http://www.springerlink.com/content/71958358k273622q/</ref>,<ref>http://calvino.polito.it/~mcrtn/</ref>
*Modelling the movement of interacting cell populations<ref>http://www.ma.hw.ac.uk/~jas/researchinterests/index.html</ref>
*Mathematical modelling of scar tissue formation<ref>http://www.ma.hw.ac.uk/~jas/researchinterests/scartissueformation.html</ref>
*Mathematical modelling of intracellular dynamics<ref>http://www.sbi.uni-rostock.de/dokumente/p_gilles_paper.pdf</ref>
*Mathematical modelling of the cell cycle<ref>http://mpf.biol.vt.edu/Research.html</ref>


'''Modelling physiological systems'''
=== Playable characters ===
<!-- Confirm character(s) on talk page before adding, your character will be added by another user, providing that your reference meets WP:VERIFY. -->
[[Image:Sonicchronicles shade.jpg|thumb|Shade, the new character in her battle armor]]
The playable characters are separated into three "classes", depending on their role in battle. A character can be either a power class, (such as Sonic), a shifter class, (such as Amy), or a support class, (such as Tails).{{Fact|date=September 2008}} The characters are listed by the order in which they are found and added to the party in the game.


*Modelling of [[artery|arterial]] disease [http://www.maths.gla.ac.uk/~nah/research_interests.html]
* '''[[Sonic the Hedgehog (character)|Sonic the Hedgehog]]'''<ref name="ignfront">{{cite web|url=http://uk.ds.ign.com/objects/856/856273.html|title=IGN: Sonic Chronicles: The Dark Brotherhood|publisher=IGN}}</ref>
*Multi-scale modelling of the [[heart]] [http://www.integrativebiology.ox.ac.uk/heartmodel.html]
* '''[[Amy Rose]]'''<ref name="ignfront"/>
* '''[[Miles "Tails" Prower]]'''<ref name="ignfront"/>
* '''[[Rouge the Bat]]'''<ref name="ignfront"/>
* '''[[Big the Cat]]'''<ref name="ignhandson2"/>
* '''[[Knuckles the Echidna]]'''<ref name="ignfront"/>
* '''[[Cream the Rabbit]]'''<ref>http://www.worthplaying.com/article.php?sid=53897</ref>
* '''[[Doctor Eggman]]<ref name="review_ign">{{cite web | url = http://ds.ign.com/articles/913/913197p1.html | publisher = IGN | accessdate = 2008-09-25 | date = 2008-09-24 | first = Mark | last = Bozon | title = IGN: Sonic Chronicles: The Dark Brotherhood Review}}</ref>
* '''[[Shadow the Hedgehog]]'''<ref name="ignfront"/>
* '''[[E-123 Omega]]'''<ref name="cvgngamer">{{cite web|url=http://www.computerandvideogames.com/article.php?id=190471|author=NGamer Staff|title=DS Previews: Sonic Chronicles: The Dark Brotherhood|accessdate=2008-06-09|quote=Fiddling with the item menus revealed that new character Shade will be playable, as will E-123 Omega, the clanking bot star of Sonic Heroes and Shadow The Hedgehog.}}</ref>
* '''Shade'''<ref name="cvgngamer"/>


===Population dynamics===
==Development==
[[Population dynamics]] has traditionally been the dominant field of mathematical biology. Work in this area dates back to the 19th century. The [[Lotka–Volterra equation|Lotka–Volterra predator-prey equations]] are a famous example. In the past 30 years, population dynamics has been complemented by [[evolutionary game theory]], developed first by [[John Maynard Smith]]. Under these dynamics, evolutionary biology concepts may take a deterministic mathematical form. Population dynamics overlap with another active area of research in mathematical biology: [[Mathematical modelling in epidemiology|mathematical epidemiology]], the study of infectious disease affecting populations. Various models of viral spread have been proposed and analysed, and provide important results that may be applied to health policy decisions.
[[Image:ZoahTSS.jpg|thumb|Promotional artwork for the Zoah]]
Development began in 2006 when BioWare began to look into a handheld video game project to start, and at that point, had over 30 people involved.<ref name="igninterview" />


===Mathematical methods===
During February 2008, Sega announced that it would be giving fans the opportunity to vote on the name of a hostile alien race that would appear within the game.<ref>{{cite web|url=http://blogs.sega.com/sonic/2008/02/19/sonic-chronicles%E2%84%A2-a-race-to-name/|title=SONIC CHRONICLES™ A RACE TO NAME|author=ArchangelUK|date=2008-03-19|publisher=Sega Europe}}</ref> From March 4, fans were invited to vote on Sega Europe's Sonic portal ''Sonic City'' for their preferred name.<ref>{{cite web|url=http://blogs.sega.com/sonic/2008/03/04/sonic-chronicles-race-to-name-poll-is-live/|title=Sonic Chronicles: Race To Name - POLL IS LIVE|author=ArchangelUK|publisher=Sega Europe|date=2008-04-03}}</ref> The winning entry in the contest was announced as '''The Zoah''',<ref name="zoahname">{{cite web| author=Sega Europe|url = http://blogs.sega.com/sonic/2008/03/19/zoah-far-zoah-good-the-winner-is-announced/| title = Sonic City Blognik| date = 2008-03-18| publisher = Sega Europe| accessdate = 2008-03-19}}</ref> a reference to the town in [[Panzer Dragoon Saga#Solo_Wing|Panzer Dragoon Saga]]. The name was suggested as a potential name initially by the Sega fan blog '''Sega Nerds'''.<ref>{{cite web| author=Sega Nerds|url = http://www.seganerds.com/2008/03/19/sonic-chronicles-naming-contest-results-now-in/|title = seganerds| date = 2008-03-19| accessdate = 2008-03-19}}</ref><ref>{{cite web| author=SEGA America Blog|url = http://blogs.sega.com/usa/2008/03/21/sonic-chronicles-a-race-to-name-final-name-announced/|title = SEGA America Blog| date = 2008-03-21| accessdate = 2008-03-21}}</ref>


A model of a biological system is converted into a system of equations, although the word 'model' is often used synonymously with the system of corresponding equations. The solution of the equations, by either analytical or numerical means, describes how the biological system behaves either over time or at [[equilibrium point|equilibrium]]. There are many different types of equations and the type of behavior that can occur is dependent on both the model and the equations used. The model often makes assumptions about the system. The equations may also make assumptions about the nature of what may occur.
[[Richard Jacques]], while not the main musical composer, has composed some tracks for Sonic Chronicles.<ref name="chaogarden">{{cite web| author=SpinDash.de|url = http://spindash.de/index.php?option=com_content&task=view&id=689| title = BioWare Event in Munich| date = 2008-08-08| publisher = SpinDash.de| accessdate = 2008-08-08}}</ref>


The following is a list of mathematical descriptions and their assumptions.
The game went [[Gold release|gold]] on September 5, 2008.<ref name="playtm_gold">{{cite web|url=http://www.gamezone.com/news/09_05_08_05_54PM.htm|title=Video Game News - SEGA Announces that Sonic Chronicles Goes Gold|publisher=play.tm|accessdate=2008-08-26|date=2008-09-05}}</ref> Sonic Chronicles won the award of best DS Game at the Games Convention 2008 in Leipzig, Germany. <ref name="gc_best_game">{{cite web| author=SpinDash.de|url = http://spindash.de/index.php?option=com_content&task=view&id=739&Itemid=1156| title = Sonic Chronicles Best DS Game| date = 2008-08-22| publisher = SpinDash.de| accessdate = 2008-08-22}}</ref>


'''Deterministic processes ([[dynamical system]]s) '''
A sequel to the game is a possibility if the game performs well enough, and the developer, BioWare, has "a precise idea" for how the sequel should be made;<ref name="palgn_interview">{{cite interview |last=Kieken |first=Dorian |subject=Sonic Chronicles: The Dark Brotherhood |interviewer=Joseph Rositano |title=Sonic Chronicles: The Dark Brotherhood Interview |date=2008-09-06 |accessdate=2008-09-26 |url=http://palgn.com.au/article.php?id=12392&sid=248d0156da654cf21610c3b51cb3ab3f}}</ref> another indication of a potential sequel is the game's ending, which leaves the player with the message "to be continued".<ref name="review_ng">{{cite web | url = http://www.computerandvideogames.com/article.php?id=197716 | first = Matthew | last = Castle | title = Review: Sonic Chronicles: The Dark Brotherhood | accessdate = 2008-09-26 | date = 2008-09-24 | publisher = [[NGamer]]}}</ref> Producer Dorian Kieken later confirmed that the sequel to the game was already being written.<ref name="sequel_interview">{{cite interview|last=Kieken |first=Dorian|subject=Sonic Chronicles: The Dark Brotherhood|date=2008-09-28|accessdate=2008-09-30|url=http://savygamer.co.uk/2008/09/savygamer-podcast-episode-6.html}}</ref>


A fixed mapping between an initial state and a final state. Starting from an initial condition and moving forward in time, a deterministic process will always generate the same trajectory and no two trajectories cross in state space.
==Reception==
{{VG Reviews
|EuroG = 7/10<ref name="review_eurog">{{cite web | url = http://www.eurogamer.net/article.php?article_id=243188 | title=Sonic Chronicles: The Dark Brotherhood Review | publisher = [[Eurogamer]] | accessdate = 2008-09-24 | date = 2008-09-23 | first = Dan | last = Whitehead }}</ref>
|GI = 8.5/10<ref name="review_gi">{{cite web | url = http://www.gameinformer.com/NR/exeres/A2567996-CC5B-4431-B71A-D33536F0CE50.htm | first = Brian | last = Vore | accessdate = 2008-09-25 | title = Sonic Chronicles: The Dark Brotherhood | work= [[Game Informer]]}}</ref>
|1UP = A<ref name="review_1up">{{cite web | url = http://www.1up.com/do/reviewPage?cId=3170205&p=39 | title = Sonic Chronicles: The Dark Brotherhood Review | date = 2008-09-26 | accessdate = 2008-09-27 | first = Ryan | last = Scott | publisher = 1UP}}</ref>
|IGN = 6.5/10<ref name="review_ign"/>
|NG = 7.9/10<ref name="review_ng" />
|NP = 8/10<ref name="review_np">{{cite journal |last=Loe |first=Casey |year=2008 |month=November |title=The Wild Blue Yonder |journal=Nintendo Power|pages=p.&nbsp;96|url=http://www.tssznews.com/2008/09/18/nintendo-power-sonic-chronicles-review-scan/|accessdate=2008-09-20|volume=234}}</ref>
|ONM = 93%<ref name="review_onm">{{cite web | url = http://www.officialnintendomagazine.co.uk/article.php?id=5942 | title = Review: Sonic Chronicles: The Dark Brotherhood | accessdate = 2008-09-27 | date = 2008-09-25}}</ref>
|GR = 76%<ref name="agg_gamerankings">{{cite web | url = http://www.gamerankings.com/htmlpages2/941521.asp | title = Sonic Chronicles: The Dark Brotherhood | publisher = [[Game Rankings]] | accessdate = 2008-09-20 }}</ref>
|MC = 77/100<ref name="agg_metacritic">{{cite web | url = http://www.metacritic.com/games/platforms/ds/sonicchroniclesthedarkbrotherhood | title = Sonic Chronicles: The Dark Brotherhood (ds: 2008): Reviews | publisher = [[Metacritic]] | accessdate = 2008-09-20 }}</ref>
}}


*[[Difference equations]] Discrete time, continuous state space.
''Sonic Chronicles: The Dark Brotherhood'' has generally been received well by critics, garnering a 77 out of 100 average and "generally favourable reviews" at aggregate site [[Metacritic]], based on 23 reviews,<ref name="agg_metacritic" /> and a 76% average at [[Game Rankings]] based on 14 reviews.<ref name="agg_gamerankings" />
*[[Ordinary differential equations]] (Continuous time. Continuous state space. No spatial derivatives.) See also [[Numerical ordinary differential equations]].
* [[Partial differential equations]] (Continuous time. Continuous state space. Spatial derivatives.) See also [[Numerical partial differential equations]].
* Maps (Discrete time. Continuous state space)


'''Stochastic processes (random dynamical systems)'''
The game received positive reactions in the area of graphics and environments, being "bright, colorful, and cheery",<ref name="review_1up" /> "undeniably nice-looking" with "slick presentation",<ref name="review_eurog" /> and with cut-scenes that are "visually sharp".<ref name="review_gamepro">{{cite web | url = http://www.gamepro.com/article/reviews/207226/sonic-chronicles-the-dark-brotherhood-ds/ | title = Review: Sonic Chronicles: The Dark Brotherhood (DS) | publisher = GamePro | date = 2008-09-25 | accessdate = 2008-09-27 | first = Emily | last = Balistrieri }}</ref>


A random mapping between an initial state and a final state, making the state of the system a [[random variable]] with a corresponding [[probability distribution]].
As Sonic's first transition to the RPG genre, the overall scenario, including the game's story and dialog, were received with mixed reception. Critics acknowledged that, compared to other games in the genre, the "chapter-based story is predictably lightweight and cartoony",<ref name="review_1up" /> and though interesting enough, it "takes a backseat to the actual gameplay".<ref name="review_gamepro" /> Indeed, combined with a generally low difficulty level (also described as "phenomenally easy"<ref name="review_eurog" />), the story and scenario were cited as a good reason that this entry would be suitable as a role-playing game for younger gamers.<ref name="review_gamepro" /> Reviewers also agreed that the story only begins to become more elaborate and interesting in the second half of the game.<ref name="review_ng" /><ref name="review_ign" /> The cast of playable characters, who some reviewers acknowledged were less than well received in previous Sonic games,<ref name="review_ng" /><ref name="review_gi" /> were generally received well, and were "well written and likeable".<ref name="review_gi" />


* Non-Markovian processes -- [[Master equation|Generalized master equation]] (Continuous time with memory of past events. Discrete state space. [[Waiting time]]s of events (or transitions between states) discretely occur and have a generalized [[probability distribution]].)
Further to the reception of the RPG design, gameplay elements of the role-playing genre also received mixed reception. Some reviewers cited the "overbearing" battle system becoming "an annoyance",<ref name="review_np" /> whilst others found the combat system a "joy",<ref name="review_1up" /> with a good difficulty level and "some great tension".<ref name="review_gi" /> The lack of multiplayer options, aside from the ability to share Chao, were also seen as a disappointment.<ref name="review_3dj">{{cite web | url = http://www.3djuegos.com/juegos/analisis/2744/0/sonic-chronicles-la-hermandad-siniestra/ | title=Sonic Chronicles: La Hermandad Siniestra | publisher = 3DJuegos | accessdate = 2008-09-22 | date = 2008-09-22 | author = Jesús Bella Ceacero}}</ref>
* Jump [[Continuous-time Markov process|Markov process]] -- [[Master equation]] (Continuous time with no memory of past events. Discrete state space. Waiting times between events discretely occur and are exponentially distributed.) See also [[Monte Carlo method]] for numerical simulation methods, specifically [[Continuous-time Monte Carlo]] which is also called kinetic Monte Carlo or the stochastic simulation algorithm.
{{-}}
* Continuous [[Markov process]] -- [[stochastic differential equation]]s or a [[Fokker-Planck equation]] (Continuous time. Continuous state space. Events occur continuously according to a random [[Wiener process]].)


'''Spatial modelling'''
==Other media==

[[Archie Comics]] has made an adaption of Sonic Chronicles titled '''"Invaders from Beyond"''', featuring the kidnapping of [[Knuckles the Echidna|Knuckles]].<ref>{{cite journal|title=Invaders from Beyond|issue=#191|date=2008-08-13|journal=[[Sonic the Hedgehog (comic book)|Sonic the Hedgehog]]|publisher=[[Archie Comics]]}}</ref> The short strip only covers the beginning of the story, and ends by urging readers to buy Sonic Chronicles to find out "what happens next".
One classic work in this area is [[Alan Turing]]'s paper on [[morphogenesis]] entitled ''The Chemical Basis of Morphogenesis'', published in 1952 in the [[Philosophical Transactions of the Royal Society]].
*Travelling waves in a wound-healing assay [http://www.maths.ox.ac.uk/~maini/public/gallery/twwha.htm]
*[[Swarming]] behaviour [http://www.math.ubc.ca/people/faculty/keshet/research.html]
*The mechanochemical theory of [[morphogenesis]] [http://www.maths.ox.ac.uk/~maini/public/gallery/mctom.htm]
*Biological pattern formation [http://www.maths.ox.ac.uk/~maini/public/gallery/bpf.htm]
*Spatial distribution modeling using plot samples [http://links.jstor.org/sici?sici=0030-1299%28199008%2958%3A3%3C257%3ASDOTMU%3E2.0.CO%3B2-S&size=LARGE&origin=JSTOR-enlargePage]

==Example of a model: The Cell Cycle==
The eukaryotic [[cell cycle]] is very complex and is one of the most studied topics, since its misregulation results in [[cancer]].
It is possibly a good example of a mathematical model as it deals with simple calculus but gives valid results.<ref>{{cite web|url=http://mpf.biol.vt.edu/Tyson%20Lab.html |title=Tyson Lab |publisher=Mpf.biol.vt.edu |date= |accessdate=2008-09-10}}</ref> and the Bela Novak lab (Oxford University)[http://cellcycle.mkt.bme.hu/] have produced several models of the cell cycle simulating several organisms. They have recently produced a generic eukaryotic cell cycle model which can represent a particular eukaryote depending on the values of the parameters, demonstrating that the idiosyncrasies of the individual cell cycles are due to different protein concentrations and affinities, while the underlying mechanisms are conserved (Csikasz-Nagy et al, 2006).<br />
By means of a system of [[ordinary differential equation]]s these models show the change in time ([[dynamical system]]) of the protein inside a single typical cell; this type of model is called a [[deterministic system|deterministic process]] (whereas a model describing a statistical distribution of protein concentrations in a population of cells is called a [[stochastic process]]).<br />
To obtain these equations an iterative series of steps must be done: first the several models and observations are combined to form a consensus diagram and the appropriate kinetic laws are chosen to write the differential equations, such as [[Reaction rate|rate kinetics]] for stoichiometric reactions, [[Michaelis-Menten kinetics]] for enzyme substrate reactions and [[Goldbeter–Koshland kinetics]] for ultrasensitive transcription factors, afterwards the parameters of the equations (rate constants, enzyme efficiency coefficients and Michealis constants) must be fitted to match observations; when they cannot be fitted the kinetic equation is revised and when that is not possible the wiring diagram is modified. The parameters are fitted and validated using observations of both wild type and mutants, such as protein half-life and cell size.<br />
In order to fit the parameters the differential equations need to be studied. This can be done either by simulation or by analysis. <br />
In a simulation, given a starting [[vector]] (list of the values of the variables), the progression of the system is calculated by solving the equations at each time-frame in small increments.<br />
[[Image:Cell cycle bifurcation diagram.jpg|thumb|500px]]
In analysis, the proprieties of the equations are used to investigate the behavior of the system depending of the values of the parameters and variables. A system of differential equations can be represented as a [[vector field]], where each vector described the change (in concentration of two or more protein) determining where and how fast the trajectory (simulation) is heading. Vector fields can have several special points: a [[Fixed point (mathematics)|stable point]], called a sink, that attracts in all directions (forcing the concentrations to be at a certain value), an [[unstable point]], either a source or a [[saddle point]] which repels (forcing the concentrations to change away from a certain value), and a limit cycle, a closed trajectory towards which several trajectories spiral towards (making the concentrations oscillate).<br />
A better representation which can handle the large number of variables and parameters is called a [[bifurcation diagram]]([[Bifurcation theory]]): the presence of these special steady-state points at certain values of a parameter (e.g. mass) is represented by a point and once the parameter passes a certain value, a qualitative change occurs, called a [[bifurcation]], in which the nature of the space changes, with profound consequences for the protein concentrations: the cell cycle has phases (partially corresponding to G1 and G2) in which mass, via a stable point, controls cyclin levels, and phases (S and M phases) in which the concentrations change independently, but once the phase has changed at a bifurcation event ([[Cell cycle checkpoint]]), the system cannot go back to the previous levels since at the current mass the vector field is profoundly different and the mass cannot be reversed back through the bifurcation event, making a checkpoint irreversible. In particular the S and M checkpoints are regulated by means of special bifurcations called a [[Hopf bifurcation]] and an [[infinite period bifurcation]].

==See also==
{{columns-list|2|
*[[Abstract Relational Biology]]
* [[Biocybernetics]]
* [[Bioinformatics]]
* [[Biologically-inspired computing]]
* [[Biostatistics]]
* [[Cellular automata]]
* [[Charles S. Peskin]]
* [[Coalescent theory]]
* [[Complex Systems Biology]]
* [[Computational biology]]
* [[D'Arcy Thompson]]
* [[Dynamic Systems in Biology]]
* [[Epidemiology]]
* [[Ewens's sampling formula]]
* [[Excitable medium]]
* [[Mathematical model]]
* [[Metabolic-Replication Systems]]
* [[Morphometrics]]
* [[Neighbour-sensing model]]
* [[Organismic systems (OS)]]
* [[Organismic supercategories]]
* [[Population dynamics of fisheries]]
* [[Population genetics]]
* [[Protein folding]] (also [[BlueGene]] and [[Folding@home]])
* [[Relational Biology]]
* [[Self-reproduction]]
* [[Software for molecular mechanics modeling|Software for molecular modeling]]
* [[Systems biology]]
* [[Theoretical biology]]
}}

==Notes: Inline and online==
<references/>


==References==
==References==
=== Bibliographical ===
{{reflist|2}}
*S.H. Strogatz, ''Nonlinear dynamics and Chaos: Applications to Physics, Biology, Chemistry, and Engineering.'' Perseus, 2001, ISBN 0-7382-0453-6
*N.G. van Kampen, ''Stochastic Processes in Physics and Chemistry'', North Holland., 3rd ed. 2001, ISBN 0-444-89349-0
*P.G. Drazin, ''Nonlinear systems''. [[Cambridge University Press|C.U.P.]], 1992. ISBN 0-521-40668-4
*L. Edelstein-Keshet, ''Mathematical Models in Biology''. SIAM, 2004. ISBN 0-07-554950-6
*G. Forgacs and S. A. Newman, ''Biological Physics of the Developing Embryo''. C.U.P., 2005. ISBN 0-521-78337-2
*A. Goldbeter, ''Biochemical oscillations and cellular rhythms''. C.U.P., 1996. ISBN 0-521-59946-6
*L.G. Harrison, ''Kinetic theory of living pattern''. C.U.P., 1993. ISBN 0-521-30691-4
*F. Hoppensteadt, ''Mathematical theories of populations: demographics, genetics and epidemics''. SIAM, Philadelphia, 1975 (reprinted 1993). ISBN 0-89871-017-0
*D.W. Jordan and P. Smith, ''Nonlinear ordinary differential equations'', 2nd ed. O.U.P., 1987. ISBN 0-19-856562-3
*J.D. Murray, ''Mathematical Biology''. Springer-Verlag, 3rd ed. in 2 vols.: ''Mathematical Biology: I. An Introduction'', 2002 ISBN 0-387-95223-3; ''Mathematical Biology: II. Spatial Models and Biomedical Applications'', 2003 ISBN 0-387-95228-4.
*E. Renshaw, ''Modelling biological populations in space and time''. C.U.P., 1991. ISBN 0-521-44855-7
*S.I. Rubinow, ''Introduction to mathematical biology''. John Wiley, 1975. ISBN 0-471-74446-8
*L.A. Segel, ''Modeling dynamic phenomena in molecular and cellular biology''. C.U.P., 1984. ISBN 0-521-27477-X
*L. Preziosi, ''Cancer Modelling and Simulation''. Chapman Hall/CRC Press, 2003. ISBN 1-58488-361-8

=== External ===
*F. Hoppensteadt, ''[http://www.ams.org/notices/199509/hoppensteadt.pdf Getting Started in Mathematical Biology]''. Notices of American Mathematical Society, Sept. 1995.
*[[Michael C. Reed|M. C. Reed]], ''[http://www.resnet.wm.edu/~jxshix/math490/reed.pdf Why Is Mathematical Biology So Hard?]'' Notices of American Mathematical Society, March, 2004.
*R. M. May, ''[http://www.resnet.wm.edu/~jxshix/math490/may.pdf Uses and Abuses of Mathematics in Biology]''. Science, February 6, 2004.
*J. D. Murray, ''[http://www.resnet.wm.edu/~jxshix/math490/murray.doc How the leopard gets its spots?]'' Scientific American, 258(3): 80-87, 1988.
* S. Schnell, R. Grima, P. K. Maini, ''[http://eprints.maths.ox.ac.uk/567/01/224.pdf Multiscale Modeling in Biology]'', American Scientist, Vol 95, pages 134-142, March-April 2007.
*Chen KC et al. Integrative analysis of cell cycle control in budding yeast. Mol Biol Cell. 2004 Aug;15(8):3841-62.
*Csikász-Nagy A et al. Analysis of a generic model of eukaryotic cell-cycle regulation. Biophys J. 2006 Jun 15;90(12):4361-79.
*Fuss H, et al. Mathematical models of cell cycle regulation. Brief Bioinform. 2005 Jun;6(2):163-77.
*Lovrics A et al. ''[http://www.biomedcentral.com/content/pdf/1471-2105-7-494.pdf Time scale and dimension analysis of a budding yeast cell cycle model.]'' BMC Bioinform. 2006 Nov 9;7:494.


==External links==
==External links==
{{linkfarm}}
*[http://www.sega.com/sonicchronicles/us/index.php Official website]
*[http://biocyb.cs.ucla.edu/research.html UCLA Biocybernetics Laboratory]
*[http://sonic.bioware.com Official Developer's Community site]
*[http://www.tucs.fi/research/labs/combio.php TUCS Computational Biomodelling Laboratory]
*[http://www.agr.nagoya-u.ac.jp/english/e3senko-1.html Nagoya University Division of Biomodeling]
*[http://www.bmi2.bmt.tue.nl/Biomedinf/ Technische Universiteit Biomodeling and Informatics]
*[http://wiki.biological-cybernetics.de BioCybernetics Wiki, a vertical wiki on biomedical cybernetics and systems biology]
* [http://www.biostatsresearch.com/repository/ The Collection of Biostatistics Research Archive]
* [http://www.bepress.com/sagmb/ Statistical Applications in Genetics and Molecular Biology]
* [http://www.bepress.com/ijb/ The International Journal of Biostatistics]
*[http://www.smb.org/ Society for Mathematical Biology]
*[http://www.esmtb.org/ European Society for Mathematical and Theoretical Biology]
*[http://www.math.canterbury.ac.nz/bio/ Biomathematics Research Centre at University of Canterbury]
*[http://www.maths.ox.ac.uk/cmb/ Centre for Mathematical Biology at Oxford University]
*[http://mathbio.nimr.mrc.ac.uk/ Mathematical Biology at the National Institute for Medical Research]
*[http://www.imbm.org/ Institute for Medical BioMathematics]
* [http://eqworld.ipmnet.ru/en/solutions/syspde/spde-toc2.pdf ''Mathematical Biology Systems of Differential Equations''] from EqWorld: The World of Mathematical Equations
*[http://sbw.kgi.edu Systems Biology Workbench - a set of tools for modelling biochemical networks]


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Revision as of 17:16, 12 October 2008

For use of statistics in Biology, see Biostatistics.
For use of basic artimethics in Biology, see relevant topic, such as Serial dilution.

Mathematical biology, biological mathematical modeling, biomathematics or computational biomodeling is an interdisciplinary field of academic study which aims at modeling natural, biological processes using applied mathematical techniques and tools. It has both practical and theoretical applications in biological research: In cell biology, protein interactions are typically expressed as “cartoon” models, which, although easy to visualize, do not fully describe the systems studied: to do this, mathematical models are required, which, by describing the systems in a quantitative manner, can better simulate their behavior and hence predict unseen properties.

Importance

Applying mathematics to biology has a long history, but only recently has there been an explosion of interest in the field. Some reasons for this include:

  • the explosion of data-rich information sets, due to the genomics revolution, which are difficult to understand without the use of analytical tools,
  • recent development of mathematical tools such as chaos theory to help understand complex, nonlinear mechanisms in biology,
  • an increase in computing power which enables calculations and simulations to be performed that were not previously possible, and
  • an increasing interest in in silico experimentation due to ethical considerations, risk, unreliability and other complications involved in human and animal research.

Areas of research

Below is a list of some areas of research in mathematical biology and links to related projects in various universities. These examples are characterised by complex, nonlinear mechanisms and it is being increasingly recognised that the result of such interactions may only be understood through mathematical and computational models. Due to the wide diversity of specific knowledge involved, biomathematical research is often done in collaboration between mathematicians, physicists, engineers, biologists, physicians, zoologists, chemists etc.

Computer models and automata theory in biology and medicine

A monograph on this topic summarizes an extensive amount of research in this area up to 1987, including subsections in the following areas: computer modeling in biology and medicine, arterial system models, neural models, biochemical and oscillation networks, neural nets, genetic networks, abstract relational biology, metabolic-replication systems, category theory applications in biology and medicine, automata theory and complete self-reproduction, chaotic systems in organisms, relational biology and organismic theories.[1]; this published report also includes 390 references to peer-reviewed articles by a large number of authors [2], [3] (See also the 300+ References List in: "Bibliography on category theory applications", including mathematical biology applications).


Modeling cell and molecular biology

This area has received a boost due to the growing importance of molecular biology.[4]

  • Mechanics of biological tissues

[5]

  • Theoretical enzymology and enzyme kinetics
  • Cancer modelling and simulation [6],[7]
  • Modelling the movement of interacting cell populations[8]
  • Mathematical modelling of scar tissue formation[9]
  • Mathematical modelling of intracellular dynamics[10]
  • Mathematical modelling of the cell cycle[11]

Modelling physiological systems

Population dynamics

Population dynamics has traditionally been the dominant field of mathematical biology. Work in this area dates back to the 19th century. The Lotka–Volterra predator-prey equations are a famous example. In the past 30 years, population dynamics has been complemented by evolutionary game theory, developed first by John Maynard Smith. Under these dynamics, evolutionary biology concepts may take a deterministic mathematical form. Population dynamics overlap with another active area of research in mathematical biology: mathematical epidemiology, the study of infectious disease affecting populations. Various models of viral spread have been proposed and analysed, and provide important results that may be applied to health policy decisions.

Mathematical methods

A model of a biological system is converted into a system of equations, although the word 'model' is often used synonymously with the system of corresponding equations. The solution of the equations, by either analytical or numerical means, describes how the biological system behaves either over time or at equilibrium. There are many different types of equations and the type of behavior that can occur is dependent on both the model and the equations used. The model often makes assumptions about the system. The equations may also make assumptions about the nature of what may occur.

The following is a list of mathematical descriptions and their assumptions.

Deterministic processes (dynamical systems)

A fixed mapping between an initial state and a final state. Starting from an initial condition and moving forward in time, a deterministic process will always generate the same trajectory and no two trajectories cross in state space.

Stochastic processes (random dynamical systems)

A random mapping between an initial state and a final state, making the state of the system a random variable with a corresponding probability distribution.

Spatial modelling

One classic work in this area is Alan Turing's paper on morphogenesis entitled The Chemical Basis of Morphogenesis, published in 1952 in the Philosophical Transactions of the Royal Society.

  • Travelling waves in a wound-healing assay [3]
  • Swarming behaviour [4]
  • The mechanochemical theory of morphogenesis [5]
  • Biological pattern formation [6]
  • Spatial distribution modeling using plot samples [7]

Example of a model: The Cell Cycle

The eukaryotic cell cycle is very complex and is one of the most studied topics, since its misregulation results in cancer. It is possibly a good example of a mathematical model as it deals with simple calculus but gives valid results.[12] and the Bela Novak lab (Oxford University)[8] have produced several models of the cell cycle simulating several organisms. They have recently produced a generic eukaryotic cell cycle model which can represent a particular eukaryote depending on the values of the parameters, demonstrating that the idiosyncrasies of the individual cell cycles are due to different protein concentrations and affinities, while the underlying mechanisms are conserved (Csikasz-Nagy et al, 2006).
By means of a system of ordinary differential equations these models show the change in time (dynamical system) of the protein inside a single typical cell; this type of model is called a deterministic process (whereas a model describing a statistical distribution of protein concentrations in a population of cells is called a stochastic process).
To obtain these equations an iterative series of steps must be done: first the several models and observations are combined to form a consensus diagram and the appropriate kinetic laws are chosen to write the differential equations, such as rate kinetics for stoichiometric reactions, Michaelis-Menten kinetics for enzyme substrate reactions and Goldbeter–Koshland kinetics for ultrasensitive transcription factors, afterwards the parameters of the equations (rate constants, enzyme efficiency coefficients and Michealis constants) must be fitted to match observations; when they cannot be fitted the kinetic equation is revised and when that is not possible the wiring diagram is modified. The parameters are fitted and validated using observations of both wild type and mutants, such as protein half-life and cell size.
In order to fit the parameters the differential equations need to be studied. This can be done either by simulation or by analysis.
In a simulation, given a starting vector (list of the values of the variables), the progression of the system is calculated by solving the equations at each time-frame in small increments.

In analysis, the proprieties of the equations are used to investigate the behavior of the system depending of the values of the parameters and variables. A system of differential equations can be represented as a vector field, where each vector described the change (in concentration of two or more protein) determining where and how fast the trajectory (simulation) is heading. Vector fields can have several special points: a stable point, called a sink, that attracts in all directions (forcing the concentrations to be at a certain value), an unstable point, either a source or a saddle point which repels (forcing the concentrations to change away from a certain value), and a limit cycle, a closed trajectory towards which several trajectories spiral towards (making the concentrations oscillate).
A better representation which can handle the large number of variables and parameters is called a bifurcation diagram(Bifurcation theory): the presence of these special steady-state points at certain values of a parameter (e.g. mass) is represented by a point and once the parameter passes a certain value, a qualitative change occurs, called a bifurcation, in which the nature of the space changes, with profound consequences for the protein concentrations: the cell cycle has phases (partially corresponding to G1 and G2) in which mass, via a stable point, controls cyclin levels, and phases (S and M phases) in which the concentrations change independently, but once the phase has changed at a bifurcation event (Cell cycle checkpoint), the system cannot go back to the previous levels since at the current mass the vector field is profoundly different and the mass cannot be reversed back through the bifurcation event, making a checkpoint irreversible. In particular the S and M checkpoints are regulated by means of special bifurcations called a Hopf bifurcation and an infinite period bifurcation.

See also

2

Notes: Inline and online

  1. ^ Baianu, I. C.: 1986--1987a, Computer Models and Automata Theory in Biology and Medicine., in M. Witten (ed.), Mathematical Models in Medicine, vol. 7., Ch.11 Pergamon Press, New York, 1513 -1577. http://www.springerlink.com/content/w2733h7280521632/
  2. ^ http://www.springerlink.com/content/w2733h7280521632/
  3. ^ Also currently available as an updated PDF at: http://cogprints.ecs.soton.ac.uk/archive/00003718/01/COMPUTER_SIMULATIONCOMPUTABILITYBIOSYSTEMSrefnew.pdf
  4. ^ "Research in Mathematical Biology". Maths.gla.ac.uk. Retrieved 2008-09-10.
  5. ^ http://www.maths.gla.ac.uk/~rwo/research_areas.htm
  6. ^ http://www.springerlink.com/content/71958358k273622q/
  7. ^ http://calvino.polito.it/~mcrtn/
  8. ^ http://www.ma.hw.ac.uk/~jas/researchinterests/index.html
  9. ^ http://www.ma.hw.ac.uk/~jas/researchinterests/scartissueformation.html
  10. ^ http://www.sbi.uni-rostock.de/dokumente/p_gilles_paper.pdf
  11. ^ http://mpf.biol.vt.edu/Research.html
  12. ^ "Tyson Lab". Mpf.biol.vt.edu. Retrieved 2008-09-10.

References

Bibliographical

  • S.H. Strogatz, Nonlinear dynamics and Chaos: Applications to Physics, Biology, Chemistry, and Engineering. Perseus, 2001, ISBN 0-7382-0453-6
  • N.G. van Kampen, Stochastic Processes in Physics and Chemistry, North Holland., 3rd ed. 2001, ISBN 0-444-89349-0
  • P.G. Drazin, Nonlinear systems. C.U.P., 1992. ISBN 0-521-40668-4
  • L. Edelstein-Keshet, Mathematical Models in Biology. SIAM, 2004. ISBN 0-07-554950-6
  • G. Forgacs and S. A. Newman, Biological Physics of the Developing Embryo. C.U.P., 2005. ISBN 0-521-78337-2
  • A. Goldbeter, Biochemical oscillations and cellular rhythms. C.U.P., 1996. ISBN 0-521-59946-6
  • L.G. Harrison, Kinetic theory of living pattern. C.U.P., 1993. ISBN 0-521-30691-4
  • F. Hoppensteadt, Mathematical theories of populations: demographics, genetics and epidemics. SIAM, Philadelphia, 1975 (reprinted 1993). ISBN 0-89871-017-0
  • D.W. Jordan and P. Smith, Nonlinear ordinary differential equations, 2nd ed. O.U.P., 1987. ISBN 0-19-856562-3
  • J.D. Murray, Mathematical Biology. Springer-Verlag, 3rd ed. in 2 vols.: Mathematical Biology: I. An Introduction, 2002 ISBN 0-387-95223-3; Mathematical Biology: II. Spatial Models and Biomedical Applications, 2003 ISBN 0-387-95228-4.
  • E. Renshaw, Modelling biological populations in space and time. C.U.P., 1991. ISBN 0-521-44855-7
  • S.I. Rubinow, Introduction to mathematical biology. John Wiley, 1975. ISBN 0-471-74446-8
  • L.A. Segel, Modeling dynamic phenomena in molecular and cellular biology. C.U.P., 1984. ISBN 0-521-27477-X
  • L. Preziosi, Cancer Modelling and Simulation. Chapman Hall/CRC Press, 2003. ISBN 1-58488-361-8

External

External links

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