This month's topics:
Linear algebra and belief systems
Math and the perfect cup of coffee
A natural knot in an important protein
A research report in Science (October 21, 2016) bore the title "Network science on belief system dynamics under logic constraints"; the authors are Noah E. Friedkin, Anton V. Proskurnikov, Roberto Tempo, and Sergey E. Parsegov. They quote a definition of a belief system: "a configuration of ideas and attitudes in which the elements are bound together by some form of constraint or functional interdependence." In the figure, the belief system is represented by $m$ truth statements $b_1,\dots, b_m$ and their interdependence is represented by the magenta matrix $C$, where the $i,j$ entry, a number between 0 and 1, gives the strength of the logical implication from $b_j$ to $b_i$.
The authors investigate the dynamics of the beliefs of $n$ individuals $p_1, \dots, p_n$ in the various items in the system. The beliefs are modeled by a $n\times m$ matrix $X$ (blue in the figure) where the entry $x_{ij}$ is 0 if individual $p_i$ believes that statement $b_j$ is false, 1 if $p_i$ believes it is true, and $0.5$ if $p_i$ has "maximum uncertainty" about the matter. The dynamics studied is the process $X(0), X(1), ..., X(k), ...$ by which this pattern of beliefs can change.
What makes it change? Besides the implementation of the logical implications encoded in the matrix $C$, an individual's beliefs are also shaped by those of his or her peers. The red matrix $W$ describes by a number $w_{ij}$ between 0 and 1 the weight which $p_i$ assigns to beliefs held by individual $p_j$. Finally the (orange) diagonal matrix A characterizes by a number between 0 and 1 each individual's "level of openness".
The authors' model involves only matrix addition and multiplication. They posit that for each individual $p_i$ the row $x_{i1}, ..., x_{im}$ representing $p_i$'s beliefs at step $k+1$ will be an average, weighted by that individual's "openness," between $p_i$'s original set of beliefs and the set given by the $i$th row of the matrix product $WX(k)C$: $$X(k+1) = AWX(k)C + (I-A)X(0).$$
Friedkin et al.'s model for the dynamics of a belief system under logical constraints, and incorporating peer-to-peer influence. Individual $p_i$'s beliefs at step $k+1$ are represented by the $i$th row of the blue matrix $X(k+1)$. The equation states that they are a weighted average between a row that represents the effect of peer influence and logical implications on the beliefs at step $k$ (the $i$th row of the matrix $WX(k)C$) and a row representing $p_i$'s original pattern of beliefs, or "prejudices." The weighting is given by the $i$th entry in the diagonal matrix A: it is 1 if $p_i$ is maximally open to change, and 0 if $p_i$ is maximally stubborn.
The authors spend some time on an example from recent history, involving the run-up to the invasion of Iraq and Colin Powell's speech to the General Assembly about weapons of mass destruction.
A "perspectives" piece in the same issue of Science gives some background for this research. Carter E. Butts (UC Irvine) explains in "Why I know but don't believe: Individuals hold interdependent beliefs that affect whether or not they accept scientific findings" that in current models of the dynamics of belief systems "cognitive structure has been given short shrift," and gives an example to show the importance of Friedkin et al.'s novel incorporation of an inter-propositional influence matrix ($C$ in the figure) in their model. "For instance, an individual who believes that human civilization is too insignificant to alter the global environment will tend to reject evidence for anthropogenic change in atmospheric carbon dioxide conventration, because the former belief undermines the latter. If this same persom is later persuaded (perhaps in an entirely different context) that human civilization can alter the global envoronment, he or she may become more likely to accept the notion that human activity has substantially increased atmospheric carbon dioxide levels."
It's a Science & Environment item on the BBC News website: "Maths zeroes in on perfect cup of coffee," by Paul Rincon, November 15, 2016. "Brewing the perfect cup of coffee is always going to be a subjective endeavour. But the work by Kevin Moroney at the University of Limerick, William Lee at the University of Portsmouth and others offers a better understanding of the parameters that influence the final product." Rincon quotes William Lee: "Our overall idea is to have a complete mathematical model of coffee brewing that you could use to design coffee machines, rather like we use a theory of fluid and solid mechanics to design racing cars." The current work focuses on the effect of the grain size of ground coffee. "It had previously been known that grinding beans too finely could result in coffee that is over-extracted and very bitter. On the other hand not grinding them enough can make the end result too watery." Dr. Lee: "What our work has done is take that [observation] and made it quantitative." The work, "Asymptotic Analysis of the Dominant Mechanisms in the Coffee Extraction Process," appears in the SIAM Journal on Applied Mathematics, 76, 2196-2217. Abstract here.
The article "Methyl transfer by substrate signaling from a knotted protein fold," by Thomas Christian (Thomas Jefferson University) and eight collaborators, appeared in Nature Structural and Molecular Biology, October 2016. The research concerns the properties of the bacterial enzyme TrmD, and particulary the importance of the overhand knot which is part of its folded structure. "TrmD catalyzes methyl transfer from AdoMet to the N1 of G37 on the 3' side of the tRNA anticodon" As the authors remark, the product of this enzyme (m1G37-tRNA) is essential for life.
The overhand knot in the protein TrmD. Image of protein courtesy of Ya-Ming Hou.
"Using an integrated approach of structural, kinetic, and computational analysis, we show that the structurally constrained TrmD knot is required for its catalytic activity." Specifically, "[the] knot is necessary to fold AdoMet into the bent conformation for methyl transfer." The authors remark that "TrmD is a leading antimicrobial drug target, owing to its essentiality for bacterial growth, its broad conservation across bacterial species, and its substantial differences from the human and archaeal counterpart Trm5." And draw the conclusion for medical applications: "the best inhibitors of TrmD should target the knot."
Tony Phillips
tony at math.sunysb.edu